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hM(x|θ) =dPθ (x < X < x+ dx)
dxm [VCJDGv²JMSJTSF"[RLEYZD>[Z^µCEDGWX ^XEYZ[\Y\MKF/SJTSEF* ^YZ[Z[Z^LR¶J/^ Ot^LNM x > [±^ WLR¶MKYZWLSJeDKFICR^DKMKY\MKYZWL F @Pθ(X ≤ x) = FM(x|θ) = 1 − exp
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hM(u|θ) du).
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¿ WQRHÈ^[Z[\WLRHJIL CR^DKMKY±¶QR[\YZJDÈLEWQRHY\LNMGFDGJIHGH|JID 8²MGDKWYZHvµWSI[\J'HJIvXW ±MKF'H~SWLNM[ZJIHSJIQ É CEDKJIv²Y\JIDGHÈH|WLM¶JDKÇMÃ^YZLEJv²JLNMe[\J'HeCE[ZQRHeQMKYZ[ZYZHKFIHeJIL SQRDG^XEY\[ZY\MKFµJM9[ ^XRY\[ZYÁMGF º ½ LEJwDKF FDGJLR¶Jvw^t¸oJIQEDGJ®JIL#[±^Pv²^MKYDGJ®J'HoM/[ZJ²[\YZOiDKJSJ p ^ >[\J'HKH9>o³'´ ZN @ º ¿ WQRH W¶^[\Y±HKWLRH©LEWMKDGJFMKQRSEJeHKQED>QEL¶Wv²C WNHK^LNM ∑ RH|WQEv²YZH 8®QELEJ/OWYZDKJeSJIQ É ¶^QRHKJIHSJ/SF ^YZ[\[±^L ¶JiJMLEWL 8QRL@HK¹HoMv²JevQE[\MKY±¶Wv²C WNHK^LMÃH º p JIHnMG^Q É SJTSF ^YZ[\[±^L ¶JeH|WLMSWLR¶eSJIH WLR¶MGY\WLRHHKY\v²CE[ZJIH>SEJ x RSJeMo¹iCJeC W[\¹iL vµJeWQPJ É C WLEJLNMKYZJ[ º3 45"463 7 98;:6<=<?>9@/BA&<CAEDEzF< ^¶ÃuR^LNM λ > 0 [\J/v²WSI[\JeJ É CWLEJILNMKYZJ[ E(λ) SJ/SJILRH|Y\MKF
f(x|λ) = λ exp(−λx) 0 x≥0
v²WSF[ZY±H|J®[Z^PSQEDGFJ²SJµONYZJ²SVU QRL#HK¹iH|M"Iv²J ∑ H|WQEv²YZH98SJIHeSF ^YZ[\[±^LR¶JIH #$# ( *% % SEJµC ^DeH|WLMG^Q É SJSF ^YZ[\[±^L ¶J/¶WLRHoMÃ^LNM hE(x) = λ º p ^²SQEDGFJ/SJ/OiYZJ/vµWt¹JLELRJÈJ'HoM[5U J'H|CFDÃ^L ¶J ¼ [X ] = η = 1/λ º3 45"4,G 7 98;:6<H8&<=IJ<zFKMLK ^¶ÃuR^LNM (η, β) > 0 E[ZJev²WiS [ZJTSJ Å JIY\XEQR[\[ W(η, β) ^²CWQEDSJILRHKYÁMGF
f(x|η, β) =β
η
(x
η
)β−1
exp
−(x
η
)β0x≥0.
WLMG^Q É SJ²SF* ^Y\[Z[±^LR¶J hW (x) = (β/η)(x/η)β−1 JIH|Me¶DKWYZHGHG^LNMÇ>JMev²WiSEF[ZYZHKJµSEWLR¶®QEL ∑ xNQEY¤OiY\JIY\[Z[ZYÁMÇxNQR^LRS β > 1 º m [J'HoM®SEFI¶DGWY±HGHK^LMeÇJMv²WSF[ZY±H|J QRL ∑ xNQEYnDG^¸oJQELEY\M/ÇxNQR^LRS β < 1 º m [J'HoM®J É C WLEJLNMGY\JI[xNQR^LRS β = 1 º LµLRWMKJÈ^QRHGH|Y λ = 1/η JM[±^/CR^DG^v²FMKDGY±HK^MKYZWL µ = λβ H|JIDG^eXJI^QR¶WQEC²QMKYZ[\Y±HKFJSE^LRH[ZJDGJIH|MKJSJ/¶JTSW¶QEv²JLNM º U Y\[R¹^OiYZJYZ[\[ZY±HKHKJv²JLNM ¶J[ZQEY\Çc¶YEHU ^¶I¶FI['IDKJlxNQR^LRS β > 2 º ¼ LµJ JM [±^OiYÁMGJIHGH|JlSQ²OiY\JIY\[Z[\Y±HGH|JIvµJILNMCJQM MKDGJCJDÃËQEJCR^Dn[±^®SEFDGY\OFJÈSEQ£MÃ^Q É SJeSF ^YZ[\[±^LR¶J h′W (x) RxNQEY~JIH|M>^[ZWDÃHCWHKYÁMGY\OJ º YZLRHKY [±^®¶WLEL ^Y±HKHG^LR¶JSJ βW DGJQELRJ®YZLRSY±¶^MKYZWLSJµLR^MKQEDGJ % H|QED[ZJµ¶Wv²C WD|MGJv²JLNMeSJ ∑ ~CEQEY±HKxNQqU Y\[¤CJDGv²JM/SJµSEYZH|MKYZLE]QRJD
QELP¶WvµCWHG^LNMlxQRYOiYZJYZ[\[ZY\M WD|MGJv²JLNMlSVU QRLP¶Wv²C WNHK^LNMnXEYZJLJLNMKDGJMGJLiQ SWLNMl[\J'HlSF* ^Y\[Z[Z^LR¶J'HnHKWLNMlH|QEDKMKWQM^¶¶Y±SJILMGJ[Z[\J'H?> β ' 1 @*qxQRJ[Z[\J xNQEJwHKWY\MT[Z^POt^[ZJQEDTSEQ C ^DÃ^v MGDKJ²SVU FI¶ÃuEJI[\[ZJ η º k J SJDGLEYZJD/JIH|MTxNQR^LNMG8P[\QRYuEWv²W]ILEJ98 X R^[ZWDÃHlxNQEJ β JIH|MHK^LRHSYZv²JLRHKYZWL EJM¶WDKDGJIHKC WLRS£^Q ·<6ON PQ R CJDöJILMGY\[ZJTSJ/[±^wSY±HoMGDKYZXEQMGY\WL HKWY\M
P (X < η) = 0.63.
3 ³ − !#"%$&(' )*+ $,-./)0 .+ 1
p J²[ZY\OiDGJ²SEJ À WSEHKWLX> · @¶WLRH|MKY\MKQEJ²QELRJwDKF FDGJLR¶J®YZLNMKFDGJIHGHG^LNMKJµCEQEY±HKxNQqU YZ[ WQEDGLEY\MeQRLEJwDKJIOiQEJµ¶Wv²CE[MGJSJIH®CEDGWCEDGYZFMKF'H®SEQ7v²WS[ZJ£JM²SJ@HKWLJIvµCR[\WY©SE^LRH²SJ£LRWvXRDKJIQRH|J'HTFMKQRSJ'H²SJ [^XEYZ[\Y\MKF º ¿ WMGWLRHµxNQEJP[\J'Hv²WiS [ZJIHl]^v²v²^µJMl[ZW]LEWDGvw^[!SJTSJL H|Y\MKF'HlDKJ'H|CJI¶MKYZOJIHf(x|α, γ) =
γα
Γ(α)xα−1 exp (−γx) 0 x≥0,
f(x|m,σ) =(xσ
√2π)−1
exp
− log2(x/m)
2σ2
0x≥0
H|WLNMJQ É ^QRHKHKYMGD"'HqQMGY\[ZY±H|F'H H|WYÁM¯C WQED¯v²WSFI[\Y±H|JID[ZJOiY\JIY\[Z[ZYZHGH|JIv²JLNM H|WYÁM¯C WQED¯v²WSFI[\Y±H|JID[ZJMKJIv²CRHvµWt¹JLev²YZHC WQED¤DKFICR^DGJD¯QEL®HK¹HoMv²J©JLC ^LELEJ9> s r>r> @ º k JIC JILRSE^LNM t[\JlH|JILRHSJIHCR^DG^v MKDGJIH¯JIH|MXRY\JILv²WYZLRHFOiY±SJLNM º ³ º© ( % * #& &( ! %' #&( ( ( "( $ ( * )% ( %#$( $*()#q () * ( &( %' ∑ q () ( (&(#$ ) ( ∗ ∑
* ( *() *() * % !& "*( () #$( "( &( #$ !" % ∑ % % $*(
3 4,5"45 7 98&:F< RzK L&<7~EA?L<CAEDE½ LEJOt^DGYZ^XE[\JÈ^[ZFI^tMGWYZDKJ X H|QRY\Ot^LNM©[\Jv²WiS [ZJ 8TDGY±HKxNQEJ'Hn¶WLR¶QEDKDGJLNMÃH B(η0, η1, β) FMGQRSYZFWDGYZ]YZLEJ[Z[\JIv²JLNMCR^D Â JID|MGuEW[ZWL > ³c@J'HoMSF"[RLEYZJ/CR^D
X = min(E,W )
WE ∼ E(1/η0),
W ∼ W(η1, β)
Y\L SFCJLRSR^v²v²JLNM>[ U QELEJTSJe[5U ^QMKDGJ º p JMG^Q É SJ/SF* ^Y\[Z[±^LR¶JSJT¶JÈv²WSI[\JT¶WLRHKYZH|MKJJLP[±^µHKWv²v²J/SJIHnMÃ^Q ÉSJ/SF* ^Y\[Z[±^LR¶J/J É C WLEJLNMGY\JI[VJMSJ Å JYZXEQE[Z[
hB(x|η0, η1, β) = η0 +β
η1
(x
η1
)β−1 0x≥0.
YZLRHKY i[ZWDÃHKxNQEJ β ≥ 1 ¶JÈvµWSI[\JCJDGvµJMlSEJÈv²WSF[ZY±H|JIDnQEL ∑ H|WQEv²YZH 8®SEQ£OiY\JIY\[Z[ZYZHGH|JIv²JLNMnvw^Y±HnF]^[\JIv²JLNMH|Q HK¶JCMGY\XE[ZJSJ>MGWv®X JIDJL²CR^LELEJC ^D^¶¶Y±SJILM º p U ^DKMKY±¶[ZJSJ  JID|MGuEW[ZWL  WQRHKxNQEJM Ì k J[ZJQ É > · @CE[±^¶FJL LRLEJ É JIH qW DGJwQEL_DGFIHKQEv²F SJIHTCEDKWCEDGY\FMKFIH/SEJ£¶JM|MKJ SEYZH|MKDGY\XRQMKYZWL_JMFMGQRSYZJw[ U JIH|MKYZvw^tMKYZWLSEJIH/CR^DG^Çv MKDGJIH>SR^LRHlQEL¶I^SDGJ/SQEDÃ^XEYZ[\Y±H|MKJ'> CR^D>vw^ É Y\v²Y±HK^MKYZWLPSJ/[Z^µOiDÃ^Y±H|JIvXE[±^LR¶JÈJMY\L FDGJLR¶JX ^ ¹FIHKYZJLELEJc@ ºÀ ^LRH¶JM®^D|MGYZ¶[\Jq[\J£ONYZJYZ[Z[\Y±HKHKJv²JILMµSJ [5U WX¸oJM®SqU FMKQRSJ£JIH|MC JIDGËQ¶Wvµv²J [±^@CEDGY\L ¶YZCR^[ZJ ¶^QRH|JSJ£SFÇ
^YZ[\[±^L ¶J º YZLRHKY ¯WLHKQECECWHKJ β > 1 JM η0 > η1¶J£xNQEY©YZvµCR[\Y±xNQEJ£xNQEJ£[±^ SQRDKFIJ SJ£OiYZJ v²Wt¹JLRLEJ ^ Ot^LNM
^¶I¶Y±SJLNMlJIH|MlCR[\QRHl[ZWLR]QEJ/xNQEJe[Z^µSQEDGFJ/SJeOiYZJv²Wt¹JLELEJe^ Ot^LNM>SF* ^Y\[Z[Z^LR¶JCR^D>QRH|QRDKJ º LPv²WLNMKDGJ SJIHMKJ'HoMÃHeLNQRvµFIDKY±xNQEJIHJM/SEJIHv²JIHKQEDKJ'HeSJ²]DÃ^LRSJIQEDÃHCEDGFISEYZ¶MKYZOJIH VxNQEJw¶JMKMKJ²v²WSF[ZYZHG^tMGY\WL ^YÁMHKJLRHeSE^LRHeSJIH¶^HW VCR^DuR^XEYÁMGQRSJqH|JIQE[\Jµ[±^[ZWYSJ Å JYZXEQE[Z[JIH|MQEMKYZ[\Y±H|FIJ º p U YZLNMKFDMeJIH|M/SWLR¶²SVU ^v²FI[\YZWDGJD WDKMKJv²JILM/[±^CEDGFISY±¶MGY\WLPSQ¶Wv²C WD|MGJv²JLNMSJ ∑ º
! O% )( '()*,+ - -1 6- 3/f
º ( * #$* ( $ *%'* () *&( &$%| * %' % % ( & %' " *( $**( #$&*()#$ * !" ( #$* *%'* () #& *( * (
¿ WMKWLRH®¶JCJLRSR^LNM²xNQEJPSE^L H[ZJPH|MKDGY±¶Mµ¶I^SDGJSqU ¼À Ê ¶JPvµWSI[\JJIH|M²^CECR^DKQ_CJQ8QMKYZ[\J@^Q%¶WQEDGH®SJLEWMKDGJlFMKQ SJJIL²J JM N¶Wvµv²JCRDKF'¶Y±H|F>^QRCR^DÃ^ Ot^LNM [ZJIHSWLELEFIJIH ¼À Ê H|WLNM]FLRFDÃ^[ZJv²JLNM¤MKDGY\FIJIH^Q²HKJL HW[5U WL¶WLELR^ ZM SJ ^ËWL^HKHKJÄCEDKF'¶Y±H|J [ZJQRDÈWDKYZ]YZLEJ º À U ^QMKDGJIHSWLRLEFJ'HYZLRSQRH|MKDGYZJ[Z[\J'HWLMeCJDGvµY±HSVU QEMKYZ[\Y±H|JID¶JÈv²WS[ZJe^ OJI¶ÈH|Q ¶¶*'HiJIL£CR^D|MGYZ¶QE[\YZJDlSJ'H>SQEDGFJIH©SJeOiY\JeSJv²Wt¹JL HnSJMGDG^LRH|CWDKM> ¿k Ê JILMGDKJICEDGYZHKJIH©SJ[ZW¶^tMGY\WLSJ/OWY\MKQEDGJIHA@ º ¨T¢¤ª Te¨T*q¯e¯V¯2²¡P¡d§n£¢¤¥|©¯¡P3 4;463 Rzo KÈ6<=8&< 8 L C< 8&<z6<p J'HnSEWLELEFIJIH©H|WLNMlSF"[RLEYZJIH©H|QRDnQELJIHKCR^¶Jv²JIHKQEDÃ^XE[ZJ (χ,A) xNQEJÈ[5U WL v®QELEY\MlSVU QELEJev²JIHKQEDGJÈSWv²Y\LR^LNMKJ
ν CWHKYÁMGY\OJ º À ^LRHLEWMKDGJ/FMGQRSJ A JIH|M[±^µMKDGY\XRQ X WDKFI[\YZJLRLEJ ∗ B(χ) JM ν JIH|M%> [\JTCR[\QRHÈH|WQEOJILNMA@l[±^²v²JIHKQEDKJ®SJp JXJIHK]QEJ º À ^LRHl[\JT¶I^SDGJeSJ/[Z^²SQRDKFIJeSEJ/ONYZJ χ = IR+ º LLRWMKJl^[ZWDÃH X QELEJ©O ^DKY±^XR[\J©^[ZFI^tMGWYZDKJnSF [RLRY\JlH|QRD[5U JIHKCR^¶JCEDGWXR^XEYZ[\Y±H|F (χ,A,Mθ : θ ∈ Θ ⊂ IRd) º LLEWMKJIDG^CE[\Q HHKYZvµCR[\JIvµJILNM X ∼ M(θ) JM M(θ) HKJDÃ^SF'H|YZ]LEFIJ©HKWQ H[ZJ©MKJDGv²J©SEJ &( > SJlCEDGWXR^XEYZ[\Y\MKFc@
SJ X º k JMKMKJOt^DGYZ^XE[\J^[ZFI^MKWYZDGJÈHKJDÃ^^HGHKWi¶Y\FIJJIL£]FILEFDÃ^[ 8®QELEJeSQEDGFJeSJeONYZJiJ É ¶JCMGF^Q¶ÃuR^CEY\MKDGJ : W£[ZJ¶I^SDGJTSJ[5U FMKQ SJµHIU F[±^DG]Y\MG> JMLEJ®¶WLR¶JDGLEJTCE[ZQRHH|JIQE[ZJv²JLNMÈ[±^ SQRDG^XEY\[ZY\MKF@ º p ^ SJILRH|Y\MKFµSJ X C ^DDÃ^CRC WD|M8 ν H|JIDG^LRWMKFIJ f(x|θ) JM²HK^ WLR¶MKYZWLSJDKFICR^DKMKY\MKYZWL F (x|θ) º p ^ WLR¶MGY\WL7SJPH|QEDGOiY\J > WQ % @®HKJDÃ^SF'H|YZ]LRFJCR^D
S(x|θ) = Pθ(X > x) = 1 − F (x|θ).½ L®F'¶ÃuR^LNMKYZ[Z[\WL
xn = (x1, . . . , xn) Y º Y º S º J'HoM^[\WDGH¯[Z^ÈDGJCEDGFIHKJLNMG^MKYZWLSEJ Xn = Xn 'QRLEJ>H|QEY\MKJ>SJ©Ot^DGYZ^XE[ZJIH^[\F'^tMKWY\DGJIHY\LRSEFCJLRSE^LNMKJIHnJMlY±SJLNMKY±xNQEJv²JILMlSY±HoMGDKYZXEQEFIJIH©SJ (χ,A,Mθ)º À ^LRHn[\J'Hn^CECE[ZYZ¶I^tMKYZWL HWL LEWMKJIDG^HKWQEOJLNMlCR^D¶Wv²v²WiSEYÁMGF
x1, . . . , xni.i.d.∼ M(θ).
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JLQRLEJTWX H|JIDKOt^tMGY\WLC ^DKMKYZJ[Z[\J yiºÀ U QRLEJ ^ËWL ]FILEFDÃ^[ZJWL LEWMKJIDG^
yn[ZJIHnFI¶ÃuR^LMGY\[Z[ZWLRH©SJÈSEWLELEFIJIHlSY±H|CWLRY\XE[ZJIH> ¼¾ WQ£HKY\v®QE[±^tMKYZWL H @ º LHKC F'¶Y5[RJ/SJQ É Mo¹iC J'H>SJ/v²FI¶I^LEY±H|v²J'H
³ º ( " "!"$ º m [nHU ^]YÁMTSQ v²FI¶I^LEY±H|v²J²CEDGFCWLRSEFDÃ^LNMeSR^LRH/¶JM|MGJwFMGQRSJ¯xQRY¶WDKÇDKJ'H|CWLRS 8QELEJ SQEDGFJ SVU WXRHKJDGOt^tMKYZWLCRDKF'SF [RLRY\J º WYÁM C = Cn QELEJ HKQEYÁMGJ SEJOt^[ZJQEDÃH [ É JIHJMY\LRSEFCJLRSE^LNMKJIHeSJ
Xnº L#SF"[RLEY\MT^[ZWDÃH Yi = min(Xi, Ci) ∀i ∈ 1, . . . , n º p WDGHGxNQEJ®¶JIHe¶JILRH|QRDKJ'HH|WLMlQELEY±xNQEJIH iWLCR^DG[\JIDG^H|YZv²CE[\JIv²JLNMlSEJ r ¹iC J m º À U QELPC WY\LNM>SJONQRJÈYZLRSQRH|MKDGYZJ[
3 ³ − !#"%$&(' )*+ $,-./)0 .+ 1
>¼¾ SEJMo¹iCJ m CEDGW]DKJ'HKHKY' > WQ $# %'(( @nJIH|M©CE[ZQRHnDKF'^[ZYZH|MKJN^Q£HKJLRHnWP¶JIHnYZLRHoMÃ^[Z[Z^MKYZWLRH©HKWLNMnv²Y±H|J'HJIL@HKJDGOiYZ¶J8wSEJIH>SE^MKJIH>SEY\OJDÃH|J'H ºº ( " º LwSF [ LEYÁMnQEL²LEWv®XEDKJ [ É J r ∈ 1, . . . , n YZLRSFIC JILRSE^vµv²JILM©SJ Xn
º LwSF"[RLEY\M^[\WDGHC = X∗
r [\JHKYZLE][ZJMGWL/DGJIH|MKDGJYZLM8>[Z^HoMÃ^tMKY±H|MKY±xQRJSVU WDÃSDKJ r SJ X º h QEYZH Yi = X∗i ∀i ∈ 1, . . . , rJM Yj = X∗
r ∀j ∈ r + 1, . . . , n º k J/Mo¹iC JµSJ¶JLRHKQEDGJT^[ZFI^tMGWYZDKJ/¶I^DÃ^¶MGFDGYZHKJeJL vw^¸oWDGYÁMGFTSJIHJIHGHK^YZHSJ/[±^XWDÃ^tMGWYZDKJNxNQEJ/[5U WL@HoMGWCECJT^QPX WQMSJ/[±^ r N PQ R SEF* ^YZ[Z[Z^LR¶J/¶WLRH|MG^tMGFJ º
3 4 ;45 D LDL< 8BA&A < A L ABD(<p ^²HoMGDKQ ¶MKQRDKJeSJIHlSWLELEFIJIHlWX H|JIDKOt^XR[\J'HYZLRSQEY\MSWL ¶ÈJIL]FLEFIDG^[ [5U J É YZH|MKJILR¶JeSVU QEL@JLRHKJv®XE[ZJSJ (( $
&(( * NLRWMKFe[ZJeCE[\Q H>H|WQEOJILM zJMSWLNM>[Z^µSJILRH|Y\MKF/¶WLRSY\MKYZWLRLEJ[Z[\JIv²JLNM 8[5U F'¶ÃuR^LNMGY\[Z[\WL£WXRH|JIDKOFHKJDÃ^
LEWMGFJ k(z|θ,yn) º k JMJLRHKJv®XE[\J/SY DGJTSJ ^ËWL WLRSE^vµJILNMG^[ZJ/H|JI[\WL[ZJIHvµWSI[\J'H º³ º À ^LRH®[ZJ¶^HSVU QELv²WSI[\JPH|WQEv²YZHR8QELEJPH|WQEDG¶JwQELEY±xNQEJSJPSF* ^Y\[Z[±^LR¶J > J É CWLRJLNMKYZJ[©JM Å JIY\XEQR[\[ @¶JIHSEWLELEFIJIHv²^LRxNQR^LNMGJIHHKWLNM[ZJIHO ^[\JIQEDÃHSJSF ^YZ[\[±^L ¶J¶JLRHKQEDGFJIH RHKWY\M[ZJIHSEWLELEFIJIH>YZLR¶Wv²CE[MKJ'H º Y\LRHKY
z ∼ M(θ) ºº p WDÃHKxNQEJn[\J©v²WiS [ZJnJ'HoM 8DGY±HKxNQEJ'H¯¶WLR¶QEDKDGJLNMÃH'¶ÃuR^xNQEJlSWLRLEFJ WXRHKJDGOFIJJIH|M^tM|MGDKYZXEQEFIJ 8È[5U QELµSJIHv²WS[ZJIHeJIL_¶WL ¶QEDGDKJILR¶J º ¼ [\[ZJ ¶JILRH|QRDKJwSWLR¶²SJIHeDGFI^[\Y±HK^MKYZWLRHÈCEDGWtOJILR^LNMSqU ^QMKDGJIHev²WS[ZJIH º YZLRHKY SE^LRHn[\J¶^H©SQvµWSI[\J B(η0, η1, β) iQELRJOt^[ZJQEDlSJSF* ^Y\[Z[Z^LR¶JÈWXRHKJDGOFIJNSVU WDKYZ]YZLEJ^¶¶YZSJILNMKJ[Z[ZJR> J É ÇCWLEJILNMKYZJ[Z[\Jc@ ¶^¶ÃuRJ QELEJ>SWLELEFIJ©SJ>SF ^YZ[\[±^L ¶JlSJlONYZJYZ[Z[\Y±HKHKJv²JILM > SEJ Å JYZXEQE[Z[ @JMDGFI¶Y\CEDGWxNQEJv²JLNM ºh WQRD¶JwMo¹iCJ SJ v²WS[ZJ¯[\J'HTSWLELEFJ'H/vw^LRxNQR^LNMKJIHH|WLMTFI]^[\JIvµJILNMe[ZJIHWDGY\]Y\LRJIH/SJ'HTSF* ^Y\[Z[±^LR¶JIH ºÂ YZJLFOiY±SJv²v²JLNM R[ZJIHSWLELEFIJIHSJT¶JLRHKQEDGJ/LEJTCJDGv²JM|MGJLNMCR^HSJTCWQROWYZDSY±HoMGY\LR]QEJID[Z^wHKWQRDG¶JeSEJSF ^YZ[\[±^L ¶J/xNQEY^QRDG^YÁM>CEDGWSQEY\Ml[±^wSWLELEFJeYZLR¶WLELiQEJ º
p JµMKJIDKv²J (( $* ()*&( > $ ( & @J'HoMTHKWQEOJLNMeY\LNMKDGWSQEY\MeSR^LRHe[ZJIHe^DKMKY±¶[ZJIHeHG¶YZJLNMKY5[ xNQEJ'HC WQED>YZLR¶[ZQEDGJ[5U QEL@WQ@[ U ^QEMKDGJeSEJT¶JIHlCRDKW¶J'HKHKQRHlSE^LRHlQELEJT¶JDKMG^YZLEJe]FILEFDÃ^[ZY\MKF º À ^L Hl[Z^²HKQEYÁMGJTSJT¶JMGDG^ Ot^YZ[!WLMKJILMGJDÃ^²SVU QEMKYZ[\Y±H|JID^Q@vw^ É YZvQEv [ZJeMKJIDKv²J (#$& )%'* C WQEDFOiY\MKJID>[\J'H>¶WL QRH|YZWL H º
Chapitre 2 < X O
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real dataweibull dist.B dist.
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[\W]Ç5MKJIvµC HnW¶Wv²v²JL ¶JeOFIDKY\MG^XE[\JIv²JLNM©[ZJ/OiY\JIY\[Z[ZYZHGH|JIv²JLNM > WQ@[ZJeDG^¸oJQELEY±HGH|JIvµJILNMA@©SJ ∑ º & &( k JMKMKJ²SFvw^DöÃuEJTJ'HoMe¶JIC JILRSE^LNMJ É MKDv²Jv²JILM[\YZv²YÁMGFJ º ¼ L#J JM [5U JIH|MKYZv²^MKJIQEDÈSJ ^CR[Z^LÇs JYZJD HU Y\[NJIH|MHG^L HXEY±^Y±H IJ'HoM¤H|JILRH|YZXE[ZJK8[Z^ ^YZXE[\JMG^Y\[Z[\JnJM 8[±^¶JLRHKQEDGJSJ©[ U FI¶ÃuR^LNMKYZ[\[ZWLS> OWYZD¯^Y\L H|Y h JMKJIDGHKWL ³I´ NJM k uRJL W[Z[Z^LRSJD Ì p ^L SX JIDK]V³I´ Z @M[±^µCEDKF'H|JILR¶J/SVU QELPC WY\LNM^Mo¹NCRYZxNQEJNC ^D>J É JIvµCR[\JeJL@xNQEJQRJSJ SY±HoMGDKYZXEQMGY\WL > SFIOJDGv²Y\L ^]J@WQEXE[ZY\FSWLRLEFJ ¶JLRHKQEDGFJO8_SDGWY\MKJ@JLX[RLSVU FI¶Ãu ^LNMKYZ[\[ZWL JMG¶ º @*CJDKMKQEDGXJ¶WLRH|Y±SFIDG^XE[\JIv²JLNM©[5U ^t¸oQRH|MKJIvµJILNMl[ZYZLEFI^Y\DGJ º **()#* p JIH/v²FMKuEWSJ'He]DÃ^CEuEY±xNQEJIHeDGJIH|MKJILNMeCR[\QRHSJIH/CRDKF'^[±^XE[ZJIHe^Q É MKJ'HoMÃHT¶[±^HGHKYZxNQEJ'HSqU ^SFIxNQR^MKYZWLSJv²WS[ZJR> CR^DnJ É Jv²CE[ZJ W[Zv²W]WDKWtONÇ v²Y\DGLEWtO) k DÃ^v²JID|ÇcOWL s Y±HKJIHA@xNQEJÈSJ'Hv²FMGuEWSJIH©SVU JIH|MKYZvw^tMKYZWL º À JLEWv®XEDGJQRHKJIH©DGF* FDGJL ¶JIHlHKWLNMSY±H|CWLEYZXE[ZJIH>CWQRD>[ U ^CECR[\Y±¶^MKYZWL@SJ/¶JIHvµFMKuEWSJ'H>¶Wvµv²J/CEDGFI^[Z^XE[ZJ^QP¶ÃuEWY ÉSJ>v²WS[ZJ º À ^LRH[ZJ¶WLMGJ É MKJSEJ[Z^/SEQEDKFIJSJOiYZJ¶YÁMGWLRH p ^ >[\J'HKH >|³I´ Z ¶Ãu ^C º· @ À U ]WH|MKYZLEW Ì0 MKJCRuEJLRH>o³'´ Z· @JM  JDKMKuEW[\WL > ³iCR^]JIH Ç @ º3 3 , 3 + I(*RO&)(W&) G'12- ,$½ LEJw^QEMKDGJ ^ËWLHKYZvµCR[\J²SJR[ É JD[\J'HÈC ^DÃ^v MGDKJ'HSJ'Hv²WS[ZJIHJ'HoM/SVU JIH|MKYZvµJIDeJv²CEYZDKY±xNQEJv²JLNM[\J'HÈv²WÇ
v²JLNMGH/WQ_[\J'HTxNQR^LNMGY\[ZJIHTSEJ [Z^SEYZH|MKDGY\XRQMKYZWLSJ X ∼ M(θ) º p JwC ^DÃ^v MGDKJ η SQ_v²WiS [ZJ£SJ Å JYZXEQE[Z[J'HoM^YZLRHKY[ZJ 63 PQ R CJDöJLNMGY\[ZJ£SJ[Z^ SEYZH|MKDGY\XRQMKYZWL º k JCJLRSR^LNM ¤¶JwMo¹iCJSVU JIH|MKYZvw^tMGY\WL_JIH|MµHKWQEv²Y±HG8CE[ZQRHKY\JIQEDÃHSY54w¶QE[\MKFIH
• [5U J É CEDKJ'HKHKYZWL1SJIH xNQR^LNMKYZ[\J'H²WQSJIH vµWv²JLNMGHwLU J'HoM CR^H WDöFIv²JLNMwJ É CE[ZYZ¶YÁMGJ¶U JIH|M [ZJ ¶^H£^ OJI¶[ZJv²WS[ZJ B 8µDKY±HKxNQEJ'Hl¶WL ¶QEDGDKJILNMGH9>  JDKMKuRW[ZWL &%~· @ º h ^D>^YZ[\[ZJQEDÃH N[5U JIH|MKYZv²^MKYZWLPJv²CEYZDKY±xNQEJ/SJIHxNQR^LMGY\[ZJIHlJ'HoM ^QRHGH|FIJC ^D>[±^®CRDKF'H|JILR¶JeSJT¶JILRHKQEDKJ º• [ZJIHY\LNMKJIDKOt^[\[ZJIH>SJ®¶WL [ ^LR¶JTSJeMKJI[ZHJIH|MKYZvw^tMKJIQEDÃH>Jv²CEYZDKY±xNQEJIHH|WLM]FLEFIDG^[\JIv²JLNM^HK¹Nv²CMGWMGYZxNQEJ'H > JMSWLR¶eLEWLP^SE^CMGFIH 8µLEWMKDGJe¶WLNMKJ É MKJeY\L SQRH|MKDGY\JI[ @ º
B ? C %"!# M *! (, ]*
3 3U] +I(*??&)(W!#GNH-.!c!c+p J'Hv²FMGuEWSJIHnSJIHnv²WYZLRSDGJIH¶I^DGDKF'HWLNMCWQRDWX¸oJ'¶MGY SVU JIH|MKYZv²JD θ JLwv²YZLEY\v²Y±HK^LNMnQELEJJ É CEDGJIHGH|YZWLµMKJI[\[ZJxNQEJ
φ =
n∑
i=1
[ψ∗(X∗i ) − ψ(X∗
i |θ)]2
W X∗iJIH|M[Z^HoMÃ^tMKY±H|MKY±xQRJSVU WDGSEDKJ ∗ i SJ>[5U FI¶ÃuR^LMGY\[Z[ZWL ψ(x|θ) JIH|MMo¹iCEYZxNQEJIv²JLNMQELEJ WLR¶MGY\WLwSVU FIOW[\QMGY\WLwSJ
X > SJLRHKY\MKF WLR¶MGY\WLSJDGFC ^DKMKY\MKYZWL º\ºZº @lJM ψ∗(x) J'HoMÈC ^DJ É Jv²CE[ZJQELJIH|MKYZv²^MKJIQEDÈLRWLCR^DG^vµFMKDGYZxNQEJ®SJψ(x|θ) º k ^L.[RJI[ZS Ì< WDK]v²^LO>o³I´ 7; @¤WLNMnCRDKWC WNH|FÈ[\J¶ÃuEWY É ψ(x|θ) = logS(x|θ) W S(x|θ) J'HoM©[Z^ WLR¶MKYZWLSJ®H|QEDGOiY\JRSWLM[Z^wHKQEDGOiY\JTJIv²CEY\DGY±xQRJTJIH|MQEL J'HoMGY\vw^MKJQRDÈHG^LRH>XRYZ^YZHÈSYZHKCWLEYZXE[ZJRYZLRSFIC JILRSE^LMeSQ¶ÃuEWY É SJv²WS[ZJeCR^DÃ^v²FMKDGYZxNQEJ ºk JIHv²FMKuRWiSEJIH®CEDKF'H|JILNMKJLNM®¶JDKMG^Y\LEJ'HSY54 ¶QE[\MKF'HV8 ¶Wv²v²JLR¶JDC ^D[ZJ£]DG^LRS LEWv®XEDGJ£SJHKW[ZQMGY\WLRH 8
DGJo¸oJMKJD¶^DLqU ^CEC ^DKMKJL ^LNMCR^H 8 Θ > CR^DÃ^v MGDKJ'HSVU FI¶ÃuEJI[\[ZJLRF]^MKY' H JMG¶ º @ º m [¯J'HoMeSWLR¶TLEFI¶JIHGHK^Y\DGJTSJ[ZJQRDDÃ^t¸oWQMKJIDeSEJ LEWv®XEDGJQRHKJIH/¶WLNMKDÃ^YZLNMKJIH º  JI^QR¶WQEC SqU ^QMKJIQEDGHTWLNMTFv²Y±H 8@[\JIQEDHKQt¸oJMµSJ WDKMKJ'HeDKJ'HoMGDKY±¶MGY\WLRHSVU Jv²CE[ZWYqSR^LRH>SJ'H¶^H>SJ ^YZXE[ZJeMG^YZ[Z[\J/JM>¶JLRHKQEDGFIHB> p ^ >[ZJIHGH~³I´ Z NC ^]J'H 6<6N Ç 6<6<6 @CC ^D>J É JIvµCR[\J Ê DGY\J'SiÇvw^L Ì » JID|MÃH|XR^ iu >o³'´ Z @Y\LRSEYZxNQEJILM xNQEJ@[5U J'HoMGY\vw^tMGY\WL%SEJIH²v²WYZLRSDGJIH²¶^DKDGFIHµSVU QEL%vµWSI[\J B 8#DKY±HGxQRJIH¶WLR¶QEDKDGJLNMÃHÇ©W@[5U QEL SEJIH>CR^DG^v MKDGJIHlJ'HoM [ É FeÇJIH|MXEYZJLCE[ZQRH>XRYZ^YZHKFJTxNQEJT[5U JIH|MKYZv²^MKYZWLSQvw^ É YZvQEvgSJOiDÃ^Y±H|JIvXE[±^LR¶Ja> OWYZD § Eº Eº Eº ³c@ º m LRSY±xQRWLRH®¶JCJL SE^LNMµxNQqU QELEJPCEDGW¶FISEQEDKJSJ£v²WYZLRSDGJIH®¶^DKDGFIHC WQED®SJ'HSWLELEFIJIHvw^LRxNQR^LNMKJ'HCRDKWC WNH|FIJCR^D J'^[Z¹ Ì!Å JIH|MKvw^¶WMKM>|³I´ ;· @HIU ^CECR^DKJILNMKJ 8QEL£^[Z]WDKY\MKuRvµJ ¼ s > OWYZD§ Eº Eº Eº @ º L²MKDGWQROJDÃ^ÈCE[\Q HSJCRDKF'¶Y±H|YZWL²H|QRDn¶JHKQt¸oJMnSE^L H s ¶ p ^¶ÃuE[±^L Ì DKY±HKuELR^L >o³'´´ CR^]J'H ; ³Ç ;· @ º3 3 R(WNH$ ,(/ Yn[\J'H/v²FMKuRWiSEJIH/]DG^CEuEY±xQRJIHeCJQEOJLNM®^Y±SJID 8@[Z^@Ot^[ZY±SE^tMGY\WL SEQ¶ÃuEWY É SVU QELEJ ^v²Y\[Z[\JSJwv²WiS [ZJIH q[\J'H
v²FMGuEWSJIH¤SVU JIH|MKYZvw^tMGY\WL®CR^D¤DKFI]DGJIHGH|YZWL/WQ®CR^D¤vµWv²JLNMGHLEJ>HKWLNMC ^HQMKYZ[ZYZHG^XE[ZJIH¤SE^LRH¤LEWMKDGJ©¶I^SDGJnSVU FMGQRSJSVU ^QEMG^LNM/CE[ZQRH/xNQqU J[Z[\J'HeLqU QEMKYZ[\Y±H|JILNMxQU QELRJwCR^DKMKYZJµSEJw[ U Y\L WDGv²^MKYZWL#MKDÃ^LRHKv²YZHKJµCR^D/[5U F'¶ÃuR^LNMGY\[Z[\WL º p J'HevµFÇMGuEWSJIHSJvw^ É YZv²YZHG^tMGY\WLSJ[±^ OiDÃ^Y±H|JIvXE[±^LR¶JEFI[Z^X WDKFIJIH>JIL WL ¶MKYZWL#SJ[±^£H|MKDGQR¶MGQEDGJSJIHSWLELEFJ'H CEDGWÇv²JMKMKJILM 8µCEDGJv²YDGJeONQRJeSqU MKDGJ/X J'^QR¶WQECPCE[ZQRHlJ 4 ¶^¶J'H ºG"4G"4,G 7 >>z z|N Dz|BA 8&<<o Ez| < KÈo AÈ<3 3 3 9!#N<QEp ^7OiDG^YZHKJv®XE[Z^LR¶J L(yn; θ) ¶WDKDGJIHKCWLRSd87[Z^%SJL H|Y\MKF SJ [Z^8CEDGWXR^XEY\[ZY\MKF ¸oWY\LNMGJ SJ'HSEWLELEFIJIH
yn =
(y1, . . . , yn) SJCRDKWtOJLEYZDwSQ1vµWSI[\J M JI[\[ZJDGJCEDGFIHKJLNMKJ@SWLR¶PMKWQMKJ[±^ ¶WLEL ^Y±HKHG^LR¶JWX¸oJI¶MKYZOJHKQED θ^CEC WD|MGFJµCR^D/[±^@HoMGDKQ ¶MKQRDKJwSJ'H/SWLELRFJIH º ¼ L_SYZOiYZHG^LNM/[ U FI¶Ãu ^LNMKYZ[\[ZWL ynJL SWLRLEFJ'HeLEWL ¶JILRH|QRDKFIJIH xr
JM¶JLRHKQEDGJIH 8²SDKWYÁMGJ cn−r
¶Wv²v²JT^Q § ³ º : WLPWXEMKYZJLNM RSR^LRHlQEL¶I^H>SJT¶JLRHKQEDGJSJeMo¹iCJ m CEDGW]DKJ'HKHKY'
L(yn, θ) =
[r∏
i=1
f(xi|θ)]
n−r∏
j=1
S(cj |θ)
.
À ^LRHl[ZJT¶^HlSJT¶JILRH|QRDKJ/SJeMo¹iCJ mKm EWL@WXMKYZJLNML(yn, θ) = f(x∗1, . . . , x
∗r |θ) (S(x∗r |θ))n−r
W x∗i J'HoM[±^TH|MG^MKY±HoMGYZxNQEJSqU WDGSDGJ i SEQ H|WQRHoÇcFI¶Ãu ^LNMKYZ[\[ZWL xr> OWYZDCR^DJ É Jv²CE[ZJ » ^Q SWYZL N C WQEDSJ'HFI¶DKY\ÇMGQEDKJ'HCE[\Q HÈJ É CE[\Y±¶Y\MKJ'H @ º L SF'¶DGYÁM/SE^LRH[\J®MG^XE[\J'^Q º ³[±^£OiDÃ^Y±H|JIvXE[±^LR¶JWXRH|JIDKOFJSEJµSWLELEFIJIHe¶JILRH|QRDKFIJIH
] 3 − " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
H|JI[\WL[ZJIH©MGDKWYZHlv²WSI[\J'HlQMKYZ[\Y±HKFIH > [U8µJL ¶WDGJeC WQED>QELEJT¶JLRHKQEDGJ/SJMo¹NCJ m CEDKW]DGJIHGH|Y' @ º
;O,*,3!+%&E(η) η−r exp
(−
n∑i=1
yi/η
)
a%!+0*&+&W(η, β)
[βrη−r
r∏j=1
xβ−1j
]exp
(−
n∑i=1
(yi/η)β
)
B(η0, η1, β)
[r∏
j=1
(1/η0 + (β/η1) (xj/η1)
β−1)]
exp
(−
n∑i=1
yi/η0 −n∑
i=1
(yi/η1)β
)
º ³ =DG^YZHKJv®XE[Z^LR¶JeSJ'H>SWLELRFJIH>¶JLRHKQEDGFJ'HB> Mo¹iC J m CEDGW]DGJIHGHKY @nCWQED>[ZJIH©MGDKWYZHlv²WSI[\J'HlFMKQ SY\F'H º
p Jvw^ É YZvQRv0SJnOiDG^YZHKJv®XE[Z^LR¶J θMLJIH|M¯[±^HKW[ZQMKYZWLTSEJIH¯FIxNQR^tMGY\WLRHSEJOiDG^YZHKJv®XE[±^LR¶J ∂l(yn, θ)/∂θ = 0W l(yn, θ)
J'HoM©[Z^/[ZW]Ç5OiDÃ^Y±H|JIvXE[±^L ¶J º WQRHnSJIH©¶WLRSY\MKYZWLRHnSJDGF]QE[Z^DKY\MKFÈ^tM|MGJYZLNMKJIHC WQED©[\J'HMKDGWY±Hv²WSI[\J'HθML
JIH|M>QELEY±xNQEJ/JM>MGJ[xNQEJ√n(θML − θo
)L−−−−→
n→∞N(0, I−1(θo)
)
W θoJIH|Ml[±^µOiDG^Y\JeOt^[ZJQED > Y\L ¶WLRLNQRJ@©SQ@CR^DÃ^v MKDGJ θ JM I(θ) J'HoM>[±^µv²^MKDGYZ¶J/SVU YZL WDKvw^tMGY\WLPSEJ Ê Y±HKuEJDSQv²WiS [ZJ M(θ) ¶WLRHKYZSFIDKF
I(θ) = − ¼ θ
[∂2f(x|θ)∂θi∂θj
]
i,j∈1,...,d
.
À J'HµY\L WDKvw^MKYZWLRHµCE[ZQRHµCEDGFI¶Y±HKJIHµH|QEDw¶JM|MGJ@WCMGY\vw^[ZY\MKF@^HK¹Nv²CMGWMGYZxNQEJ¤SR^LRHµQEL%¶^SEDKJ@SJ@SWLELRFJIHµvw^LEÇxNQR^LNMKJ'HJM/C WQEDe[\J'HÈv²WSI[\J'HeQMKYZ[\Y±HKFIHeSE^L H¶JµSEWi¶QEv²JLNM qHKWLNMeCEDGFIHKJLNMKFIJIHSE^LRH W^SE[\JI¹)>o³I´ ³c@  ^¶ÃuR^>o³'´´ · iCR^]JIH ;7Z Ç ·H6 @  JDKMKuRW[ZWL % > · @ º
3 3 3 3 C JU- $ W(/!c I&)` DH'(/X+!c /+a (NI- ,12p WDÃHGxQU YZ[¤LqU ¹#^C ^HeSJ HKW[ZQMKYZWL#J É CE[ZYZ¶YÁMGJ²^Q#CRDKWXE['Iv²J²SEJµvw^ É YZv²YZHG^tMGY\WL ~[ U Jv²CE[ZWYSqU QELRJµCRDKW¶F'SQEDGJ
LiQEv²FDGYZxNQEJ>SVU JIH|MKYZvw^tMKYZWL²J'HoMLRFI¶J'HKHG^YZDGJ º p U ^[Z]WDKY\MKuEv²J ¼ s > $# & &( ( À JIv²CRHoMGJD p ^Y\DÃSÌ QRXEY\L ³I´ @J'HoM¶JDKMG^YZLEJIvµJILNM[Z^µCEDGW¶FISEQEDKJ/SVU JIH|MKYZvw^tMGY\WLP[±^²CE[ZQRH>QMGY\[ZY±H|FIJTSE^LRH[\JT¶I^SDGJ/SJIH>v²WSI[\J'H8²SWLELRFJIH©vw^L xQ ^LNMKJ'H º ¼ [Z[ZJT^®SWLELEFe[ZY\JIQ 8µ[5U FI[Z^X WDG^MKYZWLPSJeLEWv®XEDKJIQ É ^[Z]WDKY\MKuRvµJ'HlSFDGY\OFIHl^v²FI[\YZWDÃ^LNMH|J'HlC JID WDKvw^LR¶J'H ºp U ^¶¶F[ZFDÃ^tMGY\WL SJIH^[Z]WDKY\MKuEv²J'H ¼ s JMSEJ®HKJIHSFDGY\OFIH R[ZJIHÈLEWv®XEDGJIHSVU Y\MKFIDG^MKYZWLRH R[±^£H|MG^XEYZ[\Y\MKFµSJ®[5U JIH|Ç
MKYZvw^tMKYZWL+ONY±H|ÇM8Ç5OiY±H£SJ#[ U Y\LRYÁMGYZ^[\Y±HK^MKYZWL0SJIHP^[Z]WDKY\MKuEv²J'Hn[Z^8H|JILRH|YZXEYZ[\Y\MKF SJ ¶J[Z[ZJÇ ¶Y98[±^DGF]QR[Z^DKY\MKFSEJ#[±^OiDG^YZHKJv®XE[Z^LR¶JµWLNM ^YÁM[5U WX¸oJMSJ²MKDIH/LRWvXRDKJIQRH|J'HeDKJ'¶ÃuEJDöÃuEJ'H~YZvµCWHGHKY\XE[ZJIHG8@[\Y±HoMGJD/v v²J£SR^LRH/[ZJ SWÇvw^YZLEJTDGJIH|MKDGJYZLMSJ[±^ SQEDÃ^XRY\[ZYÁMGF º Y¯[ZJ[ZYZONDGJSJ s ¶ p ^¶ÃuE[±^L Ì DGYZHKuELR^L)>o³I´´ @l¶WL HoMGYÁMGQEJTQELEJ®DKF FDGJLR¶Jvw^t¸oJQRDKJ£HKQED®[ U Jv²CE[ZWY©]FILEFDÃ^[ZYZHKF SJ'H®^[\]WDGYÁMGuEv²JIH ¼ s QRL_MKDÃ^ Ot^YZ[YZvµCWDKMG^LNM®H|QRD[5U JIvµCR[\WY©SVU ¼ s JMµSEJH|J'HSFIDKYZOF'H>H|QED>[ZJIHvµWSI[\J'HSJ Å JYZXEQE[Z[J'HoM^QRHGH|YqCRDKF'H|JILMGF/SE^LRH[Z^µMGu IHKJTSJ  ^¶Ãu ^>|³I´´ · E¶ÃuR^C º ³/JM @ º½ LEJwDGJOiQEJ²]FLEFIDG^[\J²SJw[±^¶WL HoMGDKQR¶MKYZWL JMSJ'HeCEDGWCEDGY\FMKF'H/^HK¹Nv²CMGWMGYZxNQEJ'H/SQ_SFDGY\OFwHoMGWi¶Ãu ^H|MKY±xQRJ ¼ sJIH|M>CEDKF'H|JILNMKFJ/SE^LRH ¿ Y\JI[ZHKJL > @ º ¿ WQRH ^YZHKWLRHlY±¶Y[\J/DÃ^CECJ[qSEJIH>¶^DG^¶MGFDGYZH|MKY±xNQEJIH©JIHGHKJLNMKYZJ[Z[\J'H>SJIH>SEJQ ÉCEDGY\LR¶Y\C ^[ZJIH©v²FMKuRWiSEJIH º
B ? C %"!# M *! (, ]/]
9!#(*NH+!c ¼ [Z[ZJ ¶WL H|Y±HoMGJ`8#vw^ É YZvµY±HKJD²[ U JIHKCFDÃ^LR¶JPSJ@[±^OiDÃ^Y±H|JIvXE[±^L ¶J¶Wv²CE[MGJ@SEQ8v²WS[ZJ¶WLRSY\MKYZWLRLEJ[Z[\JIv²JLNMn^Q É SWLELEFIJIHWXRH|JIDKOFJ'HJMK8TQELEJOt^[\JIQEDSJ θp
¶WQEDG^LNMKJ º L ¶ÃuEWY±H|Y\MQELRJYZLEYÁMGYZ^[\Y±HG^tMKYZWLθ0º p U YÁMGFDÃ^tMKYZWL p→ p+ 1 HIU F'¶DGYÁM
θp !#"$%& '() *(+,+ -
θ → Q(θ|θp) = [l(xn, θ)|yn, θp].
./0 1*& '23+%θp+1 = arg max
θ∈ΘQ(θ|θp)
p ^HKQEYÁMGJ θp ^YZLRHKYCEDGWSQEY\MKJ ^Y\M¶DGW ±MKDGJ[±^OiDG^YZHKJv®XE[±^LR¶JJM¶WLiOJDG]JOJDÃH θML v²Wt¹JILELR^LNM¤xQRJ[±xQRJIH
ui¹iC WMKu 'H|J'H º ¼ LP]FLEFIDG^[~[Z^w¶WLiOJIDK]JLR¶JÈSFIC JILRS WDKMKJv²JILMSEJ/[ U Y\LRYÁMGYZ^[\Y±HK^MKYZWL θ0 > SJ'H^v²F[ZYZWDÃ^tMKYZWL H©HKWLNMCWHGH|YZXE[ZJIH>JIL@YÁMGFDÃ^LNM ¼ s HKQEDQEL]DG^LRSPLEWv®XEDGJeSqU YZLEY\MKY±^[ZYZHG^tMGY\WLRH @*RJM[Z^²v²FMKuEWSJTLU FIONY\MKJCR^H>[ZJTCEY]JTSJvw^ É YZvw^[ZW¶^Q É SJ/[±^®OiDÃ^Y±H|JIvXE[±^L ¶JJMCJQMSWLR¶CEDGWiSEQEY\DGJ/SJIHlDGFIHKQE[\MG^MGHlF[ZWYZ]LRFIH>SJ θML
> Å Q ¯³I´ Z<6 @ º
9!#(*NH+!c * p JTSFIDKYZOFTH|MKW¶ÃuR^HoMGYZxNQEJ/SJ ¼ s R[5U ^[Z]WDGY\MKuEv²J i¼ s > # $ #/ $# & &( ( k J[ZJQ É/ÌÀ YZJXW[\M©³'´ Z7: ³'´´ @V^^[\WDGHFMGF©¶WLRËQTC WQED¤H|QEDGv²WLNMKJID¯¶JIHSY 4 ¶QE[ÁMGFIH > WDKYZ]YZLEJI[\[ZJv²JLNMSE^LRH/QEL¶WLNMKJ É MKJ SJ v²FI[Z^LE]J'HTSJw[ZWY±HTSJ CEDGWX ^XEYZ[\Y\MKFc@ º WL CEDGY\L ¶YZC J J'HoMµSJ£¶WLRH|MKDGQEY\DGJ²QELEJ£¶Ãu ^ LEJ SJs ^D WtO²uEWv²W]LEJeJIDK]WiSEYZxNQEJE¶WLiOJDG]JLNMGJÈOJDÃHlQELEJ/[ZWYH|MG^MKYZWLELR^Y\DGJeQELEY±xNQEJSJ/v²Wt¹JILELEJ θML
EJLDGJvµÇCE[±^ËI^LNMl[\J'HSWLRLEFJ'Hlv²^LRxNQR^LNMGJIH©CR^DSEJIH>CJDKMKQEX ^tMKYZWL H>^[ZFI^tMGWYZDKJ'H º L@¶ÃuEWY±HKYÁM>QELEJ/YZLEY\MKY±^[ZYZHG^tMGY\WL θ0 º p U YÁMGFDÃ^tMGY\WL p→ p+ 1 J'HoM 4 ' i ∈ 1, . . . , n )!*,+%&5 *+,+ -6*
* -&-2,7 ki(X |Y = yi, θp)
./)894 ' i ∈ 1, . . . , n -+:2*-<;-'(
Xi ∼ ki(X |Y = yi, θp)
=/0 2, >(+:23+1 ?,@ ;-'+2AB 2 & &-C '-D
θp+1 = arg maxθ∈Θ
n∑i=1
log f(Xi|θ)
L CEDGWCWHKJµ^[\WDGH θSEM =∑P
i=m+1 θi/(P −m) m P ^ OJI¶ m JM P HKQ.4 HG^v²v²JLNMT]DÃ^LRSRHV¶Wv²vµJJ'HoMGY\vw^tMGY\WLSJ θML
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> k JI[\JIQ ÉÌ+À Y\JIX W[ÁM>³'´´ : À YZJXW[\M Ì m C³'´´ · @ ºk JCJL SE^LNM [ U FMÃ^CJ HoMGWi¶Ãu ^H|MKY±xQRJ²SEJ H|YZvQE[±^tMGY\WL SEJw[ U Y\L WDGv²^MKYZWL vw^LRxNQR^LMGJ²CR^DÃ^±MTH|QRHG¶JICMKYZXE[ZJ SJ
LEWt¹JID [ U YZL WDGvw^tMKYZWLwCRDKF'H|JILMGJNH|Y [Z^T¶JLRHKQEDGJJIH|MnCR[\QRHnYZvµCWDKMG^LNMKJxNQEJ[ZJIH©SWLELEFJ'HDGFJI[\[ZJIHJMnxNQEJÈ[ U FI¶ÃuR^LÇMGY\[Z[\WLwJIH|MnSJ ^Y\XR[\J>MÃ^YZ[\[ZJG> [Z^/Ot^[\JIQEDSJH|YZvQE[±^tMGY\WLw¶WQRDG^LMGJ θp+1DKY±HGxQ ^LNMSVU MKDGJ>MGD"'HJDGDGWLEFIJ@ º WiSEY\WLEWtO
> ; @©YZLRSY±xNQEJ®C ^DJ É Jv²CE[ZJ®xNQEJ[5U ^CECE[ZY±¶^tMGY\WL ^Q v²WSI[\JµSJ Å JYZXEQE[Z[¤DKJ'HoMGJ®^¶I¶JICMG^XE[\J¸oQRHKxNQqU 8 n = 20 CWQED/QRL MG^Q É SJ ¶JLRHKQEDGJµLRJ SFCR^HKHG^LNM/CR^H 10% ºqs ^[\uRJQEDGJQRHKJv²JILM/CJQSVU JIHGHG^Y±H/H|QEDTSEJIH/F'¶ÃuR^LNMKYZ[Z[\WLRHYZLRSQRH|MKDGYZJ[±H > SJ[ U Y\L SQRH|MKDGY\JTLiQR¶[\F'^YZDKJc@lH|WLNMOJLiQRH¶WL WDKMKJDȶJMKMKJTY±SFIJ º k JTxNQEY ^Y\M[5U FI[\FI]^L ¶J/JMÈ[ U Y\LNMGFDMCEDÃ^tMGYZxNQEJlSJ i¼ s SE^LRHQELµ¶I^SDGJ©]FLRFDÃ^[NDKY±HKxNQEJlSVU JL ^YZDKJ©QELEJlv²FMGuEWSJ 8[Z^ÈCEDGFI¶Y±HKY\WL RWQEJ> JM 8[ U ^CRC WD|M
] − " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
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3 3 3 ] QQV$NH-. ,(/X- X&)(*$, $,+ L DG^CECJ[Z[\J¶YÁÇ ^CRD"'H[ZJIHCWYZLNMGHÈY\v²CWDKMG^LMÃHSJ®[Z^£vw^ É Y\v²Y±HK^MKYZWL SJ®OiDG^YZHKJv®XE[Z^LR¶JTSEJIHv²WSI[\J'HQMGYÁÇ
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• %' ! &(( *% WYÁM T =∑n
i=1 xiº [ZWDÃH ηML = T/r JMµJ'HoM HG^LRH®XEY±^Y±H²H|Y r = n > CR^H²SJ
¶JLRHKQEDGJ@ º• %' *% % θML = (ηML, βML) J'HoMHKW[ZQMKYZWL@SJ'HlFIxNQR^tMGY\WLRH>SJ/OiDÃ^Y±H|JIvXE[±^LR¶J
1
β=
n∑i=1
xβi logxi
n∑i=1
xβi
−
r∑i=1
log yi
r, η =
(n∑
i=1
xβi
)1/β
r1/β.
k JM|MKJHKW[ZQMKYZWL%JIH|MµMKDÃ^SY\MKYZWLELRJ[Z[\JIvµJILNMw^CECEDGW¶ÃuEFJCR^DwSJ'H²vµFMKuEWSJ'H²SJSJIHG¶JILMGJPMo¹NCJ ¿ J lMKWLEÇ ^CEuRHKWL º p U J'HoMGY\vw^MKJQRD>WXMGJLiQ@JIH|MlLRWMKF θNRº
• %' 8 ** #$&(#"*( B(η0, η1, β) θML = (η0,ML, η1,ML, βML) J'HoM[±^wHKW[ZQMKYZWLSJ'H>FIxNQR^ÇMGY\WLRHSJeOiDG^YZHKJv®XE[±^LR¶J
η0 =
r∑i=1
xi
r∑i=1
pE(xi),
1
β=
n∑i=1
(yi)β
ln (yi)
n∑i=1
(yi)β
−
r∑i=1
pW (xi) ln (xi)
r∑i=1
pW (xi),
η1 =
n∑i=1
(yi)β
r∑i=1
pW (xi)
1β
,
^ OJI¶ pE(xi) = hE(xi)/[hB(xi)],pW (xi) = 1 − pE(xi),
pE(xi)JM pW (xi)
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)GL!+&#&+n4&?E 1%')(SGI,-!+&+&+,
)GL;f4'%%,*τ 6 < n ≤ 20 20 ≤ n ≤ 40 40 ≤ n ≤ 80 n ≥ 80
τ = 0%nl '%I!+]I1 nl nl nl
τ ≤ 50%pBJ pBJ pB: pBJ
50% ≤ τ ≤ 75%pBJ pBJ nl nl
τ ≥ 75%pBJ pBJ pB:Unl nl
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MKDIH[±^DG]J'H¶Wv²v²J>LEWQRH[5U YZ[Z[\QRH|MKDGJDGWL HSE^L H[Z^H|QEY\MKJÈSQ£¶ÃuR^CEYÁMGDKJ º m [±H WQRDKLEY±HGH|JILMSEJIH[ZYZvµY\MKJ'HDÃ^Y±H|WLELR^XE[\J'HC WQED[±^wDGJI¶ÃuEJIDG¶ÃuEJ/SVU QELEJJIH|MKYZv²^MKYZWL θ SJ θ RJMMKFv²WY\]LEJLNMÈSJ[ U Y\v²CEDGFI¶YZHKYZWL¶JID|MÃ^YZLEJTSJ/MGWQMGJ/MKJILMÃ^tMGY\OJSVU JIH|MKYZv²^MKYZWL@SJ θ XR^HKFJ/HKQED>[ZJDGJIHKCJI¶MSEJIHlui¹iC WMKu 'H|J'H>HoMÃ^tMKY±H|MKY±xQRJIH>¶[±^HGHKYZxNQEJ'H ºp WDÃHGxQRJ¶JIHY\LNMKJIDKOt^[\[ZJIH¤MGuEFWDKY±xNQEJIHLEJHKWLNMCR^H¶WLRHoMGDKQ ¶MKYZXE[ZJIHWQ²[\JÈH|WLM©^ OJI¶lSY54 ¶QE[\MKF%> ¶Wv²v²JSE^L H
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SJIHOt^DGYZ^XE[\J'HT^[ZFI^MKWY\DGJIHT^HKHKW¶YZFJ'H 8[5U F'¶ÃuR^LNMKYZ[Z[\WL WXRHKJDGOFyn
WL_CJQM²¶ÃuEJDöÃuEJIDG8JIH|MKYZv²JD®[Z^O ^DKY±^L ¶Jσ2 SVU QELJIH|MKYZvw^tMGJQED θ = θ(Yn) º JCEDGWSQEYZDKJTHKYCWHGHKY\XE[ZJ/[ZJIH¶WLRSY\MKYZWL HSJT[5U J É C FIDKYZv²JLNMG^MKYZWL@v²JLR^LM^Q>¼¾ q^Q MGDG^ OJDÃHSJµ[±^£DGJI¶WLRH|MKY\MKQMGY\WLSJ M FI¶Ãu ^LNMKYZ[\[ZWL HÈSJ²v v²J'H¶I^DÃ^¶MKFDGY±HoMGYZxNQEJ'HxNQEJ yn
^Y\M/HKJL HC WQED>WXEMKJLRY\DQRLEJTSY±HoMGDKYZXEQMGY\WL@SqU F'¶ÃuR^LNMKYZ[Z[\WLELR^]J/SJT¶JMJIH|MKYZv²^MKJIQED º ¼ vµCRY\DGYZxNQEJIv²JLNM WL@C JIQM^[ZWDÃHlWXEÇMKJILEY\DQELEJ/JIH|MKYZvw^tMGY\WL@SEJ σ2 º ¼ L ^Y±HK^LNMlC ^DJ É JIv²CE[\J/[5U uN¹iCWMGu IHKJTSVU QELEJ/DGFCR^D|MGYÁMGY\WLP]N^QRHGHKY\JILELEJ/SJT¶JM|MGJSY±HoMGDKYZXEQMGY\WLPSVU FI¶Ãu ^LNMKYZ[\[ZWLRLR^]J%> [\FI]Y\MKYZvµFIJTH|WQRHl¶JID|MÃ^YZLEJIH>¶WLRSY\MKYZWL HlCR^Dl[ZJeMKuEFIWDv²J/¶JLNMGDG^[[ZYZvµY\MKJc@ WLC JIQMWXEMKJLRY\D>QEL@YZLNMKJDGOt^[Z[\J/SJT¶WL.[^LR¶J^QMKWQEDSJ/[5U J'HoMGY\v²F θ(yn) ºp ^DKJICEDGWiSEQR¶MGY\WLSJ>SWLRLEFJ'H¤CEDKW¶ÃuEJ'H¯SEQ >¼¾ JIH|MSEWLR¶©QELEJ>^[ÁMGJDGLR^tMGY\OJOt^[±^XE[ZJ©C WQEDCR^[Z[ZY\JID[5U ^XRHKJLR¶J
SJIHY\LNMKJIDKOt^[\[ZJIHSEJ¶WL [ ^LR¶JlMKuEFIWDGYZxNQEJ'H º k U JIH|MLEWMG^vµv²JILM[5U JLt¸oJIQ SJIHv²FMKuEWSJ'HSJ $ " CRDKF'H|JILMGFJ'H^Q § º 6Eº º À ^LRH²¶JM|MKJC ^DKMKYZJ¤LEWQ H²CEDKWC WNH|WLRH®CE[\Q H|YZJQEDÃH²CWHGH|YZXEYZ[\Y\MKF'H²C WQED²DKJICEDKWSQRY\DGJP¶JIHw¶WLRSY\MKYZWL HSVU J É C FIDKYZv²JLNMG^MKYZWLRHl[ZJeCE[ZQRH [ SI[\JIv²JLNMC WNHKHKYZXE[\J º
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LEJLNMGY\JI[ JMnSJ Å JYZXEQE[Z[!iSR^LRHQEL£¶WLNMKJ É MGJLEWL ^H|¹iv²CMGWMKY±xNQEJJM©¶JILRHKQEDKF¶WLRH|MKDGQEY\MGHSqU ^CED"'HSJ'Hv²FMGuEWSJIHSJ WLR¶MGY\WLRHTCEYZOWMG^[\J'H ∗ º Yn[\J CRDKJIvµYZJDvµWSI[\JqMGD"'HTXEYZJL¶WLELiQ ^ ^Y\M[ U WXE¸oJM®SJLEWv®XEDGJQRHKJIHeFMKQRSJ'H[\J'HlDKF'¶JILMÃHlDKF'H|QR[ÁMÃ^tMGH>SEYZHKC WLEYZXE[\J'H>H|QRD>[\JTHKJI¶WLRSDKJ'HoMGJLNM>Jv²CE[ZWt¹F'H©SEJ/v²^LEYDGJvw^DK]Y\LR^[\J º QR¶QEL@DKF'H|QE[\MG^MLqU JIH|MSYZHKCWLEYZXE[ZJeHKQED>[ZJ/v²WiS [ZJ98²DGY±HKxNQEJ'H©¶WLR¶QEDGDKJILMÃH B(η0, η1, β) RSEJ/CR^DHG^²¶Wv²CE[ZJ É Y\MKF º3 ], , (*$ 2QE(W ,$ LH|QRCEC WNH|JnxQRJ[ZJIHSEWLELEFIJIHLEWL¶JLRHKQEDGFJ'H xr
H|WLNM¯SJIH¯DGFI^[ZY±HK^MKYZWLRHqSEJ[Z^Ot^DGYZ^XE[\J^[\F'^tMGWYZDKJ X ∼ E(η) º L£¶WLRHKYZSIDKJÈ[\J'Hl¶JL H|QEDGJIHnSJÈMo¹NCJ m > [ É Jc@JMlSEJMo¹iCJ mKm > ^[\F'^tMKWY\DGJ@ º p JIHnDKF'H|QR[ÁMÃ^tMGH©HKQEY\Ot^LMÃHlH|WLM©YZHGHKQRHnSEJ
'O% % O?M ? 6-( -O, ! (-O ]WY
[±^²¶WLRLR^Y±HKHG^L ¶JeSQP¶Wv²CWDKMKJv²JILM>JILP[ZWYqSEJ/[Z^µOt^DGYZ^XE[ZJ2
r∑
i=1
xi/r ∼ χ2(2r).
L MKDGWQROJDÃ^PCR^D®J É Jv²CE[ZJ£SE^LRH p ^LELEWt¹>o³'´´ ; qCR^]JIH ZH6 Ç Z7: @eSJIH/CRDKF'¶Y±H|YZWLRHTJMµSEJIH/J É JIv²CE[\J'H¶WLR¶DKJMGHHKQEDl[Z^¸oQRH|MKY5[ ¶^MKYZWL@SVU JLR¶I^SDGJv²JLNMÃH©¶Wv²CR^DG^XE[\J'H[ZWDÃHKxNQEJeHKJQE[ZJÈQRLEJevµJ'H|QRDKJeSQMKJv²CRHlF'¶WQE[\F T CJL SE^LNM[5U J É C FIDKYZJLR¶JeJIH|M>SEYZHKC WLEYZXE[\JS> H|WYÁM[ZWDÃHGxQU ^QR¶QELRJÈOt^[ZJQRDSJ/SF* ^Y\[Z[Z^LR¶JeCR^HKHKFJeLqU JIH|M¶WLELiQEJ@ º ( " LCWHKJ T =
∑ni=1 yi
º L+LEWMGJ χ2(2r),α
[ZJ#xNQR^LMGY\[ZJSVU WDÃSDGJ α SJ[±^[ZWY χ2(2r)> H|WYÁM P (T ∼ χ2
(2r) ≤ χ2(2r),α) = α @ º JI[\WL QEL SX JIDK] > ³@*[5U JLR¶I^SDGJv²JLNMeLEWL ^HK¹iv²CMKWMKY±xQRJ®[ZJ²CE[\Q HJ"4w¶I^¶JÈSJ[Z^®Ot^[ZJQEDlSQCR^DÃ^v MGDKJ η JIH|M WDKv®QE[ZFSE^LRHn[Z^®CEDGWCWHKYÁMGY\WL£SJ k W É >|³I´ ;76 @ WL£^Sv²JM>¶Wv²v²J^CECEDGW É YZvw^tMGY\WL 2r(η/η) χ2
(2r+1)º LWXMKYZJLNM>[ U YZLMGJDGOt^[Z[\J/SJT¶WL.[ ^LR¶J 1 − α C WQED η[
2T
χ2(2r+1),1−α/2
,2T
χ2(2r+1),α/2
]. > º ³@
( " m ¶Y T =∑r
i=1 xi + (n − r) xrº WQRHTQELEJ ui¹iC WMKu 'H|J SqU YZLRSFIC JILRSE^L ¶J£SJ'H yi
QELYZLNMKJDGOt^[Z[\J SEJ£¶WL.[ ^L ¶J 1 − α C WQED η LRWL^HK¹iv²CMKWMKY±xQRJFI]^[\JIvµJILNM¶WL H|Y±SFDGF ¶Wv²vµJ£[\J v²JIY\[Z[\JIQEDJIL]FLEFIDG^[¤SR^LRHe[5U FMKQRSEJ²SEJ QELRSXJDG] > JIL HKJ²XR^HG^LNMTHKQEDe[\J ¶I^DÃ^¶MDGJFIOJLNMGQEJ[Z[\JIv²JLNMT¶JL H|QEDGF²JMTJIL ^Y\XE[ZJLEWvXEDGJeSJIHSEWLELEFIJIHA@J'HoM
[2T
χ2(2r),1−α/2
,2T
χ2(2r),α/2
]. > º @
3U] 3 ( $ FeV$$ L HKQECECWHKJxQRJ>[\J'HSWLELEFJ'HLEWL²¶JLRHKQEDGFJ'H
xrH|WLMSEJIHDGFI^[\Y±HG^tMKYZWL HSJ>[Z^TO ^DKY±^XR[\J>^[ZFI^MKWY\DGJ W(η, β) º LPSYZHKCWHKJHKJQE[ZJv²JILMSJeDGFIHKQE[\MG^tMÃHLRWL@^HK¹Nv²CMGWMGYZxNQEJ'HnJ É ^¶MÃHlH|QED>SJ'HlSWLELRFJIHl¶JLRHKQEDGFJIHlSJMo¹iCJ mKm > JM
SJMo¹iCJ m|m CEDKW]DGJIHGH|Y' iLEWLMKDÃ^Y\MKFeY±¶Y @ º ( " Å Q 5º L8HKQECECWHKJPxNQEJ r > 2 º ½ LEJP^Y\DGJ@SJP¶WL.[^LR¶J¸oWY\LNMKJ 1 − α > LEWL^H|¹iv²CMKWMKY±xNQEJ@nCWQED>[ZJIHlCR^DG^v MKDGJIH (η, β) JIH|MSFMGJDGv²Y\LEFIJ/CR^D>[ZJIHlYZLEFI]^[ZY\MKFIH>HKQEYZOt^LNMKJ'H
ϕ(x1, . . . , xr, F((1+
√1−α)/2)(2r−2,2)
)< β < ϕ
(x1, . . . , xr, F((1−
√1−α)/2)(2r−2,2)
),
2n∑
i=1
yβi
χ2(2r),(1−
√1−α)/2
1/β
< η <
2n∑
i=1
yβi
χ2(2r),(1+
√1−α)/2
1/β
W Fα(δ1,δ2)JIH|M/[\JwxNQR^LNMGY\[ZJµSqU WDGSDGJ α SEJµ[±^[\WYSJ Ê Y±HKuEJDKÇ LEJISJ'¶WDÈSJwSJI]DGFIHeSJµ[\YZX JID|MGFIH δ1 JM δ2 > Y º J º
P (X F(δ1,δ2) ≥ Fα(δ1,δ2)) = α @RJM ϕ(x1, . . . , xr, t)JIH|Ml[±^²H|W[\QEMKYZWLPQELEY±xQRJ/JL β SJ/[ U FIxNQR^MKYZWL
r∑i=1
xβi − nxβ
1
n(r − 1)xβ1
= t WQJILR¶WDKJ r∑
i=1
(xi
x1
)β
= n(1 + t(r − 1)).
k JM|MKJTHKW[ZQMGY\WLJ'HoM ^¶YZ[\JIvµJILNMJIH|MKYZv²^XE[ZJ 8²[ U ^Y±SJTSVU QELEJ/v²FMGuEWSJSJ ¿ JlMGWLÇ ^CRuRH|WL º L@SFISQRYÁMÈSEJT¶JDGFIHKQE[\MG^tMSJIHY\LNMKJIDKOt^[\[ZJIHÈSJ¶WL.[ ^LR¶J 1 − α CWQED¶ÃuR^¶QEL SJIHÈCR^DG^v MKDGJIH º À ^LRHÈ[Z^wCRDG^MKY±xQRJR[±^wv²Y±H|J®JLWiJQRONDGJeSJ/[Z^µv²FMGuEWSJ/DGJIH|MKJ/¶JIC JILRSE^LNM>MGD"'H>¶WEMKJQ H|J º
] − " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
% !& ½ LEJn¶F[XRDKJv²FMGuEWSJnSqU ^CECEDGW É YZvw^tMGY\WLSFIOJI[\WCEC FIJJv²CEYZDKY±xNQEJv²JILM¯C JIDKv²JMSJ¶WLRH|MKDGQEY\DGJSJIHlYZLNMKJIDKOt^[Z[ZJIH>SJT¶WL.[ ^LR¶JeCWQED β HG^¶ÃuR^LNM βML
> MKWQ¸oWQEDÃHlC WQED>[ZJeMo¹NCJ m|m @MEWLHKQECECWHKJexNQqU YZ[qJ É Y±HoMGJ hJM g SJIQ É WLR¶MGY\WLRHSJ'H>¶^DÃ^¶MKFIDKY±HoMGYZxNQEJ'H n JM r SJ/[ U FI¶Ãu ^LNMKYZ[\[ZWLMKJI[\[ZJIH>xNQEJg
β
βML
∼ χ2h.
p ^ >[ZJIHGHS>o³'´ ZN qCR^]J'H ³ ;N Çó ; ´H@ WQEDGLEY\MµSJ'HTMÃ^XEQE[±^tMGY\WLRHSEJ g JM h º h ^D®^YZ[\[ZJQEDÃH [5U WQEOiDG^]J SJ p ^ >[ZJIHGHCEDGWCWHKJQELEJOiYZHKYZWL£[±^DG]JSJIH©CEDGWXR['IvµJ'HYZLEuEFIDKJILNMGH 8T[±^®¶WLRH|MKDGQR¶MKYZWLSqU YZLNMKJDGOt^[Z[\J'HnSEJ¶WL.[ ^LR¶JCWQEDlSJIHv²WiS [ZJIHSEJnSQEDGFJnSJOiYZJJM¤¶WL HoMGYÁMGQEJQELRJDGF* FDGJL ¶JYZvµCWDKMG^LNMKJSJ©¶JSEWvw^YZLEJ º ¼ L.[ L 'SqU ^QMKDGJIHvµFMKuEWSJ'HMKDIHÈH|WQEOJILMQMKYZ[ZYZHKFJ'HÈSE^LRHÈ[ U Y\L SQRH|MKDGY\J`> CR^DJ É JIvµCR[\J²SE^L H h JDGDKWQRS Ì s WQEDKJ'^Q 7: @* WQEDKLRYZHGH|JILNMÈSJIHY\LNMGJDGO ^[\[ZJIH/SEJ£¶WL.[ ^L ¶JwXEYZ[Z^MKFIDG^Q É HKQEDT[ZJIHTSEJQ É CR^DÃ^v MKDGJIHeJL LEF'¶J'HKHKYÁMÃ^LNMT[ZJ£¶^[Z¶QE[SJ [5U Y\L WDKvw^MKYZWLSJ Ê Y±HKuEJD²JL θML
º p U QEMKYZ[\Y±HK^MKYZWL1SJP[Z^#v²FMKuRWiSEJ@SqU m C > OWYZD § º 6Eº º\; @TJIH|Mw^[\WDGH®¶WLRH|JIY\[Z[ZFJ º h DGFIHKJLNMÃ^LNMQELEJ£HKJL H|YZXEYZ[\Y\MKF FI[\JIOFIJ?8 [Z^@CEDGFIHKJL ¶J SJ£¶JLRHKQEDGJ¶JIH/v²FMKuEWSJ'HTDGJIH|MKJLNMT]FLEFIDG^[\JIvµJILNMTDKJ'HoMGDKJIY\LNMKJ'HB8SJIHFI¶ÃuR^LNMKYZ[\[ZWLRHlSJ/]DG^LRSJÈMG^Y\[Z[\J ºG"4,5"4G E> C8&L < 8&<HK/l DE D E @ @/BL< DEz Dz|BA 8L&A&< z6A<(DEzDL&8&<3 ] 3 , 9!#+ - (Wp JIH©CEDGW¶F'SQEDGJIHSEJ $ $ !" iY\LNMGDKWSQEY\MKJ'H©CR^D ¼ DGWL >|³I´ Z @JM>SJCEQRYZHl¶WQEDÃ^v²v²JLNMnQMGY\[ZY±H|FIJIH9> r YZLE][ZJ¹Ì Ê YZJ[±S ³I´´ @*NCJDGv²JMKMKJLNMSJCR^[Z[\YZJD>[5U ^XRHKJLR¶JeWQSJ/CEDGWCWHKJDlQELEJT^[ÁMGJDGLR^tMGY\OJB8®[±^w¶WLRH|MKDGQR¶MGY\WL£MGuEFWÇ
DKY±xNQEJ²SVU YZLNMKJIDKOt^[Z[ZJIHSJw¶WL [ ^LR¶J º k Wv²v²JµY\LRSEYZxNQEFµCEDGFI¶F'SJv²v²JLNM V[Z^Ot^[ZYZSY\MKF²SJw¶J'HeSJDGLEYZJDÃHCJQM MKDGJDKJIv²YZHKJeJL xNQEJIH|MKYZWL R^QHKJLRH>W@[\J'HSWLRLEFJ'H>SJ >¼¾ QMKYZ[\Y±HKFJIHDKJ'H|CJI¶MKJLNMCJQ[ZJIH¶WLRSEYÁMGY\WLRHSVU Y\L SFCJLÇSE^L ¶J SJSY±HoMGDKYZXEQMGY\OiY\MKF®JMÈSJTMo¹NCJSJ®¶JL H|QEDGJTWXE[\YZ]^MKWY\DGJv²JLNMHKQECECWHKFJIH^Q § Eº 6Rº ³ º k WLNMKDÃ^YZDKJIv²JLNM 8¶JM|MKJe¶WLRH|MKDGQR¶MGY\WLwMKuEFIWDGYZxNQEJN¶JIHnCEDGWi¶FISQRDKJ'HnHKWLNMn[ZYZFJIH 8®QEL£J'HoMGY\vw^tMGJQED θ = θ(Yn) SWLMlJ[Z[ZJIHnJIH|MKYZvµJILNM[Z^µOt^DGY±^LR¶J σ2 º½ LEJ WY±H [5U WXRH|JIDKOF θ(yn) WXEMKJLiQCR^D vw^ É YZvµY±HG^tMKYZWL SEJ [Z^ONDÃ^Y±HKJvXR[Z^LR¶J n[ZJHG¶ÃuEFvw^ ]FILEFDÃ^[ÈSQX WiWMGH|MKDÃ^C¶WLRH|Y±H|MKJG8wHKY\v®QE[ZJDQRL]DÃ^LRSPLEWvXEDGJ M SVU FI¶ÃuR^LMGY\[Z[ZWLRHSEJ¶^DG^¶MKFIDKY±H|MKY±xQRJIH>HKJvXR[Z^XE[\J'H 8
yn
xNQEY) WQEDGLEY±HKHKJLNM 8µ[\JIQED>MGWQED M JIH|MKYZv²^MKYZWLRH θiSQ@vw^ É YZvQEv SJ/OiDG^YZHKJv®XE[Z^LR¶J º p U JIH|MKYZv²^MKJIQED>Jv²CEYZDKY±xNQEJ
σ2M =
1
M − 1
M∑
i=1
(θi −
1
M
M∑
i=1
θi
)2
W DKJ ^Y\L H|YQRLEJ JIH|MKYZvw^tMGY\WLSJ [±^@O ^DKY±^L ¶J SJ [5U JIH|MKYZv²^MKJIQED θ(Yn) º p JIH>¸oQ HoMGY [ ¶I^tMGY\WLRHTSJ£¶J£HG¶ÃuEFvw^ > JILCR^DKMKY±¶QR[\YZJD/[±^@¶WL H|Y±HoMÃ^LR¶Jw^HK¹Nv²CMGWMGYZxNQEJc@eH|WLM/CEDGFI¶YZHKFJ'HeCR^D/J É Jv²CE[ZJ²SR^LRHB=^L_SJD Å ^^D|M >|³I´´ Z ~CEC º6· Ç 6<6 ³c@ º À ^LRHe[\J'HeCEDGWi¶Ãu ^YZLRHCR^DÃ^]DG^CEuEJIH LEWQRHDGFIHKQEv²WLRHeJM WDGvQE[ZWLRHexNQEJI[ZxNQEJ'HeCEDKWC WNH|Y\MKYZWL HHKQED/[±^DKJ'¶WLRHoMGYÁMGQMKYZWL@SJ/¶J'HlFI¶ÃuR^LNMKYZ[\[ZWLRH º h ^D>^YZ[Z[\JIQEDGH QELEJ/v²FMKuEWSJT^[ÁMGJDGLR^tMGY\OJ¶WLRHKYZH|MKJB8µJIH|MKYZv²JD>[±^µv²^MKDGYZ¶JSJ>¶WtOt^DGY±^LR¶JSJl[ U JIH|MKYZvw^tMGJQED θ(Yn) JILYZLiOJDÃHG^LNM¯QELRJ©J'HoMGY\vw^MKYZWL²SJl[Z^Èvw^tMKDGY±¶J>SVU Y\L WDKvw^MKYZWL®SEJ Ê Y±H|uRJD ºk J[Z[\JÇc¶YVJIH|M>CEDKF'H|JILNMKFJ/^Q § Eº 6Rº Eº ;Eº½ LEJ WY±HCEDGWSQEY\MÈ[5U J'HoMGY\vw^tMGJQED σM
RCR[\QRHKYZJQEDÃHv²FMGuEWSJIHC JIDKv²JM|MKJILNMSJµ¶WL HoMGDKQEYZDGJTSJIHY\LNMKJIDKOt^[\[ZJIHSEJ¶WL.[ ^L ¶JCWQRD θ > JMlCWQRDMGWQMGJ WLR¶MGY\WLSJ θ @ º ¿ WQRHnDGFIHKQEv²WLRHnYZ¶YVQEL£FIOJILMÃ^YZ[SEJÈv²FMGuEWSJIH©CEDGFIHKJLNMGFJIHJMSY±HG¶QMGFJIH>SR^LRH ¼ DGWL Ì r Y\XRHKuEYZDG^LEY >o³'´´ Z iCR^]JIH³ ;<Z Çó · @ º
A /)*G3- 24 //65WO56./.) 0* +6D.45\[*+)*24 (4-/+NW&)2& /.W**9O356?4 #-*.4-*56? /.U*E0*4SI 8 I 87 /.2 KX &/.5W/4-*
θ(Yn)?4/.)7/F[*4)6&)7/&)0*-1)0*?4P4)*0*)E/M024 //56? /U*3078E)*-C*P)*M0!X -/.4K&K4/5W
/)*[(E/./. *30*- /.4@#SX 4)*.45\[*0*456? /U*30&0*.()*[&>#) 0\)*#.-* 4/56424 //.,)*244/ /.-0* :I )E/*-.4-*4/5W/)*983) 2U. 24&GE-*:5WO356.+,D3.45 [*+)2&[.4D7?4,0*SX ?42U#)7/.)1[(3//.
'O% % O?M ? 6-( -O, ! (-O ]
*"() ( " Y σMJ'HoM©CEDKWSQRYÁMlCR^Dn[ U QMGY\[ZYZHG^tMGY\WLSVU QEL]DÃ^LRS²LEWvXEDGJSVU FI¶Ãu ^LNMKYZ[\[ZWL HXWiWMGH|Ç
MGDG^CEC F'HWL QEMKYZ[\Y±H|JJL£]FLRFDÃ^[R[5U ^CECEDGW É Y\vw^MKYZWL (θ − θ)/σM ∼ N (0, Ip)W Ip JIH|Mn[5U Y±SJLNMGYÁMGFSE^LRH IRp º ½ LYZLNMKJDGOt^[Z[\JeSJT¶WL.[ ^LR¶JeCWQED θ EXEYZ[Z^MKFDÃ^[VJMSJTH|JIQEYZ[ α EJ'HoM^[\WDGH
[θ(yn) − uα/2σM , θ(yn) + uα/2σM
] > º 6 @W uα
JIH|M[\JTxNQR^LMGY\[ZJ/SVU WDÃSDKJ α SJ/[±^µ[\WY N (0, 1) RHKWY\M P (X ∼ N (0, 1) ≤ uα/2) = α/2 º *( ( p U ^CECEDGW É Y\vw^MKYZWLX> º 6 @/LqU J'HoM²¶JIC JILRSE^LNMOt^[\Y±SJxNQEJ£[\WDGHGxNQEJ n JIH|M]DÃ^L S> Mo¹iCEYZxNQEJÇv²JLNM n ≥ 30 @ º p WDÃHKxNQEJ n J'HoMCJMKY\M È[ U ^CRCEDKW É YZvw^tMKYZWL SJ MKQRSEJLNM WQEDGLEY\M SJ_v²JYZ[\[ZJQEDÃH@DGFIHKQE[ÁMÃ^tMÃH (θ − θ)/σM ∼ Stn−1
W Stn−1DKJICEDKF'H|JILNMKJ£[±^ [ZWY>SJ MGQRSJLNMw^ OJ'¶ n − 1 SJ]DKF'H®SEJ[ZY\XJDKMKF¤F[±^DG]YZJ 8
[±^²SY\v²JILRH|YZWL p º ¼ L@LEWMÃ^LNM t(α)n−1
[ZJTxNQR^LNMKYZ[ZJeSqU WDGSDGJ α SJ Stn−1WLPWXMKYZJLNM>QEL@YZLMGJDGOt^[Z[\J/SJTHKJQEYZ[ α
[θ(yn) − t
(α/2)n−1 σM , θ(yn) − t
(1−α/2)n−1 σM
]. > º : @
$ & !" ( [RL SVU FOiY\MKJID MKWQMKJ ^CECEDGW É Y\vw^tMGY\WL%HKQED [ZJ¶Wv²C WD|MGJv²JLNM SJ [±^ Ot^DGYZ^XE[ZJ ^[ZFI^tMGWYZDKJ(θ − θ)/σM
QELEJTv²FMKuEWSJHKYZvµCR[\JTJ'HoMSJT]FILEFDGJD>CWQRD¶ÃuR^xNQEJ/F'¶ÃuR^LNMGY\[Z[\WL@X WiWMÃHoMGDG^CECFSVU JIH|MKYZv²^MKYZWL θi[±^Ot^[ZJQRD zi = (θi−θ)/σi W σi JIH|M¤[ U FI¶I^DKM|Ç5Mo¹NCJXWiWMÃHoMGDG^CSJ θi> ¶JMKMKJ©v²FMKuRWiSEJ©LRFI¶J'HKHKY\MKJnSEWLR¶nQEL®SEWQEXE[ZJ
XWiWMGH|MKDÃ^CW@ º LDGJv²CE[±^¶J^[ZWDÃH t(α)n−1
SE^L HG> Eº : @lCR^D[\J®xQ ^LNMKYZ[\JJv²CEYZDKY±xNQEJSVU WDÃSDGJ α SJ[Z^ SEYZH|MKDGY\XRQMKYZWLSJ'H (z1, . . . , zM ) º k JMKMKJ©vµFMKuEWSJ>^ȶJCJL SE^LNM¤[\JlSF* ^QMSJ WQEDGLEYZDSJ'HDKF'H|QE[\MG^MGH¯CR^D" WY±H¯JDGDG^MKY±xQRJIHqJM¤MKDIHYZL RQEJILR¶F'H>CR^DSJIH]DGWQECJIH>SEJSWLRLEFJ'H^tMo¹iCEY±xNQEJIH E¶J®xQRY¯¶WLRH|MKY\MKQEJTQRLuR^LRSEYZ¶I^CH|FIDKYZJQ É ^QPMGDG^YÁMGJv²JLNMSJ/SWLELEFIJIHlYZLRSQRH|MKDGY\JI[\[ZJIHJL ^YZXE[\J/LRWvXRDKJ º "# %' *** & &( $ & !" ¼ L.[RL R[\J'H DG^¶MGY\[ZJIH©WQxNQR^LMGY\[ZJIHlJIv²CEY\DGY±xQRJIH>SJe[±^µSEYZH|ÇMGDKYZXEQMGY\WL SJ'H (θ1, . . . , θM ) C JIDKv²JMKMKJILMSEJ WQEDGLEY\D/SJ'HeÄWLEJIHeSJw¶WL.[ ^LR¶J²¶WvµC ^DÃ^XE[ZJIH^Q É YZLNMKJIDKOt^[Z[ZJIHCEDGFI¶FISJIv²vµJILNMGHSF"[RLEY±H º À U ^CED"'H[ZJIHMKJIH|MGHÈLiQEv²FDGYZxNQEJ'HSVU ¼ DGWL Ì r YZXRH|uRY\DÃ^LEY >o³'´´ Z R¶ÃuR^CEYÁMGDKJw³ 6 @*R[ZJDGJǶWQEOiDGJv²JLNM>SJ/[ U YZLMGJDGOt^[Z[\JeJIH|M>]FILEFDÃ^[ZJv²JLNMlX WLO8µ¶WLRSEYÁMGY\WL@xNQEJe[ZJ/v²WiS [ZJC ^DÃ^v²FMGDKY±xNQEJHKWY\M¶WDKDGJI¶M º JIvw^DÃxQRWLRHxNQEJSE^L H[±^H|MKDGYZ¶MKJÈ^¶¶JCMÃ^tMKYZWL SEQwMKJIDKv²J YZLMGJDGOt^[Z[\JSEJȶWL.[ ^LR¶J N¶JMKMKJSF [RLRYÁMGY\WL£H|JIvXE[ZJ[±^ CE[ZQRHÈOt^[ZYZSJ H|QED M JIH|MKYZvw^tMKYZWL HCWHGH|YZXE[ZJIH 100(1 − α)% HKWLNMeH|Y\MKQEFIJIHÈJLNMKDGJ[ZJIHxNQR^LNMGY\[ZJIHÈJv²CEYZDKY±xNQEJIHSJ/HKJQEYZ[ α/2 JM 1 − α/2 ºk JCJL SE^LNM ©YZ[J'HoMMKDIH¶[±^YZDxNQEJ#¶JMKMKJ SF [RLRYÁMGY\WL+SWY\M MKDGJv²WSQE[ZFJ MGWQMSVU ^XWDÃS [±^v²FMGuEWSJSVU JIH|MKYZvw^tMGY\WLCEDGWSQEY\MQEL J'HoMGY\v²FeCR^D
bM (θ) =1
M
M∑
i=1
θi − θ(yn) > º\; @xNQEYY\v²CE[ZY±xQRJÈxNQEJȶJIH DG^¶MGY\[ZJIHnJv²CEYZDKY±xNQEJIHnSWYZOJILNM MGDKJSFXRYZ^YZHKFIHONY±H|ÇM8Ç5OiY±HSJ[ U JIH|MKYZvw^tMKYZWL£Y\LEY\MKY±^[ZJ θ(yn) JILw[\JIQED MG^LNM[±^TxNQR^LMGYÁMGF bM (θ) º k JÈXEY±^Y±HHKJDÃ^e]DÃ^CEuEY±xNQEJv²JILMv²Y±HJILwFOiY±SJLR¶JSE^LRH[ZJIHJ É Jv²CE[ZJIHn¶[ MKQEÇDÃ^LNM¶J¶Ãu ^CEY\MKDGJ º ¼ L.[RL ~¶JM|MKJvµFMKuEWSJ®^µMKJL SE^LR¶J 8 HKWQRH|Ç5J'HoMGY\v²JID>[ U Y\L ¶JDKMKY\MKQ SJTDGFJ[Z[ZJTH|QED[\J'HxNQR^LNMGYÁMGFIHJ'HoMGY\v²FJ'HCEQEY±HGxQU JI[\[ZJ®LRJ®DGJv²JMeCR^HÈJIL#¶I^QRHKJ[ZJ®v²WSI[\J º k JM|MGJ®v²FMGuEWSJ²JIH|MeSEWLR¶J É MKDv²Jv²JILM/SFIC JILÇSE^LNMKJ/SJ/[±^¸oQ HoMGJIHGH|J/SQ@v²WS[ZJ/CR^DÃ^v²FMKDGYZxNQEJe^YZLRHKYqxQRJeSEQ )%'( q $# &( % %'(( ∗ SQ@XWiWMGH|MKDÃ^C º
3U] 3 3 QES eE(*(/ "!#-.QÀ JQ É Mo¹iCJIH²SJ@XWNWMGH|MKDÃ^CCJQROJLNMMGDKJ@¶WLRHKYZSFIDKF'H º p JPXWNWMGH|MKDÃ^C (W Q- !c-P&)+"!#12 H|YZvQR[\J@SJ
LEWQEOJ'^Q É FI¶Ãu ^LNMKYZ[\[ZWL HJL£CEY\W¶ÃuR^LNM©^[ZFI^MKWY\DGJv²JLNMnSEJIH©SWLELRFJIHnSE^L H yn^ OJI¶DKJIvµY±HKJ º p ^>¸oQ HoMGY [ ¶I^tMGY\WL£SJ
− " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
¶JM|MKJMKJ'¶ÃuELEY±xQRJDGJCWHKJÈHKQED[±^¶WLiOJIDK]JLR¶J^HK¹Nv²CWMGYZxNQEJSEJÈ[5U J'HoMGY\vw^MKJQRDnJIvµCRY\DGYZxNQEJSJ[±^ WLR¶MGY\WL£SJÈDKFÇCR^DKMKY\MKYZWL º p U Y\LNMKFID"MSJ¶JM|MKJ>CRDKW¶F'SQEDGJnJ'HoMSqU FIONY\MKJID[Z^eHKC F'¶Y5[ ¶^MKYZWLµFOJLNMKQEJI[\[ZJ>SVU QEL²CEDKW¶JIHGH|QRH¤SJ¶JLRHKQEDGJ]FILEFDÃ^[ZJv²JLNMY\LR¶WLELiQ º k U JIH|M¤^QRHGH|YiQEL²SF* ^QMSE^L H[±^vµJ'H|QRDKJnW®¶JlCEDKW¶JIHGH|QRHSJl¶JILRH|QRDKJ SR^LRH[ZJIHFMGQRSJIHY\L SQRH|MKDGY\JI[\[ZJIH JIH|M>[\J DGQEY\MSVU QRLEJ/HoMGDG^MKF]Y\JeDGFRF'¶ÃuEYZJxNQEYqLU J'HoM>CR^H WDöFv²JILM>YZLRSFIC JILRSE^LNMGJTSQP¶Wv²CWHG^LNM∑ SWLNM/WL#v²JIHKQEDKJ²[±^@SQEDGFJ²SJwOiY\J º J[ZWL s JJ JID Ì ¼ HK¶WXR^D'>o³I´´ Z C ^]J'H · Ç <Z @ ¶JMKMKJµMKJ'¶ÃuELEY±xQRJLqU JIH|MOt^[ZYZSJ®xQRJTC WQEDÈSEJIHF'¶ÃuR^LNMGY\[Z[\WLRHlLEWL ¶JL H|QEDGFIHSJTMÃ^YZ[\[ZJTJLv²Wt¹JLELEJTHKQECFDGY\JIQEDGJ 8³ > ^Q HKJLRHWJ[Z[\JlSWLELEJ>SJIHDGFIHKQE[\MG^tMÃHCEDKW¶ÃuEJ'H¯SEJ©[5U ^CECEDGW¶ÃuEJC ^DÃ^v²FMGDKY±xNQEJnSE^LRH¤SJIH¤¶WLRSY\MKYZWL HSEJlHKYZvQE[±^tMGY\WLY±SFI^[\J'H @ ºh WD|MGJD * % >o³'´´ @¤WLNMlDKJ'¶JILRH|FÈ[\J'H©CEDKYZLR¶Y\CR^[\J'HCRDKWCEDKYZFMGFIHnJMSF ^QMÃH©SEJIH©Y\LNMGJDGO ^[\[ZJIH©SJe¶WL.[ ^L ¶JÈSEJMo¹iC J ( ¶WLRH|MKDGQEYÁMÃHlSJ ^ËWL£LEWL£CR^DG^v²FMKDGY±xQRJYZ[±H>H|WLM WDKMKJIv²JLNMH|JILRH|YZXE[ZJIH 8®[Z^MG^Y\[Z[\JeJM[\J'HFIOJLNMGQEJ[Z[\J'HSY±HGH|¹iv²FMGDKYZJIHnSJ[±^TSY±HoMGDKYZXEQMGY\WL Jv²CEYZDGYZxNQEJSEJIHSEWLELEFIJIH º p JY Ì v²YÁMGu > <6 @CR[\QRHnDGFI¶JIvµÇv²JLNM WLNM¤CEDGFIHKJLNMKF©QELEJ©FMGQRSJlJv²CEYZDKY±xNQEJlSJIHCEDGY\L ¶YZCR^[ZJIH¯CEDGWi¶FISQRDKJ©SJlX WiWMÃHoMGDG^CTLEWLCR^DG^vµFMKDGYZxNQEJ©^YZLRHKY
xNQqU QRLEJlDKJIONQRJ©SJ'H¤CEDGY\LR¶Y\C ^[ZJIHCEQEXE[ZY±¶^tMGY\WLRHHKQED¤[\JHKQt¸oJM º ¼ L²CEDGWCWHG^LNM¯CE[ZQRHKY\JIQEDÃH¤vµFMKuEWSJ'HSVU ^v²F[ZY\WDG^MKYZWLSJT¶J'HCEDGW¶F'SQEDGJIH Y\[±H>MGFv²WYZ]LEJILNM¶JCJL SE^LNMSJT[ZJQRD ^YZXE[ZJTDKJILNMG^XRY\[ZYÁMGFTH|QRDSJTCJMGYÁMÃHFI¶Ãu ^LNMKYZ[\[ZWL HECWQRD[\J'HKxNQEJI[ZH[\J'H>Y\LNMGJDGO ^[\[ZJIHSJ¶WL.[ ^LR¶JeCEDGWSQEY\MGHDGJIH|MKJILNM DG^]YZ[\J'HlJMCR^D WYZH WD|MGJv²JLNMJIDKDGWLRFIHlOiYZH|Ç 8tÇcONY±HlSJIHY\LNMGJDGO ^[\[ZJIHnMGuEFWDKY±xNQEJIH>SY±HKC WLEY\XR[\J'H º
³ º ) %K # &( % %'( ! &(( *% #$( "!"$ y0n = (x0
r , c0n−r)
$# n = 9 r = 6 x0
r = (63, 109, 140, 21, 58, 67) ∼ E(ηo = 100) * cn−r = (50, 100, 150) %' %'$ #$*($ " * *%(#$ η = 126.33 $ & !" (( &" * ! % * 8 %| $# &( % %( y0
n#&( 8 %' M ( $ $# &( % %'( yi
n = (xir, c
in−r)i∈1,...,M
xi1, . . . , x
ir ⊂
63, 109, 140, 21, 58, 67 $# * &( ci1, . . . , cin−r !" % *() *( 50, 100, 150 $# * &( & %'&$# ( *" #$(* ci
n−r = c0n−r ∀i ∈ 1, . . . ,M
FI¶Jv²v²JLNM tCE[ZQRHKY\JIQEDÃHqMGDG^ Ot^Q É HKJlH|WLM¤^tMKMG^¶ÃuRFIH8ÈSFIOJI[\WCEC JIDSEJIH¤^CECRDKW¶ÃuEJ'HLEWLTCR^DG^vµFMKDGYZxNQEJ'HCWQRDSJIHlCRDKWXE['Iv²JIH>SVU JIH|MKYZvw^tMKYZWL@SJ/CJMGYÁMÃH>FI¶ÃuR^LNMKYZ[\[ZWLRH º YZLRHKY h WD|MGJD &% >|³I´´ @p JX WiWMÃHoMGDG^C Q- !c-P&)+"!#12 > Ê Y\] º Eº @NCEDGF* FDÃ^XR[\JSE^LRHn[ZJȶ^SDGJSJC JMKY\MGH©FI¶ÃuR^LNMKYZ[\[ZWLRH DKJ'¶DGFJY\LNMGF]DÃ^[ZJv²JILM©SJIH©FI¶Ãu ^LNMKYZ[\[ZWL HnHKJ[ZWL [ZJvµWSI[\J M(θ) º p ^SEY 4 ¶QR[ÁMGFJ'HoMl^[ZWDÃHnSEJDGJCEDGWSQEYZDKJÈ[\JCEDGWi¶JIHGH|Q HSJ¶JILRH|QRDKJ@H|WQRH|Ç\¸|^¶JLNM º p JIHw¶I^HwSJ¶JILRHKQEDKJ@^[ZFI^MKWY\DGJ > Mo¹iC J m|m @µJM£SJ@Ot^[ZJQEDwQELRYZxNQEJ`[ É J > Mo¹iCJ m @LRJ
CEDGFIHKJLNMKJILMC ^HSJÈSY 4 ¶QE[ÁMGFIH vw^Y±HSEJIHSWLELEFIJIHSEJ¶JILRH|QRDKJ>F'¶ÃuEJ[ZWLELRFJIHSR^LRHyn
> Mo¹iC J m CRDKW]DGJIHGH|Y' @¤HKWLNMCEDGWXE[ZFvw^tMGYZxNQEJ'H ºr DÃ^SY\MKYZWLRLEJ[Z[\JIv²JLNM qWL_H|J ¶WLNMKJILMGJwSVU YZLR¶[\QEDGJ YZLNMKF]DG^[\JIv²JLNM[ZJIH¶JL H|QEDGJIHSE^LRHe[5U FI¶ÃuR^LMGY\[Z[ZWL X WiWMGH|Ç
MKDÃ^CECFNWQSVU JIL£YZLR¶[ZQEDGJ[±^Tv²Wt¹JILELEJ[±^Tv²FISEYZ^LEJWQ [\JÈvw^ É Y\v®QEv > k ^HoMÃ^LEYZJDȳI´´ @ º [ZJIHFI¶ÃuR^LNMKYZ[\[ZWLRHn^YZLRHKYX WiWMGH|MKDÃ^CECFIH©LEJe¶WDKDGJIHKC WLRSJILM©CR^Hl^Q É DKF'^[ZYZHG^tMGY\WLRH©SVU QRLEJeO ^DKY±^XR[\Je^[ZFI^MKWYZDGJv²YZHKJJL@¶Wv²CFMGYÁMGY\WLP^ OJI¶QEL£CRuEFLEWv LEJSJ¶JLRHKQEDGJ º ½ LEJv²FMGuEWSJMGJLNMG^LM>SJDKJICEDGWiSEQEY\DGJ[ZJIH©¶WLRSY\MKYZWLRH©SVU WXMGJLNMKYZWL£SEJIHlSWLRLEFJ'HJIH|MSWLR¶eCRDKWC WNH|FIJxNQEYHKJDÃ^®¶Wv²CR^DGFJCR^D>[Z^²HKQEYÁMGJ 8²¶JIHlv²FMKuEWSJ'HlMKDÃ^SY\MKYZWLRLEJ[Z[\J'H º
'O% % O?M ? 6-( -O, ! (-O
A
```
!""#!$%
θM
```θ1
σ2M = 1
M−1
MP
i=1
“
θi −¯θM
”2
GU.%' ¯θM = 1
M
MP
j=1
θj
`
""#yn ∼ M(θ)
(y(1)1 , . . . , y
(1)n ) ∼ M(θ)
(y(M)1 , . . . , y
(M)n ) ∼ M(θ)
θ(yn)
Eº ¶ÃuEFIv²^²SQ@XWiWMGH|MKDÃ^CPCR^DG^vµFMKDGYZxNQEJCWQED>[5U J'HoMGY\vw^MKYZWL@SJB=^D (θ(yn)) º
Eº > HKQEY\MKJ SJ [ U J É º ³@ $ $ !" &" * ! )% 8 %| $# ( % %( y0n
#$( 8 %' M (& $& $# ( % %'&( yi
n = (xir, c
in−r)i∈1,...,M
xir ∼ E(η = 126.33) ci
n−r)"
*( q ( $# #& &( (& ( #( " %| $# ( % %( &( # " cin−r = (xi
r, . . . , xir) n − r % ( " %* % %'&(#$ È&( # #$ & #%
"!&$ &( !" #&(() $(#$ %K ( ( * %'$ #$*(" &( )" ! * %cij =
∑n−rk=1 c
0k/(n − r) ∀j ∈ 1, . . . , n − r ∀i ∈ 1, . . . ,M #$( "$ " #$( &( &%' 8 %'
(()#$*(" ( %
3U] 3U] ^` !#HQ!#(*N< (WXXQ!#(*N< ?ONH!# TeE(*(/ "!#-.QXQV-.!#- &)+"!#12 L1H|QECRC WNH|JxNQEJ@CE[ZQRH|YZJQRDGH²Ot^[ZJQRDGH²SJ ¶JLRHKQEDGJIH²SY FIDKJILMGJIHw^Q%HKJYZL%SqU QEL1F'¶ÃuR^LNMKYZ[Z[\WL7CJQEOJLNMMKDGJ
¶WLRHKYZSEFDGFJIHȶWvµv²J²SJIHÈDGFI^[ZY±HK^MKYZWLRHÈSVU QRLEJ²O ^DKY±^XR[\J^[\F'^tMGWYZDKJ C ~SEWLNMWL Ot^£J'HoMGY\v²JIDJIv²CEY\DGY±xQRJv²JLNMÈ[±^DGFCR^D|MGYÁMGY\WL ¶WLRSY\MKYZWLELEJI[\[ZJv²JILM 8T[±^/MÃ^YZ[\[ZJÈSJ'HnSEWLELEFIJIH©SY±H|CWLEYZXE[ZJIH º h ^DnQEL£CEDKW¶FISFSVU YZLiOJIDGHKY\WLwSJ¶JMKMKJDGFCR^D|MGYÁMGY\WL WL7HKY\v®QE[ZJDÃ^ QEL8FI¶ÃuR^LMGY\[Z[ZWL7SJ@¶JILRHKQEDKJ@SJMG^Y\[Z[\J n xNQEJ[ U WL8vµJM|MGDG^#JL #&(#"(#$ ^ OJI¶QEL_F'¶ÃuR^LNMGY\[Z[\WL_SJ MÃ^YZ[\[ZJ n H|YZvQR[\F£SqU ^CED"'H θ º p J'H¶WLRSY\MKYZWLRH®SJ [5U J É C FIDKYZJLR¶J£^XWQEMKY±HKHG^LNM^Q É SWLELEFIJIHxn
HKJDGWLNM²^[\WDGHwHKQECECWHKFJ'H²DKJ'¶DGFFIJIH º ¿ WMGWLRHwxNQqU QELEJ SFIvw^DöÃuEJH|YZv²Y\[±^YZDKJSJHKY\v®QE[±^tMKYZWLSJ@[±^_¶JILRH|QRDKJHKJ[ZWL²QELEJ[\WY H|MG^tMGYZH|MKY±xNQEJHKWQRH|ÇZ¸|^¶JILNMKJCEDKF'H|JILNMKFJSR^LRH À WNHKH Ì k uEY±^LR]S>o³I´´ : @CJDGv²JMn[±^/¶WLRHKYZH|MG^LR¶JSJ'HJ'HoMGY\vw^tMGJQEDÃH©X WiWMGH|MKDÃ^C º & '& %( * (& % ( * #$( "* LLEWMGJ c∗1 < . . . < c∗q
> q ≤n− r @[Z^®H|QRYÁMGJWDÃSWLELEFJCR^D©WDÃSDGJ¶DGWY±HKHG^LNMSJ'H q O ^[\JIQEDÃHSY FIDKJILNMKJIH©SJ¶JL H|QEDGJ º L LEWMKJ f1, . . . , fq
[\J'H DGFIxNQEJILR¶J'HDGJ[±^tMGY\OJIHSEJ©¶JIH¯O ^[\JIQEDÃH º ¼ [Z[\J'H¯OFDGY [ JLNM ∑q
j=1 fj = (n−r)/n. LTCWHKJF]N^[ZJv²JLNM cminJM cmaxSJIQ É Ot^[\JIQEDGHSJT¶JILRH|QRDKJ'HSF [ LEY\J'H>CR^D cmin ≤ c∗1 < . . . < c∗q ≤ cmax
º p U J'HoMGY\vw^tMGJQEDÈSJ ^CR[Z^LÇ s JYZJD>SJ[±^ WLR¶MKYZWL@SJ/DKFICR^DKMKY\MKYZWL FCSJe[Z^²SY±HoMGDKYZXEQMGY\WLSJ C HKJ/SF [ LEYÁMC ^D
P (C < cmin) = F0 = 0,P (C < c∗1) = F1 = f1,P (C < c∗2) = F2 = f1 + f2,ºººP (C < c∗q) = Fq = r/nP (C < cmax) = Fq+1 = 1.
Y©[ZJIH®CEDKWCEDGY\FMKFIH®SJ@¶WL H|Y±HoMÃ^LR¶J£SJ[ U ¼ /s CWQRDµSJ]DÃ^LRSRHTFI¶Ãu ^LNMKYZ[\[ZWL HHKWLNM®XEYZJL%¶WLELiQEJIH> QELEJPOiQEJ]FLEFIDG^[\J@JL1J'HoM£CRDKF'H|JILMGFJCR^D h JMGJDÃH|WL0³'´ @YZ[J'HoM LRFI¶J'HKHG^YZDGJ@SEJH|QRCEC WNH|JID nSE^LRH£¶J¶I^HwSJ CJMGYÁMGJ
23 − " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Censures
RANDOM 1
RANDOM 2 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
− − − − − − − − − − − − − − − − −
C1 Cq
Fréquences Cumulées Relatives
− − − − − − − − − − − − − − − − − − − − − − − − − − RANDOM 3
| | | | | | | | | | −
| | | | |
| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | |
º 6 Ê WLR¶MGY\WLSJeDGFCR^D|MGYÁMGY\WL WCEMKYZvµY±H|MKJ/SJ[Z^¶JILRH|QRDKJ º
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Censures
RANDOM 1
RANDOM 2 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
− − − − − − − − − − −
C1 Cq
Fréquences Cumulées Relatives
− − − − − − − − − − − − − − − − − − − − − − − − − − RANDOM 3
| | | | | | | | | | −
| | | | |
| | | | | | | | | | |
º : Ê WLR¶MKYZWL²SJ>DKFICR^DKMKY\MKYZWL®C J'HKHKYZvµY±H|MKJSJ>[±^¶JLRHKQEDGJ º
MG^Y\[Z[\JxQRJP[ZJIHµOt^[ZJQRDGH²SJ¶JILRHKQEDKJPWX H|JIDKOFJIH®H|WLNMµOFDGYÁMÃ^XE[ZJv²JLNMwSJIHµDGFI^[\Y±HK^MKYZWLRH SVU QELEJ@Ot^DGY±^XE[ZJ^[ZFI^MKWYZDGJ>C WQEDnWXMKJILEYZDnQELXEY±^Y±H ^YZXE[ZJÈJMn[±^¶WLRHKYZH|MG^LR¶JÈSJIH©H|YZvQR[Z^MKYZWLRHnCR^DnY\LiOJDÃH|YZWL SJ FC> OWYZD k uEJIL
* % ³'´ Z @ º p JwCEDGW¶J'HKHKQRHeSJ ¶JILRHKQEDKJµLqU ^@CR^H/SEJµDÃ^Y±HKWLC ^DKMKY±¶QE[ZY'IDKJwSVU MGDKJw^[\F'^tMKWY\DGJ~vw^Y±HeWL [ZJwDKJI[\YZJ8wQEL#CEDKW¶JIHGH|QRH^[ZFI^tMGWYZDKJ/YZLR¶WLELiQHKQRHG¶JCEMKYZXE[\JµSJ WQEDGLEY\DÈ[\J'HDGFI^[\Y±HG^tMKYZWL H (c1, . . . , cn−r)
º ½ LFI¶ÃuR^LMGY\[Z[ZWLcn = (c1, . . . , cn) SJ/¶JILRH|QRDKJ'HlC JIQM^[ZWDÃH MKDGJ/DGJI¶WLRH|MKDGQEYÁM>SJTSJIQ É vw^LRY'IDKJ'H>SY FDGJLNMGJIH º #$*(") L¶ÃuEWY±HKYÁM cmin = c1
JM cmax > cq> ¶I^HJ É MKDv²J cmax → ∞ @ º p ^ WLR¶MKYZWLSJ
DKFICR^DKMKY\MKYZWLJIv²CEY\DGY±xQRJCEDGJLRS[Z^ WDGv²J®SJµ[Z^S[R]QRDKJ º 6Eº k J²¶ÃuEWY É JIH|M¶WLRHKY±SFDGF®¶Wv²v²JWCEMKYZvµY±H|MKJ²¶^DÈY\[[Z^YZHGHKJ©[±^/CWHGH|YZXEYZ[\Y\MKF 8/SEJ]DÃ^LRSEJIH¤Ot^[ZJQEDÃHSJ>[5U FI¶ÃuR^LMGY\[Z[ZWL²X WiWMGH|MKDÃ^CwSVU MKDGJ>LEWL ¶JL H|QEDGFJ'H º k JM|MGJvw^LEYDGJSVU Y\LiOJIDGHKJDe[±^PDKFICR^DKMKY\MKYZWL SJIHT¶JLRHKQEDGJIHe^PMKJILRSE^LR¶J'8@H|WQRH|Ç5FIO ^[\QRJDe[ U YZLR¶JID|MGYÁMGQRSJ HKQED/[Z^POt^[ZJQRD9[RLR^[ZJwSJIHCR^DÃ^v MKDGJIH º ½ Lµ¶ÃuEWY É SEJ cmax
DÃ^Y±H|WLELR^XE[\JIv²JLNM¯]DG^LRS/JIH|MCR^D¤J É JIv²CE[\J cmax = 2cqxQRYEH|Q.4²M 8ÈCEDKWSQRY\DGJ
DG^CEY±SJv²JLNMTSJ'H/FI¶ÃuR^LNMKYZ[\[ZWLRHeXWiWMÃHoMGDG^C SWLNMT[\J²MÃ^Q É SJ ¶JILRH|QRDKJwSY IDKJ SEJwvµWY\L HTSJ³ SQ MG^Q É SEJ¶JILRH|QRDKJ/SJe[5U F'¶ÃuR^LNMKYZ[Z[\WLYZLEY\MKY±^[ º #$*($ * L ¶ÃuEWY±H|Y\M cmin = 0 JM cmax = cq
º p ^ WLR¶MKYZWL SJÈDKFICR^DKMKY\MKYZWL²JIvµCRY\DGYZxNQEJ>CRDKJILRS[Z^ WDGvµJSJ[±^B[R]QEDGJ º :Rº k JȶÃuEWY É JIH|M #$( "& ¶WLRHKYZSEFDGF¶Wvµv²JCJIHGHKY\v²Y±HoMGJȶ^DYZ[ YZLNMKJIDGSY\M©SEJ]DÃ^L SEHMKJIvµC HSJ/SF* ^Y\[Z[±^LR¶JLRWL¶JILRH|QRDKF'H º L@C JIQM^YZLRHKYV[\JT¶WLRHKY±SFDGJD>¶Wv²vµJe[ZJ/CE[ZQRHlDG^YZHKWLRLR^XE[ZJeSJIH>SJIQ É ºp JIH DGFIxNQEJILR¶J'HCEDGFIHKJLNMGFJIHHKQEDe[\J'H [ ]QEDGJIH º 6 JM º : HKWLNMFI]^[\J'H^Q É DG^CEC WD|MÃHȶQEvQR[\F'HSEQLEWvXEDGJSJȶJLRHKQEDGJIHFI]^[\J'HHKQED[±^eMG^Y\[Z[ZJÈSJ[5U FI¶ÃuR^LMGY\[Z[ZWL º p JQRDnHKWv²v²JLEJOt^QM©SWLR¶C ^H 1 CEQRYZHGxNQqU YZ[ ^QMnCRDKJILRSDGJ^QRHGHKYJL8¶Wv²CMGJ [ZJIH®O ^[\JIQEDÃHTLEWL¶JL H|QEDGFJ'HSJ[ U FI¶Ãu ^LNMKYZ[\[ZWL YZLEY\MKY±^[ º p WDÃHSJ[ U Y\LiOJDÃH|YZWL_SJP¶JMKMKJ WLR¶Ç
MKYZWLSEJDGFCR^D|MGYÁMGY\WL RWL C JIQMMGY\DGJDeSJ'HÈ^[\F'^HHKQECFDGY\JIQEDGH 8 [±^ DGFIxNQEJILR¶JTvw^ É YZvw^[ZJ®¶QEvQE[ZFJ®SJIH¶JILRH|QRDKJ'H> @*^Y\LRHKYRxNQEJSEJIH^[\F'^H¤YZL FDGY\JIQEDÃH 8e[Z^ DGFIxNQEJILR¶J>v²YZLEYZv²^[\JȶQEv®QE[\FIJ > @ º p ^ev²FMKuRWiSEJSVU Y\LiOJIDGHKYZWL£J'HoM>CEDGFIHKJLNMGFJ/¶Y\ÇcSEJIHGH|WQRH º ' i = 1, . . . , n +:2
Ui ∼ U [0, 1]
. 8 + j7
Ui ∈ [Fj , Fj+1]
'O% % O?M ? 6-( -O, ! (-O W]
• 8+0 < j < q +:2
ci ∼ U [cj , cj+1]
• 8+j = q +:2
ci ∼ U [cq , cmax]
• 8+j = 0 + 2
ci ∼ U [cmin, c1]
CRD"'H/^ OWYZDeH|YZvQE[ZF²QEL FI¶ÃuR^LMGY\[Z[ZWL#SEJ ¶JL H|QEDGJcn = (c1, . . . , cn) qCEQRYZHeQEL F'¶ÃuR^LNMKYZ[Z[\WLLEWL ¶JLRHKQEDGF
xn ∼ M(θ) EWL¶WLRH|MKDGQEY\Ml[5U FI¶ÃuR^LMGY\[Z[ZWL@X WiWMGH|MKDÃ^CECF yi = min(xi, ci) i ∈ 1, . . . , n º k J ^Y±HK^LM [Z^¶JLRHKQEDGJwJM[ZJIHTSEF* ^YZ[Z[Z^LR¶J'HTH|YZvQR[\FIJIHHKWLNMTv²Y±H|J'H/JL¶WLR¶QEDKDGJL ¶J º ¼ L.[RL ¯WL ^YÁM[ZJ£¶ÃuEWY É SJ LEJ¶WL H|JID|ÇOJDwxNQEJ[\J'H²FI¶ÃuR^LNMKYZ[\[ZWLRHw¶JLRHKQEDGFIHwSWLNM²[ZJ@MG^Q É SEJ@¶JLRHKQEDGJ@JIH|MwMKDIH²CJQF[ZWYZ]LRFPSQ%MÃ^Q É SJ ¶JILRH|QRDKJSJw[5U FI¶ÃuR^LMGY\[Z[ZWL DKFIJ[ > v²WY\LRHSJ@³ SJ SY FIDKJILR¶Jc@ÈC WQED^v²FI[\YZWDGJDe[5U FIO ^[\Q ^tMKYZWL SJw[5U ^[\]WDGYÁMGuEv²JwH|QED/[ZJMo¹iCJSJ/SWLELEFIJIH 8QEMKYZ[\Y±H|JID º k JM|MKJMGJI¶ÃuELEY±xNQEJC JIQM^v²JLRJD 8DGJo¸oJMKJD>¶JDKMG^YZLEJ'H©¶WL.[ ]QEDÃ^tMGY\WLRHSJ/¶JLRHKQEDGJJMTY\[J'HoMTSWLR¶²LEF'¶JIHGHG^YZDKJ²SJ HKY\v®QE[\JIDTSJwLEWvXEDGJQ É FI¶ÃuR^LMGY\[Z[ZWLRH/HKJ[ZWL M(θ) CWQED®^tM|MGJYZLRSDGJ²[\JwLEWvXEDGJSVU FI¶Ãu ^LNMKYZ[\[ZWL HÈXWiWMÃHoMGDG^CEC F'HT¶WLRHKJYZ[\[ZF²CR^DTSJwLRWvXRDKJIQ É ^QEMKJQRDGH'> JLNMKDGJ JM ; JL ]FILEFDÃ^[HKJ[ZWLs JIJJD Ì ¼ HG¶WXR^D/³'´´ Z @ ºÀ F [RLEYZD£[ZJ CEDGW¶J'HKHKQRH SJ#¶JL H|QEDGJ¶Wv²v²J ^[ZFI^tMGWYZDKJJ'HoMDKY±HKxNQEF¶^D [±^_C JMKY\MKJMG^YZ[Z[\JSEJ [ U FI¶Ãu ^LNMKYZ[\[ZWL
LqU W DGJxNQqU QRLEJÈJ'HoMGY\vw^MKYZWL²MKDIHnYZvµCRDKF'¶Y±H|JSEJHG^ WL ¶MKYZWLSJDGFCR^D|MGYÁMGY\WL º ¼ L£CR^D|MGYZ¶QE[\YZJD [±^TOt^DKY±^LR¶JSJ¶JCEDGW¶J'HKHKQRH>J'HoMeC JIQ OiYZHKYZXE[\J º ½ L DGFIHKQE[\MG^tMJIH|M¶JCJLRSE^LNMSY±H|CWLRY\XE[ZJµC WQEDÈ[ZJv²WSI[\J®J É C WLEJLNMGY\JI[ CE[±^¶F®JL^LELEJ É J/SJ/¶JT¶Ãu ^CEY\MKDGJ º3U] 3 R(WF(W?%)NH$ +?EJU$,-. ,(/ XeE((P" !c- QQV-.!#- &)+ !c,12p U YZLNOJDÃH|YZWL SEJ[±^¶JILRHKQEDKJJMl[Z^®H|FI[\J'¶MKYZWLSVU F'¶ÃuR^LNMGY\[Z[\WLRHSEJMG^Q É SJȶJLRHKQEDGJȶWLRH|MG^LMnWQ CJQO ^DKY±^XR[\J
J'HoMTQEL ^¶MGJQED/MGD"'He[\YZv²YÁMÃ^LNM®SJw[5U ^[Z]WDGY\MKuEv²J SJwXWiWMGH|MKDÃ^C SWLNMT[ZJ ¶WMTJIH|M®SF|¸b8@[\WQEDGS º p JwXWiWMGH|MKDÃ^CCR^DG^v²FMKDGY±xQRJ²¶JLRHKQEDGFwLqU JIH|Mµ^YZLRHKYQMGY\[ZYZHG^XR[\J£xNQEJ£C WQED®SJ'HTv²FMGuEWSJIH®^QMGWvw^tMGYZxNQEJ'H/DÃ^CEY±SJIH VMGJ[Z[\J'HxNQEJ¼ s WQ i¼ s_ºÀ ^LRH©[ZJ¶I^HlSVU QELEJCEDKW¶FISQEDGJWCMKYZv²YZH|MKJWLCEDKWC WNH|JSJe¶ÃuEWY±H|YZD©QEL£F'¶^DKMlv²^ É Y\vw^[~SJC WQEDöJLNMÃ^]JSJ
¶JLRHKQEDGJJLNMGDKJ®[ U FI¶ÃuR^LNMKYZ[\[ZWL#Y\LEY\MKY±^[¤JM[ZJIHFI¶Ãu ^LNMKYZ[\[ZWL HÈSJµXWNWMGH|MKDÃ^CSEJ®[5U WDGSEDKJµSJ³ £º h WQEDe¶JI[Z^WL^¶WLRH|MG^MKFxNQEJ 4 000 ≤M ≤ 10 000 HKQ.4²MJILw]FLRFDÃ^[ 8T¶WLRHKJDGOJIDCR[\QRHnSJ 3 000 FI¶ÃuR^LMGY\[Z[ZWLRH¶WDGDKJ'¶MKJIv²JLNMXWiWH|MKDÃ^CECFIH º À ^LRHÈ[ZJ®¶^HCJIHGH|YZv²YZH|MKJ~WL ¶WLRHKJYZ[\[ZJ M ' 10 000 ¶JµxNQEY¯CJDGv²JMeSVU WXMKJILEYZDÈCE[ZQRHSJ 2 500F'¶ÃuR^LNMKYZ[Z[\WLRHX WiWH|MKDÃ^CRC F'HVSJµMG^Q É SJ ¶JLRHKQEDGJµSEJwvµWY\L H/SJ ; SJ SEY FDGJLR¶Jw^ OJI¶®[ZJ®MÃ^Q É Y\LEY\MKY±^[ º ¿ WÇMGWLRH¶JCJLRSR^LNMxNQEJ[±^/HoMÃ^XEYZ[\Y±HG^tMKYZWL²SJ'HFI¶I^DKMGH|Ç Mo¹iCJIH¤XWiWMGH|MKDÃ^CµCJQM MKDGJ^HGHKQEDKFIJ>H|QEDn¶JDKMG^Y\L H¤J É Jv²CE[ZJIH [ZWDÃHKxNQEJ[Z^µSYZvµJILRHKY\WL p SQv²WSI[\JJIH|M©CJMGYÁMGJE^ OJI¶xNQEJI[ZxNQEJ'Hl¶JLNMÃ^YZLEJIH©SVU FI¶ÃuR^LMGY\[Z[ZWLRHnX WiWMGH|MKDÃ^CECFIH9> C ^DJ É JIv²CE[\JeCWQED>[ZJIHvµWSI[\J'HlJ É CWLEJILMGY\JI[VJMSJ Å JYZXEQE[Z[ @ º CRD"'HTMGJIH|MGH [ZJIH®v²FMGuEWSJIHµWCEMKYZvµY±H|MKJIH®JM²CJIHGH|YZv²YZH|MKJ'H WQEDKLRYZHGH|JILNMµSJ'HJ'HoMGY\vw^tMGY\WLRHµSJIHµFI¶I^DKMGH|Ç Mo¹iCJIH
xNQEYSEY DGJLNMCJQ HK^Q>SE^LRHT[\J£¶I^H/C ^DKMKY±¶QE[ZY\JIDTSVU FI¶ÃuR^LMGY\[Z[ZWLRHTSEJ²MG^YZ[Z[\JwMGD"'H ^YZXE[ZJ > n ≤ 15 @ WDKMKJIvµJILNM¶JLRHKQEDGFIH W1[Z^ v²FMKuEWSJCJIHGH|YZv²YZH|MKJ#^CECWDKMKJSJ'HwDKF'H|QR[ÁMÃ^tMGH CE[\Q H HK^MKY±H ^YZHG^LNMÃH º p ^ ¶JLRHKQEDGJFMÃ^LNM CE[ZQRH¶WLNMKDÃ^YZ]LR^LNMKJt[5U FI¶^D|MKÇ Mo¹iCJCJIHGH|YZv²YZH|MKJH|JIDG^eJIL v²Wt¹JLRLEJCE[ZQRH]DG^LRS N¶JÈxNQEY JIH|M©^CECEDGFI¶Y±^XR[\J º ¼ L J JM N[\J'HCEDGW¶F'SQEDGJIH¤SJ>XWNWMGH|MKDÃ^CµCR^DÃ^v²FMKDGYZxNQEJ©LEJlDKJIv²JM|MGJLNMCR^H¤JLw¶^QRH|Jl[ZJ>v²WiS [ZJ©C ^DÃ^v²FMGDKY±xNQEJlQMKYZ[\Y±HKF©JMWLNM]FLEFIDG^[\JIvµJILNMMKJILRSE^LR¶JB8²H|WQRH|Ç5J'HoMGY\v²JDl[5U YZLR¶JDKMKY\MKQRSJeDGFJI[\[ZJ/H|QED θ º ¿ WQ H>¶WL H|JIY\[Z[\WLRHlSWL ¶HKWLQMKYZ[ZYZHG^tMGY\WL º3U] 3 f ^` N<-P?Q-.!cN<$,! eK(*(P" !#-.Q $,- &0+HI(`EJ Q &( % & !"
p ^#HoMGDKQ ¶MKQRDKJ£SEJIHµSWLELRFJIH®CEDGFI¶YZHKFJ£^Q § ³ º : C JIDKv²JMwSVU JLiOiYZHG^]JDT[5U JIH|MKYZv²^MKYZWLC ^D s WLNMKJÇ k ^DK[ZW SJ[±^²vw^tMKDGY±¶JTSVU Y\L WDGv²^MKYZWL@SJ Ê Y±H|uEJID Iobs(θ|yn) SJ'HSWLELEFIJIH>WXRH|JIDKOFJ'H ynº p U YZLNOJDÃH|YZWLPSJ¶JM|MGJTvw^tMGDKY±¶J
− " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
WQEDGLEY\MQELEJ®JIH|MKYZvw^tMKYZWL SEJT[Z^£vw^tMKDGY±¶JTSJµ¶WtOt^DGYZ^LR¶J/SJ θ º p U J'HoMGY\vw^MKYZWL SJ Iobs(θ|yn) CEDKWC WNH|FIJ/CR^D m C>o³'´´ : @iDGJCWHKJeHKQED>[5U QEMKYZ[\Y±HK^MKYZWL@SJ/[5U Y±SJLNMGYÁMGFTSJ p WQEYZHB>|³I´ Z @ ºIobs(θ|yn) = Ic(θ|yn) − Im(θ|yn)
W Im(θ|yn) JIH|M[5U YZL WDGvw^tMGY\WLTSJ Ê Y±H|uEJID¯SJ'HSWLELEFIJIHqvw^LRxNQR^LNMKJ'HVJM Ic(θ|yn) JIH|M[ U YZL WDGvw^tMKYZWLTSEJ Ê YZHKuEJIDSJIH®SWLRLEFJ'H¶Wv²CE['MKJ'H º p U YZSEFJ£SJ m C CEDKF'¶Y±H|FIJwLEWMÃ^v²v²JLNM®C ^D k ^HoMÃ^LEYZJD>o³I´´ @qJ'HoM²SJ DGJI¶WLRH|MKY\MKQEJIDs H|WQRHoÇcFI¶Ãu ^LNMKYZ[\[ZWL H©SEJSWLRLEFJ'H>vw^LRxNQR^LMGJIH (z
(i)n−r)i∈[1,...,M ] ∼ M(θ) xNQEY¯¶Wv²CE[MGJLNM[\J'HSWLELEFIJIHlLEWL¶JILRH|QRDKFIJIH xr
> JIL®¶J[±^ [Z^v²FMKuRWiSEJlHIU ^CECR^DKJILNMKJK8QELEJlv²FMGuEWSJ©SEJnXWiWMGH|MKDÃ^C®CR^DÃ^v²FMKDGYZxNQEJ¶WLRSEYÁMGY\WLELEJ[ @ º½ L@JIH|MKYZvw^tMGJQED>LR^MKQEDGJ[qSEJ Iobs(θ|yn) JIH|M
Iobs,M (θ|yn) = −
1
M
M∑
i=1
∂2lc
(θ;xr, z
(i)n−r
)
∂θ∂θ′+
1
M
M∑
i=1
∂lc
(θ;xr, z
(i)n−r
)
∂θ
∂lc
(θ;xr, z
(i)n−r
)
∂θ′
θ=θ
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Iobs,M (θ|yn) = Iobs(θ|yn) +1
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!%" % ηML =)GM6`P
% ηML =BM)[6
% "" ηNR =#L[6#g
βNR =$ B@P
ηSEM =_Oc2c>
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Eº : ¼ s = SJ'H¤¸oJQ É SJTSWLELEFIJIH > ¿¿ JlMKWLÇ ^CEu H|WL/@ º ¼ L]DG^HlJIH|MCE[±^¶FIJ/[ U JIH|MKYZvw^tMGY\WLPvw^ É Y\Çv²Y±HK^LM>[±^µONDÃ^Y±HKJvXR[Z^LR¶JWXRHKJDGOFIJ º
m [ ^QM/¶JIC JILRSE^LMeDGJvw^DÃxNQEJDxQRJ[ZJIHÈMKJIvµC HSEJµ¶JLRHKQEDGJIHCRDKWC WNH|F'H~SE^L HMGWQRH[ZJIHÈFI¶ÃuR^LMGY\[Z[ZWLRH HKWLNMHKYÁMGQEFIHnXEYZJLP^QÇ SJ[ 8TSJ'H©CE[\Q HuR^QMKJ'HOt^[\JIQEDGH©SJÈMKJIvµC HnSJeSF ^YZ[\[±^LR¶Ji^YZLRH|Y!N[\JOiY\JIY\[Z[\Y±HGH|JIvµJILNM©SQ£HK¹HoMv²Jv²JL ^¶J/SVU MGDKJ/W¶I¶QE[\MKFS> [ U JIH|MKYZvw^tMKYZWLSJ β MGJLRS@SWLR¶eLR^MKQEDGJ[Z[\JIv²JLNMOJIDGH³/CWQED[ZJ/v²WS[ZJTSJ Å JYZXEQE[Z[ @ ºp ^H|MKDÃ^tMKFI]YZJSJeH|YZvQR[Z^MKYZWL£SJe¶J'HF'¶ÃuR^LNMGY\[Z[\WLRH©SJ ¼¾ DKJ'H|CJI¶MKJCJQ[Z^µSFIv²^DG¶ÃuRJxNQR^[ZYÁMÃ^tMGY\OJDGFJ[Z[ZJÈxNQEJLEWQRH©^ OWL HCEDGFIHKJLNMGFÈ^Q § ³ º ixQRYq¶WL H|Y±HoMGJ 8 FI[Z^]QEJID [ZJIHlSWLELEFJ'HlSJ >¼¾ JLP¶ÃuEJDöÃuR^LNM 8CJDöJIOWYZD[±^CJDKMKYZLEJL ¶J²SJw¶ÃuR^¶QELEJ²SVU JILNMKDGJ®JI[\[ZJIH º p JµXEQMTJIH|MeYZ¶YSJMGJIH|MKJID/[Z^PH|JILRHKY\XEYZ[ZYÁMGFµSJ'Hev²FMKuRWiSEJIHeSVU JIH|MKYZv²^MKYZWLHKQEDSJ'H>SWLELRFJIH>SEWLNM>WL@Y\]LEWDGJeSF[ZY\XFDGFv²JILM[ZJ/¶WLMGJ É MKJ/SVU FMGQRSJS>F[ ^XEYZ[ZYZH|MKJ/WQPSQRDG^XEY\[ZY±HoMGJ@ ºG"4 "4,G @&@ÈKz|~ DEzoBA L 98&:F<=<?>9@/BA;<CAEDEzF<¯ L®SEFI¶Y±SJnMKWQMSVU ^X WDGSSJ©¶I^[±¶QE[ZJD[ZJIHY\LNMGJDGO ^[\[ZJIHSJ>¶WL.[ ^LR¶JMKuEFIWDGYZxNQEJ'H¯SWLELEF'H¯C ^D[\J'H¯J É CEDKJ'HKHKYZWLRH
> º ³@JM > Eº @nSEJ[5U ¼ s = SJ'H¸oJIQ É JM > MG^XE[\J'^Q º 6 @ º k J ^Y±HG^LNM WL¶WLRHKYZSIDKJeSWLR¶[±^²¶JILRH|QRDKJ¶Wv²v²J£SJ Mo¹iCJ m WQSJ Mo¹iCJ m|m º h ^DÃ^[Z['I[\JIv²JLNM ¯WL_CEDGWCWHKJ [5U QMGY\[ZY±HK^MKYZWLSVU QEL7XWiWMÃHoMGDG^C MKDÃ^SY\MKYZWLELRJ[SJMo¹iCJ m > W@[Z^²¶JILRHKQEDKJXWNWMGH|MKDÃ^CECFJeJ'HoM[±^µvµWt¹JLELRJSJ'H>¶JILRH|QRDKJ'H>SJe[ U FI¶ÃuR^LNMKYZ[\[ZWLPY\LRYÁMGYZ^[ @ º À J/v IvµJWL#CEDGFIHKJLNMGJµ[ZJIHeDGFIHKQE[\MG^MGH/SJw[±^PvµFMKuEWSJ SqU m C º ¼ L.[RL ¯WL ¶WvµC ^DGJ²¶JIHeDGFIHKQE[\MG^tMÃHB8@¶JQ É xNQEJ²CEDGWiSEQEYÁMT[ZJXWiWMGH|MKDÃ^C/¶JLRHKQEDGF > WCMKYZv²YZH|MKJnJMCJIHGH|YZv²YZH|MKJIOWYZD § Eº 6Rº Eº 6 @ º k J'Hqv²FMGuEWSJIHSJXWiWMGH|MKDÃ^C/CJDGv²JMKMKJLNM¤SVU WXÇMGJLEYZD¤[ U FI¶I^DKM|Ç5Mo¹NCJ σM
H|QRD ηMLCEQEY±H[\J'H¯YZLNMKJIDKOt^[Z[ZJIHSJ©¶WL.[^LR¶J©HKJ[ZWLT[ZJIHCEDGW¶F'SQEDGJIHSFI¶DKY\MKJ'H^Q § º 6Eº º ³ º
p J£MG^XE[\J'^Q Eº · CRDKF'H|JILMGJw[\J'HYZLNMKJIDKOt^[Z[ZJIHSEJ£¶WL.[ ^L ¶J WXMKJILNQ HTCWQEDµ[ U ¼ s = ηML = ³ 6 ³³ : SE^LRHT[ZJ¶I^HSEQ¸oJQ º k JQ É Ç ¶YlH|WLM®CEDKF'H|JILNMKFIH®SJ ^ËWL_CE[\Q H[ZYZHKYZXE[\JPH|QRD®[±^O[R]QEDGJ º\;º QED²¶JMKMKJ [R]QEDGJ¯[\J'HYZLNMKJDGOt^[Z[\J'HlSJ/¶WL [ ^LR¶J¶WLRH|MKDGQEY\MGHlSVU ^CEDIH©[ZJIH DÃ^¶MKYZ[\J'HnXWiWMÃHoMGDG^C£LEJ/HKWLNM>CR^HlSFIXEYZ^YZHKFIH º h ^DÃ^[Z['I[\JIvµJILNM [ZJ>¸oJQ JIH|MMKDÃ^Y\MKFTHKQEDÈ[Z^%[R]QEDGJ º ·Eº p J/CE[±^L SVU FI¶ÃuR^LNMKYZ[\[ZWLEL ^]J/SJ SY IDKJTMKDIHCJQ SEJ¶JI[\QEYSJ ú JIQE[ZJÈQELRJOt^[ZJQEDlSJSF* ^Y\[Z[Z^LR¶JÈDKFIJ[Z[\JÈDKJIv²CE[Z^¶JQRLEJ¶JLRHKQEDGJN^[ZWDÃHxNQEJ[±^HKWv²v²JÈSEJIHMKJv²CRH©DGJIH|MKJHKJv®XE[Z^XE[ZJTSE^LRH[\J'HÈSEJQ É ¶^H º ½ LEJ®DG^CEYZSEJT¶Wv²CR^DG^YZHKWLPJLNMGDKJT[ZJIHÈSJQ É [ ]QEDGJIH>CJDGv²JMeSJLEWMKJID[Z^ WDKMKJHKJLRHKYZXEY\[ZY\MKFwSJIHTDKF'H|QE[\MG^MGHB8P[5U ^¸oWQMSVU QELEJwOt^[\JIQEDTSJ ¶JILRHKQEDKJO> ¶J xNQEYV¸oQRH|MKY5[RJw[±^PCEDKF'¶^QMKYZWL SQ_¶ÃuEWY É SJ'HSWLELEFIJIHA@ º %K & &( p U ¼ s = JIH|MÈXRYZ^YZHKFJ®ONY±H|ÇM8Ç5OiY±HSJµ[Z^£OiDG^Y\JTOt^[\JIQED ηo
JL#[Z^H|QEDGJIH|MKYZvw^LNMSEJ ^ËWLYZv²C WD|MÃ^LNMKJ²SE^L HÈ[ZJw¶^HSEQ£¸oJIQ µ WL#^ ηML − ηo = 4354 º k J WD|M/SF'¶^[±^]JYZ[\[ZQRH|MKDGJ²[\J²CJQ SJ²DKJÇCEDGFIHKJLNMÃ^tMKYZOiYÁMGFTxNQEJ/C WNHKH"ISEJ/[ U ¼ s = º k JTXRYZ^YZH>J'HoMÈJ JI¶MGY\OJv²JLNMÈSEFMKJ'¶MGFTCR^D[Z^²v²Wt¹JILELEJTSJ'HJ'HoMGY\vw^tMGY\WLRHXWiWMGH|MKDÃ^C> CWQEDe[ZJIHGxNQEJ[Z[\J'HWL QMKYZ[\Y±HKJP³ FI¶Ãu ^LNMKYZ[\[ZWL H @*vw^YZHYZ[JIH|Mev²YZJQ É JIH|MKYZvµF²[ZWDÃHKxNQqU WLQMKYZ[\Y±HKJµ[±^v²FMGuEWSJwSVU m C JM/[\J'HeJIH|MKYZv²^MKYZWLRHeXWiWMÃHoMGDG^CW [±^P¶JLRHKQEDGJ²JIH|MeYZLNMKFI]DÃ^[ZJv²JLNM/H|YZvQE[ZFJO> ¶ º MÃ^XE[ZJI^Q Eº ; @ ºÀ ^LRH [\J¶I^H SEQ ¸oJQ l¶JXRYZ^YZHwJ J'¶MGY TJ'HoM£¶JCJLRSE^LNMH|QRDKJ'HoMGY\v²Fn¶JxNQEY>MGFv²WYZ]LRJSVU QELvw^QEOt^Y±H¶Wv²C WD|MGJv²JLNMlSJe[Z^®v²FMKuRWiSEJeSqU J'HoMGY\vw^tMGY\WL CWQEDlQELPJL H|JIvXE[ZJSqU F'¶ÃuR^LNMKYZ[Z[\WLRHlSJ¶I^DÃ^¶MKFDGY±HoMGYZxNQEJ'HnHKJvµÇXE[±^XR[\J'H 8 [±^²H|QRDKJ'HoMGY\vw^tMGY\WL£¶WLRH|MG^tMGFJCR^D©[±^v²FMGuEWSJeJIH|M>CE[\Q HnYZv²C WD|MÃ^LNMKJexNQEJ/¶JI[\[ZJ¶WLRH|MG^MKFIJHKQED
− " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
º p U J'HoMGY\vw^MKYZWL ηML = ´£³ · NCRDKW¶ÃuEJ/SJ ηo = Zw· NJ'HoM>JIL ^YÁMlMGD"'HlC JIQPDGJCRDKWSQR¶MKYZXE[ZJHKQED¶JMo¹iCJSVU FI¶ÃuR^LMGY\[Z[ZWL > JLHKQECECWHG^LNM>xQRJ/[\JeCE[±^LSVU FI¶ÃuR^LNMKYZ[\[ZWLEL ^]JJIH|M¶WDKDGJI¶MA@ º
"1(34 %^!+ dGU!#I,4f0*!#GI!+0O-L^)GI_TO C O2c0O-L^)GIFO%!+ "!+^ E_2c2c0O-I^)GIFI!+ "!+^ E\_2\
º\; ¼ HoMGY\vw^tMGY\WLSQXRYZ^YZHSqU J'HoMGY\vw^tMGY\WLPSEQPC ^DÃ^v MGDKJJ É CWLEJILMGY\JI[ η CWQED>[ZJl¸oJQ ú
% * ( * % %'* #&( (#$ h WQED[ZJT¸oJIQ H|JIQE[\J'H/MGDKWYZHvµFMKuEWSJ'HTW DGJLNMQEL7Y\LNMKJIDKOt^[\[ZJSJ²¶WL [ ^LR¶J [5% − 95%] ¶WLNMKJILR^LNM[Z^OFIDKY\MG^XE[\JµOt^[ZJQED ηo = 8760 SQ#CR^DÃ^v MGDKJ [\J'HvµFMKuEWSJ'HCEY\OWÇMG^[\J'HSJ k W É Ç QELRSXJDG]O> § Eº 6Rº ³ º ³c@JM[Z^²v²FMKuEWSJSqU m C > § Eº 6Rº Eº ; @ º p JIH>CEDGJv²YDGJIHlW DGJLNMSJ'H>Y\LNMGJDGO ^[\[ZJIHSJ ¶WL.[ ^LR¶JµMKDIHe]DG^LRSEHvw^Y±HTxNQEYHKJv®XE[\JILNM/HKQRHG¶JICMKYZXE[ZJIHTSVU MGDKJ²QEL CJQ^vµFI[\YZWDGFIH ~FMÃ^LNMSWLRLEFwxQU YZ[ZHLEJ£CRDKJILELEJLNMC ^HTJIL7¶Wv²CMKJ[Z^OFDGY\MG^XR[\J SY±H|MKDGY\XEQEMKYZWL7SJ'HOt^[\JIQEDGH®SJ£¶JLRHKQEDGJ º p ^v²FMGuEWSJSqU m C WQEDKÇLEY\M£YZ¶YQELDGFIHKQE[\MG^tM CR[\QRH DKF'^[ZYZH|MKJ º L1C JIQMLEWMKJID vw^[Z]DGF[ZJQED XRYZ^YZHS8_SEFI¶^[\JID xNQEJ[ZJIH£Y\LNMGJDGO ^[\[ZJIHwSEJ¶WL.[ ^L ¶J¶WLRH|MKDGQEY\MGHSqU ^CED"'H[ZJIH DG^¶MKYZ[ZJIHXWiWMÃHoMGDG^C H|WLM^XEQRHKYZOJv²JILMÈDKJ'HoMGDKJIY\LNMGH ¶J®xNQEYY\L SYZxNQEJµxNQEJT[ZJCE[±^L@SVU F'¶ÃuR^LNMGY\[Z[\WLELR^]JDKJMKJLiQPDKJ'HoMGJeY\LR^SFIxNQR^M > OWY\DGJeY\LRLR^¶JIHGH|YZXE[ZJ@ º
σM " ! % !" !%!
0O-IZ`NTO Z ´ Z IZ D >YQ]`\\ D \*c2]>\]`_ C \ E3ZcF>\\ D C0O-IZ`!+ "!#^ Z·N IZ D F>YQ]`\2E C \*E2]>\]/ D \ ] D d>\Z D _I 0O-IZ`O%!+ "!+^ Z·<: IZ DC >YQ]`\2EI \*EcF>\]/ DD E D O3_>\_*O D ] "1(344@E N C ] D O D cc3\>\_ CC E O3\2E\f>Y_I D ] NTO C `+U \\ D E>6E2Ec NTO C ` C C ] D >@]3O3\
Eº · ¼ H|MKYZvw^tMGY\WLRHTSJ σ JMµSJIH/YZLNMKJIDKOt^[Z[ZJIH \; Ç ´ ; ^QMKWQEDSJ ηML = 13114 CWQRDT[\JT¸oJQ > s @ º p J'HlY\LNMGJDGO ^[\[ZJIH©CEDGWSQEY\MGHlCR^D DG^¶MKYZ[ZJIHlXWNWMGH|MKDÃ^CPSWYZOJILM MGDKJ/HKWQRH|MKDÃ^Y\M>SVU QRLXEY±^Y±HlCEDGW¶ÃuEJSEJ6Eº
σM " ! % !" !%!
0O-IZ`NTO _2E C _ D \IF>\c2EO _2OO>Yc3\*c O D c>\E C \ 0O-IZ`!+ "!#^ #6; _ D \*]>\c2E3_2c _2O%>Yc3\*cO c D OF>\E*]3E3_ 0O-IZ`O%!+ "!+^ c:i _ D ]d>\c2EOE _2Oc C >Yc3\ CC DD ] C >\ C E\\ "1(344@E N ZIE *] D3C >\OE C O_f>Y C E3_ NTO C `+U ]3E2E C > CC c2]] NTO C ` C ]`_ C > C 2O C E
º ¼ HoMGY\vw^MKYZWLRH²SJ σ JM SJ'HYZLNMKJDGOt^[Z[\J'H \; Ç ´ ; ^QEMKWQRDµSJ ηML = 9162 CWQED²[ZJ¸oJIQ > s @ º p JIHlYZLMGJDGOt^[Z[\J'HlCEDKWSQRYÁMÃH>CR^D DÃ^¶MKYZ[\J'HlX WiWMGH|MKDÃ^C@SWYZOJILM MKDGJeHKWQ HoMGDG^YÁMSVU QEL@XEY±^Y±HlJIH|MKYZvµF/JILv²Wt¹JLRLEJ8 H;Rº
%( "- ! ,+(-
050
0010
000
1500
020
000
eta
vraievaleur
ML
type−IIp
boot.opt.boot.pess.
cox type−Icox type−II
approx.normale
approx.Student
intervallesbootstrap
intervallesthéoriques
º\; k Wv²CR^DÃ^Y±H|WL£SJ'H>Y\LNMGJDGO ^[\[ZJIHSJT¶WL.[ ^L ¶J [5% − 95%] ^QMKWQEDSJ ηML = ³ 6 ³³ : C WQED>[ZJ>¸oJQ ú
050
0010
000
1500
020
000
eta
vraievaleur
ML
type−IIp
boot.opt.boot.pess.
cox type−Icox type−II
approx.normale
approx.Student
intervallesbootstrap
intervallesthéoriques
º · k Wv²CR^DÃ^Y±HKWLSJIHÈYZLMGJDGOt^[Z[\J'HÈSJµ¶WL.[ ^L ¶J [5% − 95%] ^QMGWQEDSJ ηML = ´ ³ ·N C WQED[\JȸoJQ º
f − " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
G"4"45 @ @È|zK~ Dz|BA L 98;:6<H8&< IJ<zFKMLK QED©[\J'HJIH|MKYZvw^tMGY\WLRH ¿È SQ vw^ É YZvQEv°SJONDÃ^Y±HKJvXR[Z^LR¶JSJ'HV¸oJIQ É SJ Å JYZXEQE[Z[ JM > MÃ^XE[ZJI^Q
º 6 @*WL_CRDKW¶*'SJ£^Q É v v²J'H®¶Wv²CR^DÃ^Y±HKWLRH/SJ'HTF'¶^DKM|Ç5Mo¹iC J'H/JM²SJIHY\LNMGJDGO ^[\[ZJIHSJ¶WL.[ ^LR¶J HKQED (η, β)xNQqU ^Q § º\;º º L@LEJ/CEDGFIHKJLNMKJTxNQEJe[ZJIH>DGFIHKQE[\MG^tMÃH>CEDGWtOJILR^LNM>SQ@X WiWMÃHoMGDG^CPSEJeMo¹NCJ m RSEJ/[Z^²v²FMGuEWSJTSVU m CJMSQ@XWiWMGH|MKDÃ^CC J'HKHKYZvµY±H|MKJ º p JIHlDGFIHKQE[\MG^MGH>HKWLNM>CE[Z^¶F'H>SE^LRHl[ZJeMG^XE[ZJI^Q º ZEº QQ$,NH-. ,(/ - p ^ [ ]QEDGJ º C JIDKv²JMPSJOiYZHKQR^[ZY±H|JIDwCE[\Q H ^¶Y\[ZJv²JILM [ZJIH£Y\LNMGJDGO ^[\[ZJIHwSEJ¶WL.[ ^L ¶J®SEQMÃ^XE[ZJI^Q º Z ~v²JIHKQEDÃ^LNMe[ U Y\L ¶JDKMKY\MKQ SJwH|QEDe[5U J'HoMGY\vw^MKYZWL (ηML, βML) = (8687, 1.94) º k JCJLEÇSE^LNM EQELEJ ^ËWLCE[ZQRHDKF'^[ZYZH|MKJ/SJTDGJCRDKF'H|JILMGJDl[±^²ÄWLRJTSJT¶WL.[ ^L ¶Jl¸oWYZLNMKJTHKQED (η, β)ML
JM (η, β)SEMJIH|M
SJTDGJCEDGFIHKJLNMGJD>[5U ( *% #$( ! SJ'HJ'HoMGY\vw^tMGY\WLRH>XWiWMGH|MKDÃ^CSJTMo¹NCJ m JMC J'HKHKYZvµY±H|MKJ º ¼ LNMKWQEDG^LNM´ SJIHlJ'HoMGY\vw^MKYZWLRHlXWiWMGH|MKDÃ^C E¶JMKMKJ/JILNOJ[ZWCECJeJIH|M>CEDKF'H|JILNMKFJ/HKQED>[±^G[ ]QEDGJ º ZEºp JIHqY\LNMKJIDKOt^[\[ZJIHqSEJ¶WL [ ^LR¶J¶WLMGY\JILELEJILMqJILe]FLRFDÃ^[t[ZJ¶WQRCE[\JSEJOt^[ZJQEDÃH~DKFIJ[Z[\J'H (ηo, βo) = (8760, 2.15) º¼ L MKJIDKv²JIH/SEJ®MÃ^YZ[\[ZJwSVU Y\LNMKJIDKOt^[\[ZJV[ZJ²X WiWMÃHoMGDG^C SEJ®Mo¹iCJ m ^CEC WD|MGJwH|QED η QELEJwCEDGFI¶YZHKY\WLv²JYZ[Z[\JIQEDKJwxNQEJ²[ZJX WiWMGH|MKDÃ^C CJIHGH|YZv²YZH|MKJ~JMeY\LiOJIDGHKJv²JILMeH|QRD β º À U QELEJ²vw^LEYDGJ]FLEFIDG^[\J ¶J'HY\LNMKJIDKOt^[\[ZJIHH|WLNMCE[ZQRH]DÃ^L SEHxNQEJ®CWQRD[ZJµvµWSI[\J²J É CWLRJLNMKYZJ[ º p JQRDFI[Z^DK]YZHGHKJv²JLNMJ'HoMT¶WLRHKFI¶QEMKY'K8£[5U ^QE]vµJILNMG^tMGY\WLSJµ[Z^PSYZv²JLRHKY\WLSJ θ HKQED©SJIHnSWLELRFJIHnSJv IvµJMG^YZ[Z[\JÈJMnSJÈv v²JMÃ^Q É SJ¶JLRHKQEDGJxNQEJ[ZJIHSEWLELEFIJIHJ É C WLEJLNMGY\JI[\[ZJIHCEDKF'¶FÇSJLNMGJIH º ¿ WMGWLRH¶JCJLRSR^LNMxNQEJ/[±^wv²FMGuEWSJ®SVU m C WQRDKLEY\MÈSJ'HYZLNMKJDGOt^[Z[\J'HSJ¶WL.[^LR¶JTH|YZ]LEY5[ ¶I^tMKYZOJIv²JLNMCE[ZQRHlCEDGFI¶Y±H>xNQEJ/[ZJIH>^QEMKDGJIHB> ¶JTxNQEYVJIH|M>MKDIHlOiY±H|YZXE[ZJeHKQED>[±^%[R]QRDKJ º @ º QQ$,NH-. ,(/ - h ^DÃ^[Z['I[\JIv²JLNMµ^Q8¶^HµCEDKF'¶F'SJLNM ¤[5U JILiOJ[ZWCRC J@¶WLNOJ É JSEJIHµJIH|MKYZvw^tMKYZWL HX WiWMGH|MKDÃ^C > [R]QEDGJ º ´H@µCJDGvµJMSVU ^CECRDKF'¶YZJDw[5U YZLR¶JID|MGYÁMGQRSJSEJ [ U JIH|MKYZvw^tMGY\WL º À J LEWQROJI^Q QELRJ ^YZXE[ZJ SYÁÇv²Y\LiQMGY\WL+SQ CWQEDöJILNMG^]J SJ#¶JLRHKQEDGJ CJDGvµJM@¶JCJLRSR^LNMSVU ^v²F[ZYZWDGJDSEDG^HoMGYZxNQEJIvµJILNM£[Z^CEDKF'¶Y±H|YZWL SEJ[ U JIH|MKYZvw^tMGY\WL > LEWMG^vµv²JILM²¶JI[\[ZJPSEJ β @C¤WLOFIDKY5[RJ^Y\L H|Y©[Z^CEDGFIHKJLR¶J£SJ (ηo, βo) = (8760, 2.15) SE^LRH[±^CE[ZQECR^DKM®SJ'HYZLNMKJIDKOt^[Z[ZJIHSEJ¶WL.[ ^LR¶J º L7¶WLRH|MG^MKJ£xNQEJ[Z^v²FMKuEWSJ@SVU m C7H|WQRH|Ç5J'HoMGY\v²J [5U YZLR¶JDKMKY\MKQRSJPH|QRD[ U JIH|MKYZvw^tMGY\WL²SEJ θ = (η, β) [ZJIHYZLNMKJDGOt^[Z[\J'HSJ¶WL.[ ^LR¶JlLEJ¶WLNMKYZJLRLEJLNMCR^H[Z^eOiDÃ^YZJlO ^[\JIQEDSQwCR^DG^v MKDGJ º½ LEJMKJL SE^LR¶J>LR^tMGQEDGJ[Z[\JSJ[ U JIH|MKYZvw^tMKYZWL DGFIxNQEJILMGYZH|MKJÈSQwv²WiS [ZJSJ Å JYZXEQE[Z[RJIH|Mn¶JCJLRSE^LNMnSJHKQEDKJ'HoMGY\v²JIDQELµCR^DÃ^v MGDKJ©JLwHKWQRH|ÇcJIH|MKYZv²^LNM[5U ^QMKDGJB> MGJLRSE^LR¶J>xNQEJ>[5U WL®CJQEMDKJIv²^DGxNQEJID¯JIL²LEWMÃ^LNM[Z^ WDGv²J>SJIH¤ÄIWLEJ'HSJ/¶WL.[ ^L ¶J©¸oWYZLNMKJTSJ'H [R]QEDGJIH º Z JM º ´ º
σM ! ! ! % !"
η β η β η β0O-LZ`_TO E2c D E` c E2ccF>@]E*] c3` c2cF>_-` ]O \ D _ C >YQ] C _2O ` c2c>_-` C \ 0O-LZ`*O%!+ F` \*cE2] ` c2] c3` cc>\_ D Z c3` c2EF>$E3` O\ _\2E\f> C \*O c` __f>6E` \2c "1(344\E N OI C c`+_ _-cF>@c C 2O I` C > C ` C
η β η β η β0O-LZ`_TO C*D ] C c` OO E2E3\2OF>@]]3_ c3` c> C ` O C E%>YQ]/_ C c` C >$E3` C 0O-LZ`*O%!+ "!+^ EI2] C c` D C c3_\>@\]3E2c c3` c> C ` ]3_ _3\L>YO2]2] c` \ D > C ` _] "1(344\E N Ec C c` C E 2E3\>@I]c c3` O>@I` 2E
Eº Z ¼ H|MKYZv²^MKYZWLRH>SEJ ¿ JlMGWLÇ ^CEuRHKWL£CWQRD[\J'H¸oJQ É > uR^QM @©JM tº
) #14-*145WG3-? )E-*56?4GE-*4564)E/1GE-* &1)E/4D4 24)*/- /. 0 X *V4 4 4/5W/.)* I .)7/ 4)V4=6=?&)?&W56)*@*?424\GE-*64)E/4D&,24)*/-* /.,0!X *V4\44/5W/.) H:\. -/\0*<[-* /,# I 4/*?-0*24+[*N +\?424.) 0SX &/.5W/)W /@N +D0* /?0*4)E/4D&0*24)>()2&
%( "- ! ,+(- f2
0 5000 10000 15000 20000
02
46
810
(eta0,beta0)
0 5000 10000 15000 20000
02
46
810
eta
beta
ML
type−I Studenttype−I boot.quant.
boot.pess. Studentboot.pess.quant
Ip−Student
º k Wv²CR^DÃ^Y±H|WL®SJ'HYZLMGJDGOt^[Z[\J'HSJ¶WL.[ ^L ¶J 8 [5%−95%] ^QMGWQEDSEJ (ηNR, βNR) = (8687, 1.94)CWQED>[ZJ©¸oJQ º
5000 10000 15000 20000 25000
02
46
81
0
eta
be
ta
5000 10000 15000 20000 25000
02
46
81
0
eta
be
ta
5000 10000 15000 20000 25000
02
46
81
0
eta
be
ta
5000 10000 15000 20000 25000
02
46
81
0
type−I boot.boot. pessim.
MLVraie valeur
º Z WLEJSJ¶WL.[ ^LR¶J¤¸oWY\LNMGJ 8T´ CWQED(ηNR, βNR) = (8687, 1.94) > ¸oJQ @/JIH|MKYZvµFIJCR^D>[\J/XWiWMÃHoMGDG^C@SEJeMo¹NCJ m JM[ZJTXWNWMGH|MKDÃ^CPC J'HoÇHKY\v²Y±HoMGJ º
10000 15000 20000
01
23
45
6
eta
be
ta
10000 15000 20000
01
23
45
6
eta
be
ta
10000 15000 20000
01
23
45
6
eta
be
ta
10000 15000 20000
01
23
45
6
type−I boot.boot. pessim.
MLVraie valeur
Eº ´ ~WLEJSJȶWL [ ^LR¶J¤¸oWYZLNMKJ 8T´ C WQED(ηNR, βNR) = (8752, 1.21) > ¸oJQ @/JIH|MKYZv²FJCR^D>[ZJ/X WiWMGH|MKDÃ^CSJeMo¹iC J m JM[\J/XWiWMÃHoMGDG^CCJIH|ÇH|YZv²YZH|MKJ º
fW3 − " ! ( " '!$&( ' )/+ 1 $ ( )( )0 $ -.!
G"4"4 L DBA 8;< ?D 98;< @ D(<CL 8 < <CL
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X ∼ M(θ)7"PV>?(*)+57"$&(*)+M J &)0".A21.W7"P θ 5!1*57"$A21.$*)0" ,HJ A230( , <"$&(*)E X 2"P@)9&$NX2)+. , !
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r )sN1.A!$2)="$E#<+1 , Y". , <utL v , ZxwyA21.2)="$Z!#"$&%'"$&(*)+zX(=(*".#"$1ND<>Q05k"$!<1/ J 21.1$!<1 ρE(X)
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tL v E]".@%4".@(0)+]9(U("".1.*N l ]uU]spq<6>Q05k"$!<1T J !1$1.2<+1 ρE(X) ,HJ !$NXA21)+52 , Y<1.A2z
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EGFIHJHLK&M9NMOHPEn0LQ8R & h3ST&dd($ &U d U c d)T VXW Y & cyf U<Z b5[a U c T$T &U h*dqcea T$T&d n yn = (xr, cnr)&\
xr = (x1, . . . , xr) ∼E(ηo)
d2ccn−r = (c1, . . . , cn−r)
f U d U4 d]B^T&d/h*dbd U-4 fg$d`_h4g & cedh*d c8a)$+d7b $yg &c g$d 4 4 d WeY & c ηMLT@df]a( $g
]Yfh] h0d g$a 4 d)]<^)T@a U bd&WGY & c σ2cT@a a4g a U bd $U b &UU f+d¡h4f $yg & b.d 4 4 f 4 h*dsb.d U-4 fg$d¡h0a U4 T@a¥b2g Z a4c '&U h0d 4
Z b[+a U c T$T &U-4 ^ & & c 4 cHg.a)$WGi U 4 f $ $ & 4 d $ & f "& g 4 ]EfhT@d2g M Z b5[a U c T$T &U4 ^ & & c 4 cHg$a $h*d'cea4f(sh*d6bd U-4 fg$d Z,c aT_¡bdTf h*d<T$j Z b5[a U c T$T &U $U c aT'WkjT & g 45l T & g 4)m f+d M → ∞ l T#j d 4 c ]a4ced7fgB^ && c 4 cg$a)$h0d<T@a a4g a U bdh0d ηMLb &U d2g c d d7g 4
(n− r)2
r2σ2
c +η2
ML
r.
npoq0rLsJqut r )u$<NNX(0$Y:=<'5K+*:=<YA!5K+*)="$ ,@, (*)X(U(0#"$1NY
(j)n = (Y
(j)1 , . . . , Y
(j)n
∀i ∈ 1, . . . ,M N9(=G#C , '%'W2%'')(0%i1. r (*)+)A2/572)9#<1.A2 l (*) $<NNX(0$E$@%Y< , 21$<¤6$*%E%'!)0" J A!5K9)="$ ,@, (*)9zNX(*<+1) J <"$ , $21:=<'572<~u:=<M1$#NX!57"$!)0"527"#".'1$C!S , pkL[\]K+0:=<'A!5K+*)0". ,&, (*)sX(U("".1.*Nu$EA57(0%ENX(0$'2)="$1.E<)#(0<+#GA5K+)="$ ,&, (0)
X(j)n E(ηo)
2"<)D$(*<9GA!5K+*)="$ ,@, (*)s6572)9#<1.!C
(j)nr Fc
LVU(*@"σ2
C, -341.)+52 l$a
$Xg '& g X&)+52(*))U<^pT<IN1$(52!.#<+T/522)+$<1.Z2" ec#(0)`#NXA21)957*L
(0$(*)+Tj =
∑ni=1 Y
ji
L M1/&)+A!N9!)+*)+57 A!5K+*)0". ,&, (*)+ X(U(".#"$1Nu2)="$1.'2<~ , ! Tj#(0)0"/L L L r )
N9!<" , (*1q)(".21 Tj = Aj +Bj^3*5
Aj =
r∑
i=1
X(j)i G(r, η−1
ML)
Bj =
n−r∑
i=1
C(j)i .
Aj
7"Bj
$(*)=" W W hXW 2"i&)+A2NX2)9)=".i2)="$1.2<~yLV[I341.)+526 Bj"E , (*1 (n − r)2σ2
C
2"Y$(*)!$N9A!1.*)+57Z34<"
(nr)ecLY34*1$)+52 9(U("".1.*N-
ηML J A571.x" , (*1
P*1[ηML] =
1
r2M(M − 1)
M∑
j=1
(MTj −
M∑
i=1
Ti
)2
=1
r2M2(M − 1)
M∑
j=1
[(MAj −
M∑
i=1
Ai
)+
(MBj −
M∑
i=1
Bi
)]2
=1
r2M2(M − 1)
M∑
j=1
(M − 1)Aj −M∑
i=1,i6=j
Ai
+
(M − 1)Bj −M∑
i=1,i6=j
Bi
2
(0$(*)+Uj = (M − 1)Aj
7"Vj =
∑Mi=1,i6=j Ai
L9\]!2<~34*1$ , !T$(*)="O@)9A2NX2)++)="$ 2"
Uj ∼ G(r, [(M − 1)ηML]−1),
Vj ∼ G(r(M − 1), η−1ML).
(0$(*)+ A2S= , 2%'2)=" Wj = (M − 1)Bj2"Qj =
∑Mi=1,i6=j Bi
L\] !<~¡3^*1$ , $(*)=" &)+A2NX2)9)="$!)0".1$ ,@, ! 7"&)+A!N9!)+)=".!O Uj
7"VjL¢ , (01.
[Wj ] =
[Qj ] = (M − 1)(n− r) ec, P1
(Wj) = (M − 1)2(n− r)2σ2c , P*1
(Qj) = (M − 1)(n− r)2σ2c .
[Uj ] =
[Vj ] = (M − 1)r ηML, P*1 (U2
j
)= (M − 1)2r η2
ML, P1 (V 2
j
)= (M − 1)r η2
ML.
r )-2)¡A<@"
P1[ηML]
M→∞' 1
r2M2(M − 1)
M∑
j=1
[M(M − 1)r ηML +M(M − 1)(n− r)2σ2
c
]
→ (n− r)2
r2σ2
c +η2
ML
r.
Chapitre 3 ! "#
$&%'()
* ,+z.- OyM/-02143 576987:<;2=?>(@A8CBD;E6 11F11F111F11F111F11F111F1F1F11F111F11F111F11F111F1 GIH021DJ KL;2=(MONPBPQ7RS8CBT;E6U6V;E6)B46VWX;I:ZY#R[87BT\9]^11F11F111F1F1F11F111F11F111F11F111F1 _23021D0 `a6V]bYb;V=VMSNTBPQ7R[8CBD;E6cB46VWX;I:ZY#R[87BT\9]d=V]SQe(R[:<R9YbfO8g:h]OQi=V]kjl]OBPm?>(NPN11F11F111F1 _IG021on pq62RINTrVQ7]tsvuOwZx/yz|~wh|~1F111F11F111F11F111F1F1F11F111F11F111F11F111F1 HI_021DG 576(=(BP@AR[8g][>2:ZQ=(]k8gRIBPNPND]d=(]OQi=V;E66VM]OQ;9mQ7]:<\9M]OQ$1F11F111F11F111F11F111F1 I0021D_ pq6(6V]V]OQ1F1F111F11F111F11F111F11F111F1F1F11F111F11F111F11F111F1 9
S
m − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* ; +
I+z; - + =.- -/ !"$#Q + # -%&'+/-()E-* , s$<@"$`!'57(0)+"4"4"$&(*)9<F5K9N@"$1.-t,+ , " , A!S*@"$&%'->Q&1$-NNX , +V2)F$<+E`(*))A!! yn>Q& , 2%'2)="P&)>?(*1.%4"$&3*+0B ,HJ 4|(0<"; J <) 52(*))+*&..)+52q*<"$1.<657(0%'N9(01#".2%'2)=";<57(0%'N9(=$*)0" ∑ +0$ ,&,
7~".*L r )6!$N9C!1$Z , (*1+0Bi57(0)+@"$&(*) NX(*<30(*&1;%'(A , &$21q1#(0))+ , !%E!)="]572"#"$Z57(0))+.$*)+57-+*2)6"$&1$!1<)N+1$A57#&(*)'#<¤.)=". )9 ,HJ !#"$&%4".@(0)'O>?(*)+57"$&(*)+] J @)="$A!1$W2"; θ L *)>Q*R7(0)657(0<1.*)="$O J S*S , (0%'A21.2157E2<~F52(*))9$.)+52!Y!#"6->Q&1$INNX , *<~¥"$5K):=<!'+^_0A!$@!))+V2) (*)+)+)="<N+1%'C7".1$ θ, `)+"$<1.E J <)+Y341. , E , A4".(*&1$i$<1 Θ
L *)'1.A7>?A!1$!)+57E%4|2<+1$Y2) , `%4"$&C21.i" , , @3U1.E (*X21$"l t*o*on^pkL+U ,HJ (*)¡NX2<"z57(0)+".1$<+@1./<)%'#<1. a $Xg '& g π(θ)
$ , (*)I<)+i57(0))+.$*)+57 &)+A2NX2)9)="$Yyn
+, E%'#/!)/./<3U1./<IN+1*&S*%'Z+^_*A#&2)`N9!1$%'2" /A78+)+@1<+)/%'!$<1$ a $ & 4 ced2g '& g #<+1 Θ
π (θ|yn) =L (yn; θ)π(θ)∫
ΘL (yn; θ)π(θ) dθ
:=<!#"Z<)+E!)+$x".AEEN1.(*+* , @"$AE# ∫ΘL(yn; θ)π(θ) dθ <∞ L\]Y>Q.)="0+ , z>?(*)+57"$&(*)+Z J @)="$A!1$W2" θ:=< ,HJ (*)'5K!1.5K+*x"P*<N+1^34)="VB ".@%'!1P23U&2))!)=";B , !<1"$(0<1P!P3^*1$ , M , A!4".(*&1$(*)="P(*)E"$!)="$1+ a
$ & 4 cd2g '& g 2+ J !#"$&%'21 , !VS*1)+2<1[5!1*5k".A21.&.)=".! l !$N9A!1.*)+57-+!%'(1+^2".5L pkL 52(*))+*&..)+52P7~N , &52x".q, Y2)9#@"$A a $ &"4 cd2g '& g π(θ|yn)
+ J <)/NN+1$(^~&%'"$&(*)6 572"#".Z!)+$x".Az(0<`<%E&)&%i<% , iN9(=$$@+ , x".AZ#&%i< , !1q!)`S01.*)+6)+(*%i+1$ !q1.A! , $"$&(*)+ (θ1, . . . , θM )
NX21.%E2"#".2)=" J ".@%'!1!TS01.*)+2<+1. $Xg Z h b7c d 4 , E>?(01$%'
gi(x|yn) =
∫
Θ
g(x|θ)π(θ|yn) dθ
:=<$(*)=" N , <+Z#&%'N , 2%'2)="Z05252!.#& , !B ,J @)S0A2)&2<+1 K+*x".<AYB"$1^3^* ,&, 21 $<1z!S01.*)+!<1.(*+$21.34 , !!Lr wX1.&1i<)521*57"$C!1$4371.*&$(*))9 , 53EBs527"$"$%'(A , $"$&(*)6+$ , (*)£i&)+52"$2<+1. , iN , <+Y(*|!5k".x>QYNX(0#G#& , !0+9" ,HJ A7"NXE baT ^2g$ac '&U+LP)uN9!<uY%E(*".0+9 ,@, '57(*)9#".iB5K+(*#&1 ,J &)+5221$"$@"$<+'$<1 θ "$1)+$%'&$N+1
π(θ)L Z)+z575K+*Nx".1$52(*%'%'i*)+ , i5K+N+x".1$ l )+ <)5201./N , <9S*A!)A21 , p7+)+(*<+z5K215K(*)9OB
#!)+$@ , &$21 ,J *)+ , _".Z&)+<9".1$& , *<~!)4|2<~-@%'NX(*1$".*)0"O/572"#"$5! , @+1."$&(*)L
J (*|5k"$@>M<IN1$A#!)="O5K9N@"$1.z" /N1$A#!)="$21T!)IA7".* ,, 5201.z+^_0A!$@!)` J <)/A2"$<+/@)+<+".1$& ,&, 2)<+1.* , @"$A1+9(18 , ! %E(C , NX( , _=GD2&< ,&, $(*)="<". , #A l 1.2S01$(0<N+*)0" , ! %'(UC , !2~N9(0)2)="$& , +9u!xG< ,&, 5 , *.$&:=<Z7" 52(*%'NXA7"$@"$&(*)I!)0".1$ D2&< ,&, p7L
(*<9&)="$1.(U<&$(*)9'"$(0<"- J X(*1 ,J 0#NX!57"6>?(*1.%' , , £%'(A , &.4".@(0) +^_*A#&2)+)¡)(*) &)>?(01$%4".@307"/)(0<+Z2)sN+1$A#!)0".(*)+Z:=< , :=<! 5!1*5k".A21.&#"$:=<!z7~N , (*@".* , !!L\]2"#"$'%'(A , $"$&(*)sNX21.%'7"9+)+Z)(*"$1.5201$6 J A2"$<+-+M`#"$1.<+5k".<1.21i>Q*R2(*)¥N1.(*+* , ". , 57(0%EN+1$A!K2)+$&(*)£>?(*)+57"$&(*)+Y J &)="$A!1$W2"Y!#"$&%'A2<N91^3^*)="T*)+T<)¡5201.z>?1.A!:=<!)="$".: , '%~U&%'&.4".@(0)- , E3U1.*&$2%Y , )+52z" *@)+$NX21R7<52(*%'%',HJ !#"$&%4".@(0)<%'(] πJ(θ|yn)
(18πJ %'(UA , &$;<)+;&S*)(01.*)+570LP1.(*NX(0$A2P*)+ , P<";1$A!S*< , 1.&$21 ,
1$A#< , "4".O>?1$A:0<+2)="$".! l 2)"3kS*(0%'%'*)=";3ZN+*17~2%'N , Y5221$".*@)+%'~@%Y<% , (U5!<~¡ , 3U1#!%i , )957p<+572"#"$%E(A , .4"$&(*)D)$21-N+*N+1$"$57< , &C21.2%'2)="A7".<+&A26*)+ , $<x".6<£5K+*Nx".1$%*&/&)="$!1$3U&2)+157(0)+"%'%'2)="<I#!@)¡<I5K+*Nx".1$ L
P)7wy7"9+ ,J 7~N9!1#".&$&)+<+#"$1.& ,&, Y7~&#"$-+9*NN9(01#"./<)Y3*A21.@". , /&)>?(*1.%4".@(0)6++2"z J 2)¡N+1$&3*!1!#"Z(0%EG%S* , *L4"$A!1$<~+52(*)+#"$1.<+5k".@(0)6+57(0)+@"$&(*)+M J 2~UN , (0x"4"$&(*)=:*O)+(*%i+1$!<+#57(*)+)+$.)957!VKU<%&)#(0)="T%'#O2)>./<3U1./)+ ,J @)+<+".1$&NX(*<1OA2".* , &1 O".1."$A!S*&!O J 2)="$1.7".@!)I(*<¡/1.2)(0<3* ,@, 2%'2)="O!57(0%ENX(0.)="2L\]!57(0))+*&.$*)+57+^$= ,@, !M J NN<+@!)0"2)ES*A!)A21 , $<1M!57(0)="$1&)0".!(*|5k".@30! l 52*1.05k".A21.&#G"$:=<!VNKU_#:=<!V%'"$A!1$<~+N+12~U!%'N , ^p(*<Y!(*)+)A2V*)+57&2)+)! l $<1MV52(*%'NX(0.)=".V)+ , (0S*<
B ∑ p<+1$".2)="/1$ , "$&3*zB-<)'NX21#(0))'(*)+)A2-+)+Z<)+'$@"$<+"$&(*) (*))+A2*L[!Z1.A2NX(*)+$! :=< , 7~UGNX21$".M&)+<9".1$& , PNX2<302)="]*NN9(01#".21;B <+) $A21.@z:0<+!#"$&(*)+P ,HJ )9 , _U#"$O9^_*A!$&2)#(0)="P2)=".05KA2P J <)$<|5k"$&3Ux".A N+1O$$2)+52/x¤652 , 2%'2)="%'(A , $* , 0L
r ) 57(0)>?(*)+1. , (*1*)+657"$21.%'IG a U aT a 4 ced ^.aa Z 4 d U¥G".(*<".N9!1.$(*)+):=<&)0".21.1$(0S* , ¡(*< , 2~UNX21$".0+U1$57<! ,&, , 2<1 l p;1$A!N9(0)+# l p7+U<". , &$/!T521$@"$:=<!T$<1O57!T^3Uq J 2~UNX21$"0+0@"$&(*)) , !q&)>?(*1.%4G".@(0)+$<|!57"$&3*<~DA2302)="$< ,@, !@)>?(01$%"$&(*)+/(0|!57"$&3*/&$NX(*)& , ! l <".1$:=< yn
p<+52(*)+#"$1.<@"<)%'(A , $"$&(*) a1$Xg '& g π(θ)
2" , '52 , &1. 2)`>?(*)+57"$&(*)¡!T&)>?(01$%4".@(0)+T(*"$2)U<2L
J *$NX!5k" , iN , <+Z@%'NX(*1$".*)0" '57'5K+N@"$1.i" (*)95 , Y5K(0x~¡2" ,J A2"$<+E J <)Y>Q%' ,@, 'E2)+$x".A! a$yg '& g π(θ)
+#NXA!52& , !%E!)="]A!@A! B ,HJ &)>?A21.2)+52 D2&< ,&, 7"q52)9 , 5201$ J 2~UNX21$"$$ #<+1 ∑ N+1$(0N9(=#AN+*1M/10]L=P)N+*1#".&52< , @!10+525K+N@"$1.$ON , 057*)+ , /57(0)0".@)U<@"$A!".1.^34<~YT*5K+ l n!v*v0p7"]!1$S021 U<+) l nv*v m p/!)%'(A , $*)0"'`>Q*R7(0)57(0)+@"$&(*))+ ,&, , Y2<~£N+*1.*%'C7"$1.! θ = (η, β)
-u!@< ,@, 2"!)I*NN1.(>?(0)+$.)=" , !<1 *NN1.(U5K+*L(*<"z*< , (0)S6i527"#"./A7".<+1+9)(*<9 5K!1.5K(0)+q2)¡2wX2"zBA2S=S021O<)$2)+i5 , &1Y527"$"$1.2N+1$A#!)0"4".@(0) a $Xg '& g 2+2)¥)(0<+2wX(01.R!)="i1.!$NX!5k".21<+)£2)+$2%Y , 61.C2S , 52(*KA!1$!)+57 ^3*5 ,J 2~N9!1#".&$ 7" , 57(*)=".7~U"$/ ,J A2"$<+ :%E(A , $21 , /3=&2 ,&, $$2%'!)0" ∑ L¢ <s8+)9 , +)(*<+/*9(0<"$.#(0)+ B`<+)u(0<"$ , 2~U& , '7"/@)=". ,&, &S*& , ENX(*<+1 ,HJ )+ , _#"$'+^_0A!$@!)L (*<+/A!$@1.(*)+0+ a 576"$1^3^* , +$2)+$@+ , &$21527"Y*)+ , _".B!i*$N95k".@%'NX(*1$".)=" , I%E(A , .4"$&(*)¥$<|5k"$&3*-+ ,J !)G|!<¡N1$&)+52@N+ , A7")=" %'(A!1$!1<)/2~UNX21$"$$i:=<%'2)+057!1.*x" J W7".1$ ".1$(0N¡@)>?(*1.%'"$&3* 7"Z J !)="$1"! )219+ a$ &"4 ced7g '& g 2+y!z1$A#< , ".". "$1.(*N$<|5k"$@>Q!LX¢ @)+$ +y*<s57(0<1 <¡".7~U"$-+9)+(*<+zN1.(*NX(0$(*)+:=< , :=<z:=<".@(0)+(0<3*!1#".!7"P J <".1$[".1$C>?(U5! , &$A2+4:=<UNX(*<1.1.*@!)="$21.3U@1TN9(0@)="VTA2N+*1#"M<E& , (0S*<q2)=".1$T)+ , _#"$2"T2~N9!1#"L
F L d 4 $yg $U b $af( $ & $U c 4 h Z d)T & $ $ Z4 h0a U-4 bd-b[+a)$ cg$d 4&U czg Z4 fh] Z4DQ d U a U0c T@a 4 Vuh*a U-4 f Ug$a $$ & gkcTh*dig$db[+d7g$b5[d Q! & f 4m f+d2c#"%$$&5V l h 4 $ &U ^)T@d'd U j UU d)(Ud 4'! W(Pd)Tf $g b $yg & $ &"4 d $aga T$T@d2fg 4 f U dih Zg]a4g$b5[d¡h0d¡baT ^2g$a4c '&U $+a4g-h Z d a4fcih*d@T&a T & a $yg '& g h4f $+a4g.a]BS7cg$d¡h*d d& g#]'d β h*d*)'d ^2fhT$T'W,+ U.- U l f Ud)(Ud] $uT@d/g Z dT l d a 4 a U c2T@d g Z ba)$ cHfhT&a4c d h*d 4 a 4 $+db7c 4 ] Z c$[ & h & T &c m f+d 4 $yg & $ &"4)Z4 h*a U4 bdb5[a $ cg$d l d 4 c] 4 d Ud)(Ud7g c f+dih0a U4 bdig$a)$ $ & gkc,W
/sOy,+(zM2-- # M1 , /$<x". l +)+P52 5K+*Nx".1$O52(*%'%' )9 , P$<@34*)0".pV)+(*<+]#!1$(0)+V>?1$A:=<2%'%'2)=";%'!)A!PB <"$ , #!1
, 3*(52*< , &1$ 57(0<1.*)="T , '"4".&#"$:=< +^_*A#&2))+*L
• "$!1$%' [ a)$+d2g $ag$a]BS7cg$d A!$@S0) , /N91%'C7".1$ J <)/%'!$<1. π(θ)(*<
π(θ|yn)L
• *)z%E#<+1$ a $Xg '& g π !#" $Xg & $Xg$d J , J *S*@"] J <) 2)+$x".A l N1.(*+* , @"$A^p7+L 0L0$ ∫Θ π(θ) dθ <∞ L• r ) x":=< J <);%E#<+1$ π !#"/$U&d& g#]'ac d $*$(*) d U cg & $ dqg$dT&a4c d;l (*</@wyA21.2)="$& ,&, ^p ∫
Θπ(θ) log π(θ) dθ!#"q89)@0L J 2)="$1.(*N+@z1$ , "$&3*Z!#"q<)-&)+524".2<1O ,J &)+5221$"$@"$<+/ J <) %'!$<1.1+57(0%'%E , i341.)+521+
%'*&Y572"#"$-21.)&C21.6NX2<"E)N+0W2"$1.A789)@0L r )N9!<"E(0)+5NX2157!3*(*&1<)+%'!$<1.6@)>?(*1.%'"$&3*57(*%'%' N9(=$$A!+)="O<)I@)+&5!4"$!<1 J @)95721$"$@"$<9/%'!$<1.* , l 7" (0)+5 52 , &1 , p7L
• (e[+a U c d)]'d U cPh0d $+a4g$a] Z cg 4 ac '&U W Z)+ , T1$".O57q"$1^34 , +^(0)'$21Z%'2)AOB 57(0)+$&A!1$!1N , <9#&2<1N+1%'A7".1$.4"$&(*)9[@wXA!1$!)="$!0+4)(*".%'%'2)="2)E1.*&$(*)Y , >Q*52 , x".AOB %)+@!1!2)9#@"$AM , (0&V&2)57(*)+)=<+!!LyUM(0)s57(0))+ !&" <)E%'!$<1$
πθNX(*<1
θ ∈ Θ ⊂ IRp +X , !#"/#AEE52(*))9"!"$1. πθ′
N9(0<1 <)
m − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* *<"$1. N+1%'A7".1$.4"$&(*)
θ′!)I52(*)+$A21)=" , Y"$1)9>?(01$%4".@(0) |!57"$&3* Φ : θ 7→ θ′
0057(*+@!)
J(θ′) = det
[∂2Φ(Φ−1(θ′))
∂θi∂θj
]
i,j∈1,...,p.
¢ , (01. πθ′(θ′) = J−1(θ′)πθ(Φ−1(θ′))
L
• g & ^aX^ T c Z 4d g Z5m f+d U c 4 cedd7c ^aa Z 4 d UU d W U(*@" t(θ) <+) >?(0)+5k".@(0)` J &)0".A21.W7"T θ L[(*1$:=< θ !#"]NX21R7<52(*%'%'/<)341. , / , A!4".(*&1$/2)+$x".A π ++(0)I)(".21 P (t(θ) ∈ A) , 'N1.(*+* , @"$A/+^_0A!$@!)) l AA2".)="Z<)sA!3*C2)+2%'2)=" 8+~UAE)9 A p 2)s(*NNX(0$x".@(0)sB , 6N1.(*+* , @"$Ai>?1$A:0<+2)="$". Pθ (t(θ) ∈ A)(18
θ"O8~A 7"A"<+)-A!3*C!)2%'2)=" , A!"$(0@1.*L
;5: ?
! #"# .e)="$1.(<@"zN+*1 07wy1$!_U l nv +[nv n^p7+ ,J 2%'N , (*i%'!$<1.! a $Xg '& g )(*)s@)>?(01$%"$&3*! πJ "$!)="$' J *NG
NX(*1$"$!1;!P1$A!N9(0)+#PB/<)!)+$2%i , z:0<+!#"$&(*)+ , A2S0x".@%' , (*1$:=< J (*)'5K215KB/%'(A , #!1P<) @S0)(*1)+52*L* )-2~2%'N , z".1$CO#&%'N , / ,&, <9".1$ , Y3U#&(*)-S*A!)A21 , :0<+A230 , (*NNX ]!1$)91( l n!v0v p7L$&%G(' m L Y & c7T@d ] & hXS)T@d6d)($ &U d U c dTMh*dh0d U4 c Z f(x|θ) = θ exp(−θx) WGi U b5[ & 4 c π(θ) b & ] ]'df U dT & &) a] ]a`h[j [ a)$+d7g $+a4g$a]<S2cg$d 4 a d7c b W Y T#j &U - (Ud T@a ] & a*d UU d m = a/b d U fc T 4 a U cqf U dYb &UU a 4 4 a U bd$ &U b7cf+dT$T&dYh0d θ Q $Xg & d U a U c $agYd(Ud)] $IT@dEhj f U d($+d7gkc'V l&U $+d2fcTaf c ]d U cd2g`T@a ag a U bd a/b2 = m2/a d Ud a 4 a U cced U h4g.d a d2g 4 $XW a-h0d U4 c Z a $yg '& g h*d d U hg$ah0d $uTf 4 d U $uTxf 4+* $uT@acd-, l g$d/. Z ca U cTf U d $yg Z0 4 '&Um f 4 j a]d U f 4 d1kf 4)m fPj _`g$d U hg$d $U fc T@d<T&a 0 &UU a 4 4 a U20 d6h0d m W d 0 a 4 T ] cdd 4 c Z h0d] ]'d U c & ^7ced U fDd U(a, b) = (0, 0)
l ]'a 4 T#j a $yg '& g d 4 cqaT & g 4 g Z hf c2_6f U d`]d 4 fg.d U+&U $U c Z,c g$aX^)T@d πJ (θ) ∝ θ−1 W
*)(3>/40R7(*)65@)87$<(5975@3:3;<3-N23215=3230(-5&1i<) a3$Xg '& g ;)(0)65&)>?(*1.%>4?7@5@>A3CB 7>4D5&)(B 5;<3 , 3`>/4D5&13`57(01$13CB$NX(*)(;136B<)(3 , 5&%E5973F; J a $yg '& g G5&)>?(*1.%>4?75x>/B!L . , N(414"!H7I32)+52(*13'N , <JBI5&):7.<5K7@5@>&;<36N1.(*NX(:B@321/NX(*<1
θ;<3CB%L3MB#<+1@3MB/<)5@G
>?(01$%E3CBLB$<1Θ l πJ A=74)87>4 , (*1BN; J 32)87.1$(0N5K3`%>44~<5@%>4 , 3^pkL;\O32N232)(;(4)870+M<)(3B$( , <<7@5&(*)PB 532~UNXAC;5975@3:3Q4-R9(0<<7@5K7
l 32)87$13F4<7$13CBpBS;<3MB $+a4g.a*h & (Ud 4 h*d $+a4gkc c '&UU d)]d U cZlT 4-BB U4:B@B@321.%>4) nv*v =p , (01B.:=< J (*)£5K(:5VB@597Y<)34*<<7$13 N(414%'A=7.1@5HB4?7@5&(*)
θ′++NX(*<1 , 3CB.:=<3 ,&, 3CB , 3CBO%E3MB#<13CB)3IB#(0)87N , <JBO(-R , 5&S:4?7.(-5&1@3!%E32)87O<)5@>?(*1.%L3MB l 5 , B$<¤E7
;<3E5K(-5HB@5@1θ′ = h(θ)
(18h) J 3CB 7ZNJ4-B , 5&)AC4-5@13p7L9 M(:5@1ZA2S84 , 32%E32)87 (-R2321@7 l to0o ,+ § m L NX(*<+1W;3CBzN1$A5=5HB 5&(*)JB
3X7Y;<3MB&32~3!%'N , 3MB&3!)¡>?14)+RC4D5HB.p * %'(-5&)(BY;<3Y5K(:5VB@5&1 <)3E<)5:=<3iNJ414%'A=7.1@5HB@4D7@5&(*)θ+ , 3Y5K(:5x~S; J <) πJ )(*)
5&)>?(*1.%>4?75x>A;(:597iN+1@3!)(;13>32)£57(0%'N<7@35X321@74-5@)3MBZ5&)=3?4*1@5H4)+5X3CBI;<¥N1$(:R , C2%E3F; J 3CB 7@5&%>4?7@5&(*)+3=7E5=3 , 4[; J 4<<74)87N , <(B :=< J 4<u8+)(4 , +5 J 3CB 7 %E(:5@)JB ,HJ 3CB 7@5&%>4?7@5&(*)uN(414%'AX7$15&:=<3E:0<(3 ,J 3CB 7@5&%E4D7@5&(*)D%>41.S-5&)(4 , 3 l B#<1 X pZ:=<5M)(0<(B5&)87$A213CBB@3*L\4'%E3MB#<13I;<3 :37wy132_<B
πJ (θ) ∝ det Iθ1/2 +9(18 Iθ 3CB 7 , 46%E4D7$15&5X3I; J 5@)>?(01$%>4D7@5&(*)[;<3 0]5HB#K3!1&;<%'(<;C , 3/57(0)(B 5H;A!1$A-+<3CB 7^4-5@)JB 5_5&)U3`4*1@5H4)873 :=<3 , :=<3IB#(:597 , 3/5K(-5@~;<3/N(4*14*%'A=7$15HB@4D7@5&(*)L
\ J 41@7@55 , 3+;<3 T 4:B@B a4-BB@321.%E4*) l n!v0v 0p 52(*)(B 7@5K7$<(36<)361.A7>?A!1@3!)+5=3%>4|b32<13>32) , 4¡%>4?75@C!1@3-+32)c5&)87$1.(G;<(5VB4)87d3X7Y32~N , 5:=<(4)87 , 3MB N15&)+5=5&N(4 , 3CBz1.C2S , 3CB>?(*1.%E3 ,&, 3MBd; J A , 55=5K74?75@(0)e3X7 , 3CBzN1.(-R , C2%E3CBY; J 5@)U3?415H4)+5X3I3=7Z;<3N(4*14:;(^~<3CB , 5@AMBTB ,J 3!%EN , (:52; J a $yg '& g +)+(*)+5&)>?(*1.%>4?75x>/B!L:fg7$<(;<5&A N(41q<)>7.1$CMB;S014*)(;)(0%hR13&; J 4<7@32<+1B0+ ,J 32%EGN , (-5O;<365=3MB/%E3CB$<1@3MBZB 31.A230C , 3'N1$A5=5V32<~DN9(0<11.A2S0< , 4*1@5HB@321W;<3>/4*R2(*)D(-R|b3575@3:3 ,HJ 5@)>?(*1.%E4D7@5&(*)u>?1$A:=<32)87@5HB 7@3>?(0<1$)(5K3NJ41 , 43U14D5HB 3!%hR , 4)95=3^;<3CBi;(0))AX3MB2L\O597.(*)(B]N(4*1j37~<3!%EN , 3Yk&41@7@5&S:4*) l n!v p7+i3!1$)(4*1;( l n!v v=p7+ i321$GS:321 i3!1$)(4*1;( l n!v*v=t*p<+ (-R2321@7 l to0on-+P\]KJ4NL m p7+ 3 ,@, )3!1 l n!v O +qnv*v0p7+]lj4*)S i321.S-3!1 l n!v0v p7+U_U3:321B#3:3X32) l n!v0v-m=p7+=N9(0<1 , 3!<15 , 41@7$AY3=7 , 32<+1qS0A2)A!14 , 5K7$A0L d4*)(BT<)3 (*N7@5:0<(3W; J 3MBb75@%>4?75@(0)RJ4^_*ACB@5V32))3-+U<)31.C2S , 3^; J A , 5&5X5974?75@(0)N1$A!3?4<<7j;<3 N(41 , 4 3?4 , 5V;597.AO>?1$A:=<32)87@5HB 7@3 :=< J 3 ,&, 3^5&)(;<(597B#<1 , 313!52(*<3U132%E32)87 a $ & 4 ced2g '& g l ]3C321Bqn!v +`nZK(:B$K <o-3!1Q|b3C3znv*v m + Y4?7@74 nZK(8B#K`n!v*v +"d4D7 74Ynv*v,+MnZK+(:B4 , n!v0v*vDRXp :^(*)E5X414*5=7$A!1@5HB 3πJ N(4*1 , 4E;A28+)5K7@5&(*)[B#<(5@3?4)873*L
pEqr M$PeM9NMOHLPEn*L Y & cXn = X1, . . . , X ∼ M(θ) WfY & c θn(α) T@d m f+a U c T@dDhj & g.h4g$d α a $ & 4 ced2g '& g @Q W d W
P (θ ≤ θn(α)|Xn) = α VXW a ]'d 4 fg$da$yg '& g πJ d 4 cEh cduhj & g$h4g$dDh*dsg$d 0 & f g$d)]'d U c i d7c U+& c Z d ( R h[j & g$hg$d i Qb0 & d7g.a c d ]a4c 0 [ $U0c $Xg '& g,V 4
Pθ (θ ≤ θn(α))︸ ︷︷ ︸sCtu=vw=vx9y xKz|(/t#~?`z9x#zH= P (θ ≤ θn(α)|Xn)︸ ︷︷ ︸sCtu=vw=vxy xKz|gvw@=xHC-
+ O(n−i/2).
g[; J 4<<7.1@3MBO7@321E3CB0+< J <<75KV5VB4?75K:;3πJ N23XE3=7A; J -R<73X5VA;<3MB$A!S-5V-(Bi;<35=:8J4-+5=3 a1$ &"4 ced7g '& g XN2:B#G
m − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* B#AM;4D87<(3?4DV5V;597.A> $A:=<3X87@5HB 7@3*L]\4[E3MB#<3
πJ N232<<7E4DV-B/W=7@3>&A2S:5975KE3CL3C87E5=:(B@5V;AC$AC365=:LE3>:5K> -E4D7@5-3TN<5HB.:0< J 3CKV3^ J 4NN2-@7@3 a $ & 4 cd2g '& g N(4:BN&<(Bj; J 5V> ->4?7@5V-':=<3^V3CBg;<:AC3CB0+*B
O(n−i/2)N.CCB!L
*A3I;A28(5K7@5V-IN&<(BT> :@E3XVV3W;3CB \q +5HBGeB4G5VB;<3W J 5VEN2-@7@3:=<3CKV3Z> -+5=7@5V-[; J 5K87.AX.W=7 a $ &"4 ced7g '& g l 3X7zN&<(BWB 3!<V3XE3X87
θp^3=7 N2*<Y J 5K'N2-@7@3Y:=<3CKV3N;5KE3XJB 5V-e;3
Θ+J3MBb7ZN.ACB@3X87$AC3h;4DJB Y4?7@74 l n!v0v 0p7LJ\ 4
5X4-4057$AC@5HB4?7@5V-DS*ACAX4DV3F;<3πJ 5X-EE3B@-&<<7@5V- ; J <3-A!:=<(4D7@5V- ;<5@wyAX3X87@5V3XVK3`NJ3C@E3X7L;<3V4s5=-((4"!H7@3
37~NV5&5X5973XE3X87O;(4D(Bg;3CBP5C4-BGB@5K'NV3CB l 3XNJ4D@7@557<V5K3CP:=<(4D(; dim Θ3MBb7PN23=75973pkL= G-5VOnZK8B#KJ4D l nv*v0vDR9pVN2*<
;<3CB32~<3X'NV3CB!L
( % .- - ( , - - "/-R & h3ST@dFd($ &U d U c d)T'W 4D(; dim Θ = 1
+πJ 3CB 7Q$AM; 5K7IB V4 E3CB 3 ;<3 :37w @3!_B l \q ; J -;<3Dt,+
3CK 3X3XB0+nv m pkL ] & J 5V'NJ: 73 3CKV3iN(4-4-'A=7@5HB@4D7@5V- θ ; J E;CXV3N; J AK3CKV3 l 4X0 aT@d<] & h*d)T p7+πJ(θ) ∝ θ−1 L
R & h3ST@d¡h0d )Ed ^7fhT$T'W l n!v0v p N(@0NJ8B 3 3A=7 ;<3F;<3MBIE3X5VKV3 BY\q NJ IV3+E;CXV3F;<3U3X5VR VHL\3>E3X5VVK3 '\q 3MBb7IV3 g$d d d2g$d U 0 d $yg '& g ;<3 i3C$S:3X O3C@J4D;< l n!v0v0t0p7+ 3> J :"!/4-BB 3>V3#K-5@~U; J (3K5$X4D%K5K3 l ; J &X-(;5975K:3XE3C:7kpA3C87@3hV3CBZN(4D4D'CX7@3CBA :L_\ J -;<@3N;<3N3X 3XE3X87d;'$2N23X(; ; K-5@~;3N(4D4D($X7@5VB4?75K:L¢A5V(B@5
πJ (η, β) ∝ (ηβ)−1 3CB 7d; J -;<@3tE>4-5VBπJ(µ, β) ∝ (µβ)−1 3MBb7d $M; 5K7)
V4LE3CB 3Z;<3-32w232_<B;+* -;<3En0L
R & h3ST@d _Dg 4)m f+d 4Q0 &U 0 fgkg.d U c 4 W \3CBLE3CB 3CBL- 5K'! -E4D7@5-3MBNB 74-(;4D;¥N2 EV3E<;,CK3Q;<3I 3C5KR KN; 5VB@3X87Z;<3CB a $ & 4 ced2g '& g 5K'N*N(@3MBzN2 E<;,CK3>;<3-=:'N.$X7@5K7@5V-U3X87@3' 3C5KR K l i3X.S-3X n!v0v m p0/\1* 7@5VV5VB4?75K:a;3+E3CB @3MBZ5VEN(@0N@3MBW3CB 7N;<:'N:B2=59733/a4DS n548B4 l to*o0o0pd-876X38723X(;4-87;$8:3XV37'79$ 3+B 7@4?72$8:-5V3E;<3#=:;b7 7@5V-a;+*=< $> '& > G: 5V! ->4?75?!@;A72 ZK3B;A7(4D4D(,X7@3;Y5V;3X87@5DCJ4DR(K3B;;+* QL<;,CK3Z;<3EX-(79$=7@5K7@5V-Q32~F72-(3X87@5V3XG/
et t t HJI'K(L q K-LNM.ONONP q K(Q q ONK rLo P q K(RSL o M9TVU q\3;W?4DV3 %;i;<3X3X; 3X$8:3X87 3XVK3CL3C87W72$;@3X87@3B;g5V 3C87Y; Iθ 3X7Z723XE3=7@7@3X87^;<-.&;+* 3XF;'$C; 5V3dH4
! -E3L;<3B;WE3; @3B;W;<3 -32w238[;5=-3; 7J:(;4D873;/+\O3X7 7@3>;$C>4D%%43N4-74DEE3X87]$=72$L3C87@3875;@37(4D d3<4D875;0+[: 73X43=7 4D;<5^t`_3_aBb0/ r c; 7'7Jd; 3+;-+4e:-5V
yn = (y1, . . . , yn)?4DV3 %;Z:R; 3C f$X3B;]7(4DE5
K3B; 3CKV3; r `4-K3 %;Z-cX3X; 2$X3B;xr = (x1, . . . , xr)
;<3;<3X;@5K7 $f(x|θ) 3=7L;<3#! :75K: ;<3+2$87(4- 75975K:
F (x|θ) / r eD73cn−r = (c1, . . . , cn−r)
K3B;X?4-K3 2;d;<3JX3X; 33/5]g* -h723 7]; 7'72f;@3X5 3hV367=3;2; ;;<3A=3C; @3W3; 7&4D$C4D7@:5K3d3X7X;79$X5?CJ3X 3Z;<3C;@5972$I;<3;^5V; 74-:7%;^;<3A=3C; @3 fc
+<-Q-7@3
δ(θ) =
∫
IR
fc(x) F (x|θ) dx.r 4-RJ 7@5K7I4DV-%;d4 ~S $B; 974?72;Y;<3B;Y7 4'$C-2,XE3;JaN3X7it+3X7Z;3;57372f;@5975K:;ZtF3=7EiS^V73 -3;Y3Cu¢A(37~<3;7'7 /+v Bj v*vdbk/l`qHLF+m 5af/onp q-q I'OUTgK q U < T W 'r W &ts > U+u ] $ & >kvw m0s w)T$TDw $ < > < ] Z vG> 48< v '&U θ l T < ](w 4ks > wyx3wXzw@y>2wa 4x s ] & x3STDw(w)($ &U w U v wT 0 w U-4ks > Z _xt> & v|ww 4 v
πJc (θ) ∝ θ−1
√δ(θ).
^@i'/aBb
EGFIHJHLK&M9NMOHP/t'/ &`s >`T < $ < > < ] Z vG> 48< v '&U λ x s ] & xXS)TDww($ &U w U v w)T l%&U & ^8v w U vY
W ( w U-4ks > wJx3wJv8a)$'wfb$9> &c > w 4)4 d δ(λ) = n−∑n−ri=1 exp(−λci)
"0W ( w U-4ks > w shU d& >#](w 4ks > [0, c∞] δ(λ) = 1 − 1 − exp(−λc∞)(λC∞)−1 W ( w U-4ks > wJx3wJv8a)$'wfb-bE δ(λ) = r/n w 4 v $U x Z $w U x <U vyx3w λ W &U20
πJc (λ) = πJ (λ) W
l`qJHF+m Ot/ Y & v γ T <c0 &U-4 v <U vwhx u + s TDw8> Qγ ' 0.57722 VXW2Y & w U v γ1 = π2/6 + γ2 − 2γ > 0
lγ2 =
−2(1− γ) < 0 W2i U U+& vw δ(µ, β) = δ2(µ, β) + [δ(µ, β) − 1](γ1 + γ2 logµ+ log2 µ
)+ π2/6− 1 WejT & > 4 T <]-w 4ks > wJx3wJzw@y>2wa 4 $ &ts >pT < $ < > < ] Z vG> 48< v '&U (µ, β) x s ] & xXS)TDw(x3w )w ^ s T$T 0 w U-4ks > Z _xt> & v|ww 4 v
πJc (µ, β) ∝ (µβ)−1
√δ(µ, β).
^@i/ tfb
EFuHLH K)M9NMOHLP]i'/&ts >`TDw ] & xXS)TDw(x3w )w ^ s T$T l%&U & ^8v w U v
W ( w U-4ks > wJx3wJv8a)$'wfb$9> &c > w 4)4 d δ(µ, β) = n−∑n−ri=1 exp(−µcβi )
"0W ( w U-4ks > w shU d& >#](w 4ks > [0, c∞] δ(µ, β) = 1 − (βµ1/β)−11 − Γ(β−1, µcβ∞) &\Γ(x, α) w 4 v T <
d&U20 v '&U cf< ] ] < $U 0)& ] $IT'S8vw W ( w U-4ks > wJx3wJv8a)$'wfb-bE δ(µ, β) = r/n W
3C 3B;d3X>4-2 3B;723 :3X87X7@36!/4-5973;]; WK3B;d2$; K74D72;Y;<3;57372f;@5975K:; t+3X7Ai/+Y4D.;WK3B;5C43;Y;<3X3X; 3F;<3+7 [F7233.A73::@3B; ;@5D!i3X7 5D! -E31+ 3 `4-K3 N;3 =3C; @3F5KCJ5K3 VV3F3X874 !H3 3 πJ
c (θ)3B;b7ZG*o< $9> '& > 1; 74-(;4-; πJ(θ)/ G4DI;'$0C(5K7@5V-+ 3-?4DV3 I;<3#=3C; @3>;<3E7 [F723 . .Y * 3B;b7O| 4->4D5o;W KV3E
5VC(5V33/\O:; 5o; 74D87Y3XeH4+2$879$=75975K: ;+* 3?4DV3 ] 5 =:@3; 7J:(; )V47 ;5V3X5VKV3E;$0!/4-5KVV4-=3N-R;@3X23$C31+X3=7 73A=3X.; 3W * 4`7'72-@7@3]7(4f;^;+* 5V! :@>4?75K: ; 7'7(D$CL3C874D5V3]; θ 3X ; ;^;<3EX3=7 73I;<-('$X3W: X3X; 2$X33/
3CK: &3<4-:75; w0v < T'W ^t`_3_aBb<+(G* 75KV5o;@4D7@5V-e;<3πJ
c
723XE3=7Z;<3N;<5VE5K 3Cd$8:f,X3XE3X87&G* 5KX3X@7@5K7 ;3; G* 3B;b75K>4D7@5V- ;<3
θ5o; j ) j 5;E;<3QG* 3C(7K:5&;+* 3QE3; 3Q;3 -32w238[;
πJ =H43;2; 5o 33/ G4D>4-5KVK3 %;+ 72 >V3E<;,XV3^37~72-3C87@5V3X + 3^:R; 3C ?4D7@5V-(=3X.; 2$X3y; ¤E7W)Y-R<73X5V < $ &"4 vw8> '& > 7(@f7@3A^/3=7j;<: 7@5VV5;4DRV372 GG* 3; 7@[email protected]<+3=3 5<* 3B;b717(4f;GV3YC43;G;<3
πJ /gCJ6+πJ
c
;@3^ $3,XV3Z7Jd; ; $C;<3C EE3X5VKV3 j@3B= 83XE3C:7! 2$ 3C:75; 7@3(< $ &"4 vw8> '& > 3 πJ +' 4D(;+K357J %=3X874`:-3Y;<3]=3X.; 3d4 :-E3C:73Y;(4D;Og* $%4J4D87@5VKV-/ J* 3!/4=- :3$C'$X4DV31+V3;>4 7@3 2;E3X>4D% 3X87- 3QV4U: 75o; 3Q3X X-(7<7@3[;<3B;-=3C; @3B;-)U;@:5973;4-;
πJ
!/4e:-5;@3< $ & 4 v|w8> '& > ;<3B;y?4DV3 %;i-R;@3X2?4DRV3;Z$XV38f$X3B;8/+3CK:QV4(=:(4-5;2;@4-=3] 3ZG* :4>; 7=3B; ; ;;<3X3X; 31+85K_3B;b7A;<:&5V87 $C@3B; ;4D87i; * 75KV5o; 3CA;<3Y7@3Xo;A< $9> '& > -5K! :@>4D7@5D!@;+<3C7(4- 75 K5V3X^;4-; X4-;@3CJ4-R5VK5o;b733/8\3 j3X(7V-5J3; 7W=37J3C(;4-:7W7J3 >4D5H4DRV36^/--723X;(7(4DO37~<3X(7V3AV3;17(@f7@5$=72$;g;3=:4| :84D5o; :72 ^V3WE;',XV3Z37~7J:3X875K3Cob<+D74DEE3X87A;4D.;K3EC43;TU3X5VR Vg/
t t t WM.TgK < $ &"4 vw8> '& > R & xXS)TDw(w($ &U w U v w)T'W ] ^V3YE<;,CK3Z32~F72-(3X87@5V3X +
πJ (θ|yn)3; 7^37~7V5X5973A7(4DX-4| :84D5o; : /¢A5V;@5
λ|yn ∼ G(r,
n∑
i=1
yi
)
-8r
3B;b7NK3+(-hR(@3F;3F;<-('$X3B;I:c=3X.; 2$X3B;#^@ 3CZ 3 ; :597hV3#7=3B; ; ;I;<3 =3C; @3ebk/ πJ(λ|yn)3; 7
737(@3X;@5r ≥ 1
/<\1* 3; 7.$C4-=3(< $ & 4 vw8> '& > X [λ|yn]3B;b7Y$8:84DV3d4 S\
λn = r/∑n
i=1 yi^ G* 3; 7.$C4-=3W;<3&H4
i − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* 7(4D4D($X7@5;4?75K:
η * $=74D87&;$8C(5V3E 3];@5 r > 1
b0/*AQ5V:73X2?4DVK3WR5VV4D7 $C4- ;3Z5D:3C4 α ^;<5K7&;<3F0 > Z x ^ T v Z2b 5;$X V3I;<3WH4NV-5Z< $ &"4 vw8> '& > 3B;b7
χ2
(2r),α/2
2n∑
i=1
yi
,χ2
(2r),1−α/2
2n∑
i=1
yi
5 =:@3; 72-(;e7@2,;Z32~4373XE3X87E)QG* 5V87@3X2?4DVK3+;3=:CJ4D.=3! $B 3X87@5o; 7@3>3X C43;I;<3#X3X; 3+;<3E7 [F7J33. .^V:-5V\W4J4`7 / t!+7 /.i ,+F! :@ V3/t'/ tfbk/
R & x3STDw-xfw )w ^ s T$T'W J* 4372,; c^ a!v0v b7+;@5πJ 3; 7&V3 > w d w0>2w U 0 w $> '& > + πJ(µ, β|yn)
3; 7X737(@3A;@5r ≥ 23=7hV3;
yi > 1/*A3 45D$C4-2%4(5K3>3C:7@3-7J4D4D(,=7@3B;E;8* $=74DRV5K7N;<3-!/4=-cJ4?7 @3CKV33/g 37w 3=79+
πJ(µ, β|yn) =
πJ(µ|β,yn) πJ (β|yn)18
µ|β,yn ∼ G(r,
n∑
i=1
yβi
),
^@i/ idb
πJ (β|yn) ∝βr−2
(r∏
i=1
xi
)β ( n∑i=1
yβi
)−r
∫∞0 βr−2
(r∏
i=1
xi
)β ( n∑i=1
yβi
)−r
dβ
.^@i/ b
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m3X7β
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r 0+?:R<7@3C5KG32~7K5o=5K7@3CE3X87π(η, β)
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X-EE3 3ZV-5 =:(;<5K7@5V-(3XVK3
π(η, β) = π(η|β) π(β).
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π(η|β) +<3XQ3873XJ4D87iV3;A5V;$C3;^;<3i3C ::3X "^a!v0v3ifb0/
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& DV
t t t DI o M Q q L q I'Q I n rWM rLo q I o IdU oq L q M o qβ
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β/
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+ 3=7A=3XH4#72 W;3 S4D5o; :;75V=57(4DV3;/
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F / b U x s 4 vg> w)T$TDw)](w U v $ < >#T <U v l T u Z0%< >kvw U vg>2w βl = 1 w0v βr = 5 $'w s v0vG> w $'w8>! se0 & ] ](w shU#" $Ug- U" W$ U β $> &C0 ['wx3w 5
0 & >k> w 4 $ &U x _ shU w 4 v s< v '&U3&\ TDw w T$T 4 4 w]-w U v w 4 v 4 <:00Z T Z > ZGm8s wT u $U x s 4 vG> wTw 4 v0 &U d > &U v Z _ shU w < e< T <U 0 [wxfw-x % 00s T v Z4<m8s > w U xfw U v2T u w($uT & v < v '&U ] $ & 4 4 ^)TDw W'& U $9> < v m8s w l β ' 3 w 4 vd& >kvw)](w U v ] $9> & ^ < ^)TDw#w8v $IT s 4 4)( > w)](w U v l β Z & T s w c3Z)UuZ > < Tw]-w U vx <U4 [1, 2.5] W w 02< 4
β ' 10 & >k>2w 4 $ &U x
< $U-4 _ shU$U x s 4 vG> wT m8s N<E0 & >k> w 0 vw)](w U v1wgXw 0 v sZ 4&U vg> < e< T x3wf] < $U v|w U9<U 0 w l w UDU wT < 4 40<U v+$ <"41 w T$T >TDw ] < v Z > w)T'W*$ U wEx % 08s T v Z ] < 18w s > wJxfw 4-W< ^ T 4 v|w 4 w 4 vZx &U20 x3w $9> Z & >`T u < $$ < > v '&U x3w 0 w w T$T 4)4 w)](w U v l0 & ] ]-w T u w)($uT m8s w U v <UU+& aw0v < T'W Q " $ $+5VXW
¢&5K;@5 +: ; ; 7'72f;@3X-.;7@ b %; 72 :-5VW%4:5;@5V (βl, βr)/ 43%4(4#^a fbN7(@f7Jd; 3A4-K:2;g; * 75KV5o; 3C (3AK:5
=74(; [βl, βr]
β ∼ Be(p, q)
;<3Z;<3C;@5972$
π(β) =Γ(p+ q)
Γ(p)Γ(q)
(β − βl)p−1(βr − β)q−1
(βr − βl)p+q−1 , [βl,βr](β)
C4DI3CKV3+3; 7 3<5KRV3F4 ; 3C;I-8c3XVK3#72f;2; ,C;<3#; ¤#;@4-EL3C87N;+* 4d[F723X27(4D4D(,X7@3;]72 =7@3C4DV5KR2$X3#)QH4! :5;h3Xc73XL3B;N;3?4D5V4-=3>3X7L;<3 ?4DV3 (=3C:74-K3-+5V(;$8723XJ;4DEE3X87>; = 7V3 (βl, βr)
+ 737(@5$=72$ 55V87 $X3;2;@3^7@ 7W7(4- 75 V5D,C@3CL3C87jg* 4DJ4D[F; 7@3&5K(; ;b7@5V3X V-%; * 5VJ;$B; 5V@3&L<;<5DC(3CO$8:f,X3XE3X87W;4ZV-5 < $9> '& > =/-g72$XE-5:-3WH4LE4-5H4DR5VK5K7 $I;3];@3;E:L3C872;^3X7^;3E; :E<;<32
-[β] = βl +
1
1 + q/p(βr − βl) ,
.d[β] = βl +
1
1 + (q − 1)/(p− 1)(βr − βl)
;@5p, q > 1,
g4-[β] =
pq
(p+ q + 1)(p+ q)2(βr − βl)
2.
?i − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* d4-;IV3C43;A7(4- 75 V5K3Ch18
p = q = 1+=3X7 73FV-5j3B;b7 5D! -L3f/ G4DL4D5VKV3 2;0+ Eπ[β]
3F;'$8723X(;c 3; 437'72-@7
q/p3=76723 7h@3B;b73X6X-; 74-87@3E5KJ;$8723X(;(4DEE3X87L;+* 3FL<;<5DC.C4?7@5V-c;<3FV4 ?4-@5H4DX3 ^/;<3+ CL3
7J ZV3>L<;<3# 5j3+;$8723XJ; 3E; 437'7J: 7(q − 1)/(p − 1)
bk/ -jC(6+ K3B;E 3Xo 3;5C: 3;Ai'/a-) i'/a_5KV ;b7@3C87dG* $-3X874D5V];<3B;! :@E3B; 3NV4;<3X;@5K7 $7J3 7Y3C(7 :73X(^ :&%4:5;@597Y5X5
[βl, βr] = [1, 5]bk/AD7-;
3WK3EC43;y= 4D87; * 5V3X5VKV5;2;@3XE3X87^D$:-3C^3; 7E<;$CK5o;$A7J4D %4-5 1 < p < q
^gk! / 0]5: 3Ai'/a3aebk/
GF ,/< $> & $> Z v Z x3w7T &cXg 0 &U 02< v Z xfwT < T & ! 8v < $w8>#](w8vx u < $$uT m0s w8>7T u < T c & > v$[h](w5x3w ) ^ ^ 4 $9> &g$ & 4)Z $ < > ! w0> c w8> Y shU Q V-x <U4 TDw 0%<"4 $ & T a g )w ^ s T$T$ &`s >5w 4 v ]-w0> < $ &"4 v|w0> '& > Tw 4 βi
xfw i = 1, . . . , Q] & xXS)TDw 4 xfw )(w ^ s T$T] 4 w U 0 & ] $ Z v v '&U W& U w@Xw8v l TDw 4 x 4 vg> ^ s v '&U-4 < $ &"4 vw8> '& > 4&U v Z,cd< TDw)](w U v T &cXg0 &U20%< w 45li0 w m8s $'w8>#]-w0v2T u s v T 48< v '&U x3w d&U 0 v '&U4 w U w)T & $ $ <U v|w 4 w0vT u <-005Z T Z > < v '&U x3w T u < T c & > v$[h](w W & >TDw 0 [ < $ vG> w +1$ &ts > shU w6x Z- U v '&U 4ks(00$U v|w6x3w T u < T c & > vO[ ]-w-x3w )% ^ ^ 4 W
-3Co5K;@- ^ _f_ b]7372f;@3 65VF-3C:74D5V@3+7@2,;=-(7V3=7>;<3B;N;<5$X3X87@3B;h73%4(5 3B;N;<3 ;$CK3B75K:6;<3;4F[d723X27(4-4-,X7@3;d;<3EV4K:5 X74 +23C&7J4D@7@5o V5K3CWK:2;2 3N;<3B;]?4DV3 %;d;3
β-87]$=72$L:R;@3X23$X3B;& 3X-; j
7@ 5K7@3;J^V7t-3C(4D877(4-A3<3C7(K3h;3h2$8:-3;2;@5K:;Y; Y;<3;&R(43;@3;&;<3N;<:'$C3; 2--5V7(4-&33C(7K3N\4D(t[ gFC43=5H4 ^ _3_ b^ V3(; 5K7@3 [FvGv $ W$^ < >k> $U0c w0> W 0 & ] bk/43%4(4&^ a +.7 / j. bY! @5K7WK3B;]:-4D(;<3B;K5:-(3;Y;+* 3E$X7 4<;<:Kf:-5V3N;<3(X4-K5VR4?75K: ;8* 4`77 [:4-875; dH4 ;79$X5?C.C4?75K: ;+* 3?4DV3 βm
%4:5;@5V367(4-;$0!/4 7X=:EL3WV3ZE5KV5K3 ;<3
[βl, βr]/
cP T@M o I'UT@M9O tA ;O3&3=74=-.; 7(43;O5o=5 V4h($=7 4(;-V3:-5V3&;<3 43%4(4W>4-5;Z; :3:3$C@:;g;+* [+4 b 7@3C (3 3; 7@5V-A723XE3=7 74D87 ;<3Z;$CK3B75K:3X
βm5K(;5K3=7@3XE3C:79+CV3;3723X@72; 72 `4-87];<5#=5VK3CL3C87 $7J:(;<3 ) (3
5K873X@f::4?75K:#72-@74D87&;<5V373XE3X87X; AV3A7(4D4D(,X7@33/ &3C>4D(;<:;A4 3F723X@72;AK3B;Y7@:R(4DR(5KV5972$;p0
3=7p17J ∑ ;<3j7@-NRJ3C3CE7(4D(3O4e?4D87V3;7@3C7.;3; 7J3B75?!@;
x03=7x1+`4e-3B
x1 > x0/AV-%;
pi = 1−exp(−µxβi )7J i = 0, 1
/A5V; 5
1 − p1
1 − p0= exp
−µxβ
0
(x1
x0
β
− 1
)= (1 − p0)
“
x1x0
”β−1.
3C[;$C; 597
βm = log
log(1 − p1)
log(1 − p0)
log−1(x1/x0).
^@i/ b-g %4:5;@5o; ;4D87#7 ;@5K3 2; = 7V3; (x0, x1)
+i: -R<75K3C87 X3X@74-5K :hR3e;<3&?4DV3 %;#723XE3=7 74D87;3=:! -@7@3C& ;<3h3=7@5DC(3X
[βl, βr]3=7βm
/d5; j ) j 85o;&;3 i5V3X(43o5 w8v < T'W ^ a -m +7 /a b 5N7372f;@3X870+ ;4D.;=3X7 7@3 XE3i37<75 3-+D;<3;<3X>4DJ;<3X L437'72-@7];<3^;<3 E?4DV3 %; (hW (x0), hW (x1))
; 74 h;<3^;$8!/4D5VKH4DX3;<3 ∑ 3X7]3CN;$M; 5o; 3C:7 βm
+t=3O7 [d723;+* 5V87@3C@3:84?7@5V-]7(4-4 !V7N7 ;4-;437<7 $ZX4D 5Kd!/4-597G4`7'723X) 3Y=-((4D5o; ;4D.=37(43;2; $X3-R;@3X2?4DRV33/ J* 4-5KVV3 %;+ #;79$X5V4-K5o;b73^; ;<->4D5V3y=:C(E3^g* 5K.X4`7J43=5K7 $>4D(5?! 3B;b73A;<3;G37J3C 7%;G; =$C4-5K3)I2$872-(;@3)hV4E 3B;b75K:F;<3 i5V3X(4D%o5 w8v < T'W ^g=:d:3X%;@4D7@5V-75Df$X3&4e:3&(;<2$&\4-t[1+ _f_ fb0/\O-EE3ZK3E;@ K5:-36\OH4DD7@7@51^ a :mfbW 5 7372f;@3] 4-87X)N 5 3Z5V:73X@f::4D7@5V-;@ ;y! -L3Z;<3 $ < > 4A^ :;<3X>4-(;<3X)ZG* 37J3C 7Z; :-7J 2X3X8743:-3A;+* 3X3 gV-%;2 * 5V'72$C;<5K7i;3;1::4-(;<3 2;j;<3AH4I; 2$X3&;3X5K3eb7+:g* 4`7'723X7-R(4-R5KR(5KV5; 7@3A)LH4X-J4D5o; ;4DX35; Rb3B75D:3Y2$C; 597^V3;^5o; 3B;i;3ZE4-D4(- X7@3X7 $W5K873XVK3B7 3XVK3f/
&
DV
beta
densite
12
34
5
0.6 0.8 1.0 1.2 1.4
Mi/Da
(p,q)
=(1,1
) /
beta
densite
12
34
5
0.0 0.2 0.4 0.6 0.8 1.0 1.2
M9i'/
(p,q)
=(1.5,1.5
) /
beta
densite
12
34
5
0.0 0.5 1.0 1.5 2.0
Mi/i
(p,q)
=(1,2
) /beta
densite
12
34
5
0.0 0.5 1.0 1.5 2.0M9i'/
(p,q)
=(3.5,1.5
) /
beta
densite
12
34
5
0.0 0.5 1.0 1.5
2Mi'/
(p,q)
=(1.5,2
) /beta
densite
12
34
5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
M9i/
(p,q)
=(1
0,1
0) /
beta
densite
12
34
5
0.0 0.5 1.0 1.5 2.0 2.5
Mi/
(p,q)
=(2,5
) /beta
densite
12
34
5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
2Mi'/m
(p,q)
=(3.5,1
0) /
beta
densite
12
34
5
0.0 0.5 1.0 1.5 2.0
Mi/
(p,q)
=(2,1
) /beta
densite
12
34
5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
M9i/DaB_
(p,q)
=(1
0,5
) /
?i − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4*
beta
dens
ite
1 2 3 4 5
0.0
0.5
1.0
1.5
beta
dens
ite
1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
beta
dens
ite
1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
2.5
beta
dens
ite
1 2 3 4 5
01
23
4
M .i/Dafa 0(-E3;i7 [F75o 3;^;<3Z;<3C; 5K7 $< $9> '& > +7@55K$8::5V4-87 !/4-5KRV3A5K3C5KVV5;2; 3CE3X87/
beta
dens
ite
1 2 3 4 5
01
23
4
beta
dens
ite
1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
beta
dens
ite
1 2 3 4 5
05
1015
2025
30
beta
dens
ite
1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
M i/Da 30(-L3B;i7 [F75 3B;^;<3I;<3C; 5K7 $#< $9> '& > _5VX34'$C@3C:73;4e-3B&V3E%4:5 ;3βr/
( & [ Z > w U20 wExfw π(β) W 5 π 3B;b7Z7 ;j5V! :@>4?75D:3X * 3&V-5 5D! -E3; [βl, βr]+8-;<-5K7i4e:-5V
π′(β) < 0^@;@-5K7π(β)
;'$=-5o;2;@4-:73Bb 4D(; β → βr/ 5V-6+_V3;E`4-K3 %;]7V4 ;@5KRV3;Z;3
β;* $=7@3C(;<3C:7h4 j ;<3Xo)Q;<3
βr/
G4DI4D5VVK3 2;0+ 3;$X@:5;2;@4-=3N7@2,;IR 74DV3E3Xc;<3B;A?4DV3 %;A4(4 7@3B;I;<3β
3; 76723 4-5;@3XNRV4-RV33/ J* (3>4D5,X3#:3$C'$X4DV31+5KO3; 7N'$B=3;2;4D5V@3+;<33$X5DC(3X 3 π(β)
3#!/4e:-5;@3(7(43;hV4SRJ:@3 ; 79$X5V3 3 βr^V:-5V
C: @3Xi/Da b0/ b 7-;Z *o< $ & 4 v|w8> '& > 2+ 35X4-4f7 $C@5o;4?7@5V--;@5KE5VV4-5K3d;3π(β|yn)
5V(;<5o 3 3 βr3; 7O>4D
%4-5o;@5K3f/-g'C(6+?V-%; 3 n 4 ::L3C87@31+ .S\ -βn /∈ [βl, βr]
5K(;5 3 π(β)5KX34'$C@3C87]4e-3BgG* 5V! :@>4?75K:
4`7'72-@7 $C3]7(4Dyn
3=7 π(β|yn)E3X(4f0$Z;+* $U20 &U4 4 v <U 0 w ^V-:5K § i/ /Daf/Daebk/
& DV _
t t t cP T@M o IUTgM.O&L q DI o M Q q L q q0ouq3o qer O n r M rLo q I o IdU oq L DPFQ q q ,B;y:3$C'$X4DV3XE3X879+K3B;A4DJ4D[F; 7@3B;^R(4e[f$;@5K3C;y; 7'7Jd; 3C87J< $> '& > 2+7J &;<3B;A4D5o; :;^;<36;@5V7(K5o=5K7 $-+. 3
V3; 7J4D4D(,=7@3B;g;<3U3X5VR V.;@-87j5V(;$7J3C(;4D87%;8/ .#'!/4 7i4?7@7@3C(;<3O3C ::3X F &^ a3idb72 j-R7@3X(5K9+:;4-; "X4:;<3J72-[ j U3X5VR V + 3-=:;b7 7@5V-a;$CK5VR.$C $CE3X87E=-J;<5975K:3CKV33/\3;Z;<3 4 7@3 2;Y3-72$X5;@3X877(4f;O;3]%4(-5 (7J4D@7@5o V5K3C^;<3
π(β)>4D5o;W7(@f7Jd; 3C:79+f72 η +;+* 75KV5;@3X (3 T & ecf< ] ] < $U w8> 4 w c0ZUIZ > < T 4)Z w /
pEqr M$PeM9NMOHLP / ∀(a, β, b) ∈ (IR∗+ × IR∗+ × IR∗+)l ∀x ∈ IR
l&U x Z- U vkT < T & ecf< ] ] < $U w8> 4 w c3Z)UuZ > < T 4)Z wQ GIG V1$ < > 40< x3w U-4 v Z
f(x|a, b, β) =baβ
Γ(a)
1
xaβ+1exp
(− b
xβ
)1[0;+∞[(x)
^@i/ db
Y &U ] & ]-w U vZx u & > x`> w k w 4 v
& [Xk] =bk/βΓ(a− k/β)
Γ(a)∀ aβ > k > 0,
Y &U ] & xfwJw 4 vWv &ts 1 &`s > 4 x Z- U w8v t<ts v Rd[X ] = ( bβ
aβ+1)1/β W
\O3=7@7@3AV-5N< $> '& > 4`77(4D4"!H71=-EE3 > 7@5V(;<3;@5K(7V5DC.X4D7@5V->;<3AG* 5V!V$C@3C=3f/ -g>3 3=79+-H4Z;<3X.; 5K7 $E^gi'/ fb3B;b7A=: : $X3JX-(;5975K:3XE3C:7E)
β^@7@f7Jd; 5K7@5V- b&3=7Z5K ; E7I;<-L;<3(=:(4"!H7@3h ;+* 4`77@%43C7J4D
;<3B;&Et[:3X;& $C@5o 3;&K3(=:(7J: 73XE3X87(< $ &"4 v|w0> '& > ;<3β72 Y-R7@3X(5KYg* 3X.; 3ChRV3N;<3B;&5V! :@>4?75K:;
; &V3;Y;<3 7(4D4D(,=7@3B;8/ G4Dd4-5KVV3 %;+ S73X %4-5 7J3C@E3=7Z;+* 5DC(3XYV3I7@4e?4D5V ;<3=:;b7 7@5V-S4e:3ZV3C43;g37J:3X87@5V3X218
β3; 7ZC$)#af/-jC(6+'X3=7 73&V-5 3; 7 ! 3C@($C3X7(4-j74-; ! :@>4D7@5V->;+* $%43CKV35X-(;5975K:3X j
V3XE3X87A)β^V7(@f7Jd; 5K7@5V- b7++X3( 5H4@3C(;S5V87 $X3;2;4D87@3J7J WL<;$CK5o;@3XZ;<3;Y5V! ->4?75K:;-< $9> '& > ;
η 7@ 7@3N@37(4D4D($=7@5o;4?7@5V-[;<3
η=:(;<5K7@5V-3CKV3J)
β++=:LE3NV4($C;<5H4D3-+JG* 3; 7.$C4-=3N V3hE<;<3E;<3
H4hV-5_;<3U3X5VR Vg/Y4D.;OH4; 5K7@3Z;<3]=3A%4(4375K7@31+8 ;i>4D(5K3C@:; 7( $8!V$X3X87@5V3XVK3CE3X87π(µ|β)
-8µ = η−β +
7 ;y;@5K(7V3I;+* 7@5VV5;4?75K: /
EFuHLH K)M9NMOHLP / Y & v η|β ∼ GIG(a, β, b) WY & v yn
shU Z0 [ <U v T$T &U 0 &U v|w U9<U v r x &UJUuZ w 4 U+&U60 w U-4ks > Z w 4xfw )(w ^ s T$T w8v n− r
t< TDw s > 4 x3w 0 w U4%s >2w - ( Z w 4 W & U U+& v <U v µ = η−β l
µ|β ∼ G(a, b),
µ|β,yn ∼ G(a+ r, b+
n∑
i=1
yβi
),
η|β,yn ∼ GIG(a+ r, β, b+
n∑
i=1
yβi
)
npoq rsuqut ] 7@ 7 t ≥ 0+
P (µ < t) = 1 − P (η < t−1/β)
= 1 −∫ t−1/β
0
baβ
Γ(a)
1
xaβ+1exp(−b/xβ) dx
= 1 −∫ ∞
t
baβ
Γ(a)ua−1 exp(−bu) du 4e-3B
u = x−β /
EFuHLH K)M9NMOHLP / Y & v X ∼ GIG(a, β, b) WejT & > 4 ∀c > 0lcX ∼ GIG(a, β, bcβ) W
9 ?i − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* npoq0rLsJqut 7@ 7 t > 0
+
P (cX ≤ t) =
∫ t/c
0
baβ
Γ(a)
1
xaβ+1exp
(− b
xβ
)dx,
=
∫ t
0
bacaββ
Γ(a)
1
xaβ+1exp
(−bc
β
uβ
)du
18u = cx.
cP T@M o I'UT@M9ONK t Y4D.;gG* $X7 ;<3Y;<3 i3X2:-3C F h^ a fifb7+-V4A; $XV3=7@5V-+;<3B;14F[F7J3C 7(4-4-(,=7@3;]3; 7@3X;@3 K3 jE3X87W4 ; 74-;3h724'$X:@5o 33/ \1* 3723X@7@5o; 3N3; 7]; @7@ 75; 7'72f; $X367J4D4D($=7@5o 3 ^g=-874-5K87@3B;X; θ = (η, β)
b3=7+V3;($B 4?7@5V-.;E5K87 $:-4DV3;()U $B; ;<3 ;@-87>:hR3 ;@3;/ G4D+4-5KVK3 %;+ =3B;+4 7@3 2;E3 7372f;@3X87-7(4f; < $9> '& > 2$X3CKV3XE3X87E=:(;<5K7@5V-(3X 7 5o; 3 π(µ|β) ≡ π(µ)
/ \O3%4:5 S3; 7@3E;<:N7@2,;W4D@7@5DC.=5V3XG/ d4-; 7(@3CL5V3Xd7@4e?4D5V +J ;d4e:-; 7@5VK5o; $X3=7 73NE<;$CK5o;@4D7@5V-S3X757(@f7Jd;$ 3J5o; 5V-&; =5V73L; ),-V3h;<3;4F[d723X27(4-4-,X7@3;y^ O ;2 3=7 _3_ bk/F\O38723X(;4-870+8K3B; 737(@5$=72$;1; 7@ 7 3XVK3B;j;<3
π5K(;5 3C:7Y 3&V3%4:5
;+* 4F[F723X27(4D4D(,=7@3b 5 ;@-5K7 35! -=7@5V-Q;$=73XL5V5o;b73Z;<3
β7 7),D7 * 3]`4-K3 y=:;b74D87@3Y2$; 973
;+* 3&37J3C 75;@3X34'$C@3C:73&4e-3B^V3&E<;,XV3 W(θ)3=7^4D($CK5V-3AH4ZE;'$XV5;4?75K:+3XF;'$0C(5o;2;@4-:7
aX-EE3
4F[d723X27(4-4-,X7@3d;<3EC4DV5KR(4D7@5V- /
-g 3 23X70+y; 7'72f;@-; 7J -:5KE<;$XV5;@3X 4e85o;;+* 37J3C 7 7(4D $%4(4-87@5VKV- C C.75?!a;<3a;<-('$X3B;-R;@3X2?4DRV3;^ : X3X; 2$X3;%bg;3 U3X5VR V
xm = (x1, . . . , xm);3dEt[-3C3
xe/&K:2;0+8; * 437 ,B;jG* 373;2; 5V-
^@i/ idb7+<-Q4
π(θ′|β) = πJ (θ′|β, xm)
18πJ 3; 7&V3 > w d w0>2w U 0 w $9> '& > 7J &V3ZE<;,XV3h;<3 U3X5VR V3X7
θ′ * 5V(7J: 736 3X %4-5 ;<3A7J4D4D($=7@5o;@4D7@5V-
η, λ, µ /&5K.; 5 +
a = m,
b = b(a, β) =a∑
i=1
xβi .
\3E%4:5 b = b(a, β)
4`7'7J4D4"!H7^;-W;<3A!/4=:(4D7 3XVV33/ 5 β 3B;b7XX- 3=7?4 7Ea#^ E<;,XV3Z372-3C87@5V3X=b7+-Q-R7@5V3X87
b = axe/(\1* 4F[F7J3C 7(4-4-(,=7@3
a;8* 5V:73X272,=7@3AX-EE3ZV4L74D5VKV3I;<3ZG* $B%4(4D87@5VVK:C.=7@5D! /J\ * 4-(4D[;b73
R(4e[3$B; 5V3X 7J3 7&;<:Z43=-;<3CZ;4(=-'CJ4DX3W3X 3E=3C 74D5V3W3723X@7@5o; 3A 4D875974?7@5-3W3CE<; H4D87 a /
.b;$C4-K3CL3C870+H4EK:5 < $> '& > 4:;<37<73 3E; 7@ 7 @3A4(5D$C4-2%45o 33/ -g ; 7'72f;4D87xm
i.i.d.∼ W(ηe, βe)18
ηe3; 7 3Z3; 7@5V>4?75K:"< $9> '& > _;<3
η+-Q4,+';43%4J4D87
π(β)+
b|β ∼ G(a, η−β
e
),
µ|β, b ∼ G(a, b).
\O38723X(;(4D870+F=3X7 7@3YE<;$XV5;4?75K:+3; 7K ;3X)I>4D(5K3C0+83=7K3]%4:5 >;+* b ;'$=7@3C@E5V5o;b73d3C#! -.7@5V-;<3a
3=7β
3; 7X; 4(4D5K74-RV33/ 3CK:G* 37J3C 75;@3I;<5o;72-(5KRV31+J ;Y7@f7Jd; :;^;4-;V4-;@ ; j 7(4D@7@5V3]; 5?4D87@3I;3;%4:5 ;<3b^V7(@f7Jd; 5K7@5V- db 5 C E5KE3X87&K3A=:(7J: 73XE3X87i; * (3X! :75K: β 7→
a∑i=1
xβi
/ +-7@3X4E7J4DiV4; 5973
BH4h;<5o; 7@5KR 7@5V-F;<3
b = b(β)^a$=74D87Z; 7'72f; $C F$ebk/ A5V;@5 +F72 j7@ 7Z$3,X(3XE3X87
A;<3 A ;4D;OG* 3B;7(4f=3
7-R(4-R5KV5o;$W;<3;^;-'$C3;E^gk! / § af/ b7+ P (A|B); 3C4NV47-RJ4DR5VK5K7 $Z;<3
A;43%4J4D87iV3E%4:5
b/
& DV
* (X ' - #-% -6!' S-&( 'b" -A ;g( ;j5K87 $C@3B; ;@-.;G5X5J4 %4-5 L;3
b = b(a, β)3C(! -=7@5V-F;<3&g* 3723X@7@5o; 3X 4-87@5K74?75D:3&;<5o;72-(5KRV31+
72-@74-:7]; dH4?4-@5H4DRV3h:R;@3X2`4-RV3X/ &3E:hR3 [4 7@3 %;; 7'72f;@3X875 * U3723X@7];@367-(-X3N;<5 j
3=7@3XE3C:7 ; OK37J4D4D(,=7@3^;+* $%43CKV3η
;<3B; ! -.7@5V-.;j;+* 5K872$X =7O;<3η/ G-5V 7(4-g3<3X(7V3 43%4J4^a fb<+
\5 b:5Y^ _f_3ifb& 5;2;@3 w8v < T'W ^ _f_ b0/+\O3- 5G3B;b7ZV-5V ;+* =7@3($85H;<3X87A)-R<73X5V0+7 5;2 3Eg* 3723X@7Y * 43X:f$X'$C4-+7(43;A;3A! ->4?7@5V- ; 74D7@5o;b75 3Z3X7&36X-(4 !V77J43;AV3ZE<;,XV3 M(θ) /.# ;* 4`:-5K77 ;;#3XE3X87d;+* 4e5o;Cb7@4D.; ! :@($CA7(4DG* 4-(4D[;b73YRJ4e[3$;@5V3X /
\1* 3723X@7@5o; 35 4D87@5K74D7@5-3d3; 7i3CF2$C4DV5K7 $7 7),D7^>4D2:-5V(4DV3 : F3F723X@7Y7J3C X-5K7OH4N; $C3d;<355K3X
3=73X;<:3 3i3; 7@5V>4?7@5V-
xe/ Y4D.; G* 5K(; ;b7@5V31+tX3X@74D5V;4DJ4D[F; 7@3B;:-5V3X87 3C
xe (3O3B;b75K>4?75K:672-=7 3CKV3^;<3
η+'4:E3:3,C3A)
X/::5;2;4D87y7(4D4J4DR5K7 ;31+X3;A4-(4D[F; 7@3B;Y7; 5;@3X87 [R5H4D5o;A;<3ZE<;$CK5o;@4D7@5V-Q;+* 4 74D87
7 ;^5V(7J: 74D87X 3IG* 5VX3X@7@5K7 ;<3J; β 3; 7:-4D(;<3f/9\O3=7@7@36X-! ; 5V- ! 2$ 3X873I;<-5K7 =7@3EX-hRJ4?7 7 33/2\4?4DV3 xe
3; 7 3A:-4DJ;<3 yC4D4372$X5;4D87iV4E;<5o;b7@5VR 75K:>4- ::5K(4-K3M
;<3Z;<3X;@5K7 $
m(x) =
∫
Θ
π(θ)fW (x|θ) dθ.
\ 4 ;79$X5?C.C4?75K: ;<3m
3C87@4"!|3F4DV-%;A=3CKV3;3π+fW (x|θ) $X74-:7=-( 33/ -j 3 3=79+ V3>7 4$X-2,XE3F;3+@3 j
72$;@3X874?7@5V- ;3 d3 0 5V3X7 7@5#^a b>5K(;5 3aG* 35o; 7@3X.=3 ; * (3 5 3a;<3C; 5K7 $π;@ ;;<3B; =-J;<5975K:;
;+* $%4J4D'::3C4DR(5KV5972$F;<3B;6?4D5H4DRV3;X1, . . .Xn, . . . ∼ W(θ)
^@=:(;<5K7@5V- 3X(7V5V37(4-NV3 %4:5 ;+* X-(72- j73XE3X87 W W=xXW b0/Y\O3=7@7@3 5;@5V- ;3G* 5K! :@>4D7@5V-6;+* 3723X@7@5o; 3[4&$=72$ 7@5VK5o;$C33C 7J4D@7@5o V5K3CJ7J4D T 4-;4-3 U-D!@; : ^ a -m3_ + a +1a-mdbZ;4D.;h;<3B;6$X7 ;<3B;N437'7V5 $C3;/ i3C ::3X^a-m + § i'/ b]! 5K7 (3F38 3F;<3B;75V=57(4DV3;Y73%4(5 3B; 7@5VV5;4DRV3;/ -gc%4(-5o; 5o; ;4D87
π(θ) = π(θ|ω);4-; 3(!/4-E5KVK34d[F723X27(4D4D($X7@5; $X3
7(4-ω+J;3;XX-87@4D5V87@3B;^>4D2:-5V(4DV3;i73XVK3B;X 3 - [X ] = xe
7J3C@E3=7@7@3C:7W;<36%4:5;@5Vω/G4D ] < ( ] s ] x u w Ug
vG> & $ w +-"7J3 7E$8:84DV3XE3X87E=:; 7@ 5K3 3+;<3C;@5972$π;4?75;!/4D5o;@4-87W;<3B;E=:87@4D5V:73;W;<3>E-E3C:79+>4D5o;Z;<3
! :@E3F4-@R5K7@4D5V31+7J4D ! -5o;h;<5#=5VK3CL3C87N>4D5H4DR(K33=7NE<;<5DCJ4DR(K3-+]3X7L;<:87NV3;J4d[F723X27(4D4D(,X7@3;Z5;2 3X87;+* 4e-:5K ;@3X;Y723 =:(7 $43X.; 5VRV33/
A ;A;$8C(5o; ;@-.;^4DV-%;i;3 7 [d723;^; * 3F723X@7@5o;@3E 4D87@5K74D7@5-31+73Xo; * -R;@3X23$B;O;4-;^V3;Y7-R(D,CL3B;^;<3; 4-R5KV5K7 $&3X7;3XCJ4-R5VK5K7 $Y@3C=:87@2$;17J4D -/ 0Z/-g A(3<3;]^@7 /a_ b7+8 ; 7372f;@-.;17(4-4-K,XV3XE3X87 3E<;$XV5o;@4D7@5V-Q;4D;V3AX4f;18g* 5K'! -E4D7@5V- 4D87@5K74D7@5-3A;* 33C2X3Y;5K3=7@3XE3C:7X; η /
t t t q0o UT@K qr I'OUT@UIUBT sJq cT@ONT I qY$ Z0 -O0%< v '&U4 W & ;y; 7'72f;@-;5o=5 * Q3F723X@70+.; 5K7@3E) (3I;<3X>4DJ;<3I; 7 [F7J3GC ! 5o; ;@3 (3A:-4D j;<3 g3872$;@3X874?7@5-3;<3^H4I; 2$X3A;3y85V3
X; =:(7Jd;@4-:7 ∑ C 3CKV3%4(4D.=3^4 ∑ ;<3^7-NRJ3Cg3X(7(4-3
4e?4D87K3Y73X(7;xe 1+<3B;b7C4`7(4-RK3Z;<35! @(5KA4 E5V5K G* 3I;<3B;y;79$X5?C.C4?75K:;;$0C((5K3B;=:LE3A; 597B/
pEqr M$PeM9NMOHLP]i'/ Y & v xeshU w e< TDw s >5x3w IR+∗ d&`s > U w/$ < > shU w 4&`s > 0 w]x u w)( $w8>kv 4 w Wi U@U+& ] ]-w 4 $ Z0 -i02< v '&U
xfw xeTw 4 w U4 x &UUIZ _ xe
45g _ g 4 x3w M W (1w0vGvw e< TDw s > & ^ 4 w8> e< ^Tw $w s v 0vG> w%$w8>! s w 0 & ] ](w T u w 4 v ] < v '&U WJx u shU m8s<U v TwJx u & > x`> w α l/ W=w W P (X < xe) = α ∈]0, 1[ "0WJx3w T u w 4 $ Z > <U20 w E[X ] WJx s ] & xfw R
d[X ] W
1 " )!'8/ &( 368/ 2 " )!,/ 8$.+ 8/ 8 ,/8" ) . " !,/&( ,/- 1)! &( ) )! 6-&( . )#% 2' #%,/8/8 #%1
?i − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* \-%;2 3 α ' 0.5
+'=3;^7@-5o;Y:-4D(;<3 %;3E; :87Y:f$X'$C4-K3CE3X870+;4-;^G* 3B;7(@5K7&;+* Q37J3C 79+ 3ZV3I@3 3=7;+* 3h CL3L5H;$X3L;<3
X^gk! / Y4D3B; oF4(434 _f_ b0/ d3E!/4=-e5KX-;2=5V3X8731+2g* 3723X@7Y4 4#;@ -3X87A73X(;(4DX3
)Q;<-(3X (3+4`7'72$X5V4D7@5V-" 5 ;@3(;@5K7 3F4 E5VK5V3 ; * a3C; 3ChRV3+;<3-?4DV3 2;W3XV5D$C3;A)QH4[; 2$X3>;<3-5V3^V7(4-g3<3C7(K3AV-%;2 * 5V.X-(4 !V717( ;@5V3 %;g33C(7V4-5K3;g5V;<3C87@5o 3;g -+;3 ∑ b0/ &5K;@5 +:5V'! @(5K4]7 7,-7 3A?4DV3 A($C;<5H4D3E * 36`4-K3 AEt[-3C3-+< CE36; 5 =3=7@7@3I;<3C@(5D,C@3Z3B;b79+(;<36;@- 72-5V:7&;3E 31+'7( ;A5K j7 5K7@5-3f/ G4DL5_V3;^4 7@3 %;i4e[84D87@3CE4-2 $W;<3A;@3XhR(V4-RK3B;Y=-(72-@7@3CL3C872;^;4-;ig* 3723X@7@5o; 3W5V(; ; 7@5K3CKV31+C((4-=5,X31+<;$CLf:-4`7'4(5 3-+-3=7%`/ +'=5K7@:; %45V3X3X&^a3ifbg \4-t[ gX43=5H4^ _3_a1+d7 / ifb0/\O3;Y;79$ j=5DC.X4D7@5V-;4-RJ 7@5o; ;@3X87=37J3C(;4-:7X)E;<3B;E;'$XV5;4?75K:;^;<59$C@3C:73;/-gVK3B;^;<-5-3C:7A;<:=7@3]72$X5; $X3B;^4 >4 <5K 2 & 3; 7@5V-((4D5V@3N3XS=3(; 3C;]; 3C4-59757(4DY3<3X(7V3 ^V72 WV3;];79$X5?C9X4?75K:;Ja) i,+2@3B;723=7@5-3 jE3X87%b :
a3/V-%; 3>G* `:-3>;+* ::@ 7J3+;3#=-(72f;4D872; ∑1, . . . ,
∑P
4?7 73X5V87xe+N 3XVK3-7(@f7J: 75K:
1 − α;<3;
X-(7Jd;@4-872;3; 7 j 3XVV3Y3C=:@3Y3X ! :75K:3XE3C:7 /H4(?4DV3 xe
;@3]! -(;3 j 7 j 3XVK3A; 3EX-(75VV4D7@5V-; * $3,X(3XE3X872;Y7J43;2;$B; i'/yX3=7 736?4DV3 =:@3; 72-(; j 3XVV357( 7),D7])>V4#`4-K3 d;3I7@3C(7;d;3h;$0!/4-5KVH4DX3IV4#7 ;;@ :3X87A-R.; 3C f$X3
;4-;V3A7(43;2;$
^3X>4D% -.;+=38723XJ;4D87 3GH4Y7J3C 75K(3XX3g;<3 =3B; 3; 7@5V-.;_3; 7@3CV4D7@5-3 ) EX3X@74D5V]=-8737@3g;+* $=7 ;<3 :V4 3; 7@5V- aE3; 757 7),D7 x s > < ^ T 4 v|w + 7 5o; 3([S2$872-J;<@3J7J3 7I5V(; 5V@3LH4 X-J4D5o; ;4DX3N;+* 3-; 7@4?72$8:-5V3;<3LE4-5K873X(4-=3f/ \ 4 3B;b75K:&i3; 7]XV4-5K3XE3X87,-W< ^ T 4 v|w ++7 5;2 * 3CKV3L@3B= 2;5) (35o; 5V- 7(43;2;$C3N;<3EH4; 2$X3I;<3J5K3h;36=:72f;4D87%;8/(\ 4- 3; 7@5V- 3B;b7dL5 7@3-+.X4-A-[(3E7J3 7d2$X3CKV3XE3C:7dV4(72f;@3X5 3IK:2;2 3K3AX-(7Jd;@4-873B;b7^3 !G K:2;2 * 5K_4>;$!)L4D7 73X5V:7 X3X@74-5K`::33/
GF / < 4 $ Z0 -i0%< v '&U x u shU ] & x3w < $ $ < > < vAw U $9> < v m8s w 0 & ] ]-w1$w s > Z%< T 4 v|w l w U v &ts v 0%<"4 x <U-40 w0>kv < $U4 x & ] < $U w 4 4 $ Z0@< T 4)Z 4pm0s &U v 4 w0> xfw 0%< xt> w`_ 0 w0vGvw Z v s xfw Q T u $U c3Z)U w8> w Us(0 T Z%< >2wVXW &ts 4 T u < "&U-40 w $w U x <U v $U20 T s w $ &ts >Exfw 4 > < 4&U-4 xfw c0ZUIZ > < T v Z W
R & x Z T 48< v '&U-4 W 1%4J43 3; 7.$B=5DC.X4D7@5V-6+D( ;1%4(-5o; 5o; ;@-.;g* 373;2; 5V-E;$X7@3XE5K(5; 7@3A;<3b^g;[NRJ:K5o;$C3
7(4Dy;4E;<5; 7@5VR 7@5V- B bO5V(;<5o $X3I;4-;iV47372f;@5K7@5V- / y=3CV4LG* 37@3B; ;@5V-4-(4D[87@5o 3Y;3;y; 7.$B=5DC.X4 j7@5V-;g3; 7j@3C(; 3^5V(;$7J3C(;4-:73&;<3B; 4F[d723X27(4-4-,X7@3;;<3
π(β)/ F 7'72f;@-;17(4Dg3<3X(7V3&4e-:5K
E[X ] = xe/
F 7'72f;@-.;>4DV-%;JX-(4 !V7@3β ∼ π1
3=7β ∼ π2
;<3 6E<;$XV5o;@4D7@5V-;(72f;2; 5VRV3;/ .#(7.$C4D7@5-3XE3C:79+G ;;<38:-;Y%4:5;@5V
b73X 3 - [X |B, π1] =
-[X |B, π2] = xe
/
d4-;^H4;@ ; j 7(4D@7@5V3]; 5?4D87@3-+H4h74D5VKV3]C.=7@5-3a
3; 7YC$X3A7(4-^g* 4DJ4D[F; 7@3WR(4e[f$;@5K3C / 5G* :[;$X5V;4-597A;<3X4-K5VR3X
π 5o 3XE3C:7^3C#! -.7@5V-;<3Wg* 3723X@7@5o; 3-+)hG* 5V; 74-;3 Y4^5H; ^ a -m b7+8V3]X 7(K3 (a, xe)E5K5VE5 j
; 3C4-597 3! :75K:+;<3723X@7@3&3C87@3M
3=7 (3&3872$;@3X874D7@5V-`Cb5H;$M4DV3 &;<3M
^|73XVK35 3AV4E7J3C X-5K7jg* 3723X@70+7(4DE33C(7K3 7(4-EK3R5H4D5o;>;+* 45o; 7@3::4-EL3S^V?4D A: 7^5 bo w0v < T'W a + Y4D3B;4oF4(434 w8v < T'W _3_fbb0/\3;@3X.;I;<3
a; 3C4-597N4-K:2;WE:5K;J=H4D5V/ 4-h4D5VVK3 2;0+X3=7 73F3872$;@3X874?7@5V-";@3#43 73X4D5K7h4 "7375D$X7 $B;
;b7 7 @3CKV3;Y;<3π: K3E5;2 3J;@3X4D5K7$CK33$E;+* -R<73X5VZ;<3;W;<5o;b7@5VR 75K:;7C 4 5X3;]X4f%4'$; ^V7(4-d3<3X(7V3
7@2,;^;<5o;2;[($=7@5o 3;%bk/
$A%G'% / \W4:5;@5;2;@-; < $9> '& > =: : $ G(a, b)72 AK367(4D4D(,=7@3
λ;+* SE;',XV3Z37J:3X875K3Cg/
c; 7'7Jd; 3 3+g* 3723X@70+ 5K87@3C@3:f$-; NK3+74 a;<3;$0!/4-5KVH4DX31+5V(;<5o 3FV3;h3; 7@5VE4D7@5V-;(0.45, 0.5)
;
& DV
E<;<3 .[<;[λ] = (a− 1)/b
3X7;3YG* 3; 7.$C4-=3 -[λ] = a/b
/\O3X!/4-5;4D879+83C4f8=37<74-87Y;4D;i3; 7@5=7@5V-+G* 4e85o;;<3EG* 37J3C 79+_-U-R<75K3C:7
a = 20/ -g"7J3C2X38?4D87
a=:EL3 (3N74D5VKV3C.75D:31+ X3=7I3723X@7A7J3 76=H4D5V@3CE3X87
X7@3=:;@5V;$C $5=:LE3&7@37+5V! ->4?75?! / .# 3; 7Z4D5o; 3ChRH4DR(K3X 35; :+5H;$X3W; E<;<3Y3=7^;<3YV4hLt[:3X(3&3; 74f; ;@3 5=-'! ; 3-+F7 5o; * 5K 4`7'7(@%43d34`:f$X2$XE3X87WX3;;<3 #`4-K3 %;8/ 5 G* :3W;$B; 5V@357(4f;jE<;<5DC(3X
a>4D5o;
G* 3Z;<3;A;3 ; 79$=5DC.C4?7@5V-.;+<H43 3CKV3A%4-5o; 5V
EFuHLH K)M9NMOHLP '/ Y & v xe ∈ IR+∗ l a > 0 w8v α ∈]0, 1] Wi U U+& v|w
b1(a, β) =((1 − α)−1/a − 1
)−1
xβe ,
b2(a, β) =
(Γ(a)
Γ(1 + 1/β)Γ(a− 1/β)
)β
xβe ,
b3(a, β) =aβ + 1
β − 1xβ
e .
i U > < $ $'w)T$TDw m8s w B x Z4 cU wBT < ] & x Z T 48< v '&U $U x s vw $ < > Tw 0 [ & (Sx3w b(a, β) WkjT & > 45l $ &ts >v &`s v 0 [ & (Sx3wπ(β) x3w 4ks $$ & >kv [βl, βr] ⊂]0,+∞[
l
(i) P (X ≤ xe|B1) = α,
(ii) & [X |B2] = xe4 a > β−1
l
l(iii)
Rd[X |B3] = xe
4 βl > 1 W
<U4 TDw 0%< 4(iii)
l TDw<] & xfw xexfw<T < x 4 vg> ^ s v '&U ] < > c$U9< TDw M w 4 v shU m0s w-w8v 4 vg> /0 v|w]-w U v $ & 4 v d W <U4
TDw 46<ts vG> w 4Z0%<"45l 4 βl ≥ 1
lM $ & 4)4 S%x3w <`s ] & $U-4AshU ] & xfw 4 vG> /0 v|w]-w U v $ &"4 v d W
npoq rsuqut G-5VA3<3; § i'/ '/a(^V7 / fb0/
w] < > m8s w 4 w8v $> & $> Z v Z 4W/af/ d4-;YK3(X4f;
(i)+2: 4
P (η < xe) = P (U > kα,a)18
U ∼ χ22a
3=7kα,a = 2b1(β)x−β
e
:+;@5g* - ;4D5K7;79$X5?C(3C 6 4D87@5VV3 ; X +C-E723 7 ; 7.$B=5DC(3X 6 4D875KV3 ; η ^@X3W 5d7J3C@E3X7 H4=:7J4D4D5o; :W4e:3;+* 4 7@3;L<;$CK5o;4?7@5V-.;2b0/G4D 4-5KVV3 %;+3CZD74D87
`α,a = kα,a/χ22a(1−α)
+C-Z4P (η < xe`
1/βα,a |β) = α
+ $B; K74D7 75KV3QK:2;2 3 β 3; 7JC F$ !/4D5VRK3CE3X87?4-@5H4DRV33/-gCJ6+1; 5
a 1+ ;@5
xe3; 7LV3 4-:75KV3
_ ^ H4>($C;5V4-3Bb<+b1(β) ' axβ
e /log 2/ [3=7@ -3AX3I $B; 974?7&V-%; * -[4:;4`7<73IH4+;$X>4-2%43I;3
5K:37 4DVV4 :':^a:m fb3X(; 7'7Jd;@4-87 3ig* 3723X@7 ! 5K7 3W?4-K3 GEt[-3X(3O;<3H4&($M;<5V4-3m = η(log 2)1/β +(4e:3 m|β ∼ GIG(a, β, b log 2)
/ / d4-;IV3 C43;
(ii)+ ;@5
βl ≥ 1+G* 3B;79$X4D.=3F;<3
η3; 7@37@%43+;3
xe::64-
[η]/xe ∈ [1, 2/√π] '
[1, 1.1284]/ -g C43;g;<3X! -@7W5V3X5VKV5;2;@3XE3X870+d; 5
a > 4+b2(β) ' axβ
e
; * 437 ,B;gH4A7372f;@5975K: #^ &33B;§ fbk/
i/ d4-;^V36X4f;(iii)
+<V4=-J;<5975K:βl > 1
* 3; 77(43;; 273X(4-:73 : 3I;5; 7@5KR 7@5V-S;<3 U3X5VR V]3; 7&;<3L<;<3J;b7@5o73XE3X87X72f;@5975?!g;,B; 3IV3E5K3C5KVV5;2; 3CE3X87A3; 7d43;2; $-+'; :597Y;,; 3 β > 1
/-g&X43;&;<3]! -@785V3X5VVK5o; ;@3XE3C:79+<-Q3=7@ -3 b3(β) ' axβ
e
/ / d4-;YK3-X4f;d; * UE<;,XV3N37J:3X87@5V3X +
β3B;b75C$-)haf/ -3B
η ∼ IG(a, b)+ -e3=7@ -3-+9;@5- C<3
-[X |B2] = xe
+-[η] = xe
/.\O38723X(;4-870+<5V4`7'7(4-4 !V7Z7 ;(4?7 3X_;<357@f7Jd; 3Cb = axe
2;4D.;ZX3AX43;0+xe
* 3; 7W7( ;W723X 3XX-EE3&G* 3B;79$X4D.=3&;<3&V4N;<5; 7@5VR 7@5V->>4- ::5K(4-K3M
>4-5;W=-EE3&H4ZEt[-3C3;<3hG* $B%4(4D875KVK: C.75?!O4 3X -S4f; ;@5KE5VK3hg* 3723X@7/ 7@3ZK36!/4D5K75 36X3=7@7@3hE<;$CK5o;@4D7@5V-[ * 5V72f;@3
?i − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* 4 (3N3; 7@5=7@5V- ; a + 3(7Jd; ;@5KR(K3L;<5;2 ; ;@5K: 4e:3ZG* 37J3C 7A723XL3X7 74-597Z;+* 3X87 $C@5V3X]=3(%4:5 7 7),D7X 3ZV3]%4(-5
(ii)/
t t t q0o UBT@K q r I'OUBTVUBIUT sJq L P sJq M P q4-&4D5VVK3 2;0+<G* 5V87@3X3::4D7@5V-Q;+* [3723X@77J3 7 =7@3Z:@5V3X872$X3Z;<3A!/4=- )(7372f;@3X^3C ; ; c^ 7( j
; 5V3 2;%b5V87@3C ?4DVV3i;<3I0 > Z x ^ T v Z [xe1 , xe2 ];+* -;<3
(α1, α2) ∈]0, 1[273X' 3Z7J i = 1, 2
+P (X < xei) = αi^ '$B=3B; ;4D5V@3CE3X870+
xe1 < xe2 ⇒ α1 < α2b0/ *A"X4f;X7 7),D7]! $B 3X87W3X ; 4-R5VK5K7 $L3; 7YG* 37J3C 75;@3E;<3NG* 5K j
7@3C ?4DVV3(50%, 90%)
/ G4->3<3X(7V31+gH4a($=724<;<3S;<3[H4 ^ 4 w 0 v '&U ^nW4D@7 4 4D5K7@3 w0v < T'W _3_ bJ7@f7Jd; 3Q;<3=:; 5H;$C@3C 3A; 5K7@3I;3E 3; 7@5V-;^; 7 [F723
a3/ -3 j : ;;$X7@3C@E5V3X (35?4-K3 ^($C;<5H4D3xe,0
72 X / F 7'7Jd; 3 X 3 X < xe0
/] -3 j - ;;$=73XL5V3C 3A( :3XVK3y?4DV3 xe−1 < xe0
7@3CKV3X 3 P (X <
xe−1 ) = 1/2 ^ xe−1
3; 7&4DV-%;Z7J3C X-EE3ZK3A 4- 75KV3W5K!V$C@5V3 A;<3Mbk/
i'/. CE3E 3; 7@5V-3X ; 7'7Jd;@4-87y 3 X > xe0
/
*A3A4 7@3$X7 4<;<3&3; 7G;3YC<3X E3X.; 3ChRV3A;+* 5K87@3C ?4-KV3; ; d-(M)
3=7j;3A;<3X>4DJ;<3X1)YG* 3F723X@7gH47-R(4-R5KV5K7 $;<3i;'$0!/4D5VVV4-=3;<3 ∑ ;4D; %4J43 3j5V87@3C ?4DVV33/-gC(+-;4D.; X-87@3873i;3WC24DR5VK5K7 $i5V(; ; 7@5K3CKV31+=3B;6 4D875972$;6;* 4`7'7(4-@3C87@3X876;@ :3X87h4 73X(7;L;<3;$8!/4D5VKH4D.=3h^ ;<3X3X; 3BbWV3;J7 ;6X @72;h3=7LK3B;7 ;WK:':f; 4-E4-5;Y:R; 3C f$;7(4DWG* 37J3C 7I;4-;YK3(7(4f; ; $3/ A5V;@5 +
(xe1 , xe2 )7(@3C3X87ZV3(;@3X.;Y;3( 4-87@5VK3B;
37@ XE3B;^|7 [F75o 3XE3X879+ j bk/ 5 +3F723X@7 723 74D5V;@5'! @5V17 ;@5K3 %; ; 7.$B=5DC.X4D7@5V-; i = 1, . . . , P+
;(7@f7Jd; :;L;<3[E;'$XV5;@3X-;@-P37J3C 75;@3Q3X =:hR5V(4D87>V3;EE<;$XV5;4?75K:;π1, . . . , πP
$X>4DJ4D87E;3%4(43 3Z;<3B;y;79$X5?C.C4?75K:;y * 5V+! @(5979+'=-.; 5H;$X2$X3W5V(;$7J3C(;4DEE3X879+(;<3WV4!/4=: ; 5`4-87@3 :
µ|β ∼ G(
P∑
i=1
Pωiai,
P∑
i=1
bi(Pωiai, β)
),
^@i/ b
; :597WH4;<5o; 7@5KR 7@5V-S$X>4D(4-87W;+* 5V!V$X3XX3;YR(4e[f$;@5K3C3;5; 8X3;2; 5-3B;; WK3B;]$%4(4-87@5VKV-;XC.=7@5D!@;-^/;<3L74-5KVV37J:(;$C $C3;%b7+')7(4D@7@5VA;<3ZH4EE3; @3W:5V! :@>4?75D:3Z;<3-3 238[;
πJ(µ|β)/J\3;
ωi > 0;@-87A;<3;7J:5V;';y 3
g* -A723 7C<3X 3XA! :75K:I;<3jH4X-CJ4-=3j@3CV4D7@5-31 3gG* :h43=:;<3g4 A;79$X5?C.C4?75K:;+C4e-3 ∑Pi=1 ωi = 1
/
^;<37<73X =3X7 7@3;$CE4-2%4(3j;<3W; $87J4D4?7@5V-I;3jG* 3F723X@7@5o;@3j3XfCb3F723X@7@5o;@3;]4?7@:E5 3B; 1723XL3X7];<3Y=:; 7@ 5K3π
3X =:; 3C ?4-:7OK3B;W4F[F723X27(4D4D(,=7@3B;ai
;$C;<5$;Y)hV4X4-K5VR4?7@5V-/ G4D4D5VVK3 2;0+83XVK357J3C@E3X7;+* 4-;(;<5975K:3C;<3 ;79$X5?C9X4?75K:;^;<59$C@3C87@3;A^/E;3Z3=7X 4D875KV3;Y7(4-^3<3X(7V3Bb0/
/+) ' - #-% -6!' S- &( ' b" -A ;7372f;@-;A5X5 [3C;@3XhR(K3L;<3I2,8:-V3;X; 5V(7K3B;X72 C<3XdV4>74D5VVK3AC97@5-3
a;+* S37J3C 7 5 3
e+ ;@3XV-cH4 ;79$X5?C.C4?75K:c * 5VW7(@f7Jd; 3f/N\O3;h@3B=3=7@7@3B;E; :876$=74DRV5K3B;6) 7J4D@7@5Vh;<3F2$; K74D72;W7 ,B;E:3$C'$X4
;+* 4D(4-D[;@3-7;[%4:Kf:-5o 3>;<3;h37J3C 7%;^@--5V Y4D3B;4(od4J4`4 _f_ 7J 3F38 3ebY3=7L;3+g* $=7 ;<3;<3 $9> '& >d w%w2x3^ <-0"4Z;<3>H4V5K7 7 $C4D7 3-; Rb3=7@585o; 7@31+_3X7I ;W3-::4-4-:75;2;@-;5=3C 74D5V3XE3X87A7(4f;WK3 A?4DV5V;<5K7 $-72 7@ ;hV3;67-R,XE3;N;<3 CJ4DR5VV5972$F5V(; ; 7@5K3CKV31+] XE3 ;@5i ;NV3;6X-;@5H;$X-;N4`7'7(@f7@5$X3B;I;4-;IV3 C4-;<3;+* $=7 ;<3A7372f; $7(4--/10W/
4-d4-5KVK3 %;+J5K !/4 7]=:; 5H;$C@3C=3B;A2,8::K3B;A;<3JC4DV5KR(4D7@5V-h=:EL3N;<3;Y4`7'7(@%43B; $ < >(x Z d0<`s v / ^5V3XS(3;@4 4D5K7N3X(7H43=3C0+ 72 >437'7F%4(3X
a+ (3Q;5;2 ;2;@5K:63X87@3g* 3723X@7>3=7>g* 4DJ4D[F; 7@3R(4e[f$;@5K3C6+G;+* 4 74-87
& DV
7 ;W7(4D%=3 3=37(4D4D(,X7@3A3; 7i4-5; $XE3X87Y=-(72$84(3X;@5KR(K37J4DiK3B;O;<3 -7(4D@7@5V3;/ &3B;W 3B;b75K:;j -3C 73;;@-87^ $: V5,X3XE3X87y7372f; $X3;Z72 d;<5K5:-3XX3=7@7@3I;<5o; ;2; 5V-6+<>4-5;;4D.;y 3Z ;7 5o; ;@5K:;y=3C 75?CJ3X 3G* 37J3C 7 7 5;2;@3X[E $7J:(;<33/ 6* 18+G* 5K(72-@74D.=3&;<3Y;<-(3X;<3;j ,:-V3;j-Rb3B7@5-3B; ; jV3;2 3XVV3; ;* 4`7'7 [-3X9+72$C4-V4-RV3XE3X87^4 F2$; K74D72;^;<3ZH4E;<5;2 ; ;@5K: /
A-7@:;X 3L;4D;5=3X7 736$=7 ;<3-+J ;A(3J%43C2%4:;y7(4f;)+L<;<5DC(3CdV3;X`4-K3 %; xe7@f7Jd;$C3;y7J4DYg* 3 j
723X@70+ 75KV5o;$C3;^;4-;V47372f;@5K7@5V- '/-d3Y(D7@3]7J:5K87&;<3] 3-+'=3;Y?4-K3 2;3 ,=7@3C87G* 375V5V- 7J3C2;@-(3XVK3;<3Ng* 3723X@70+ 3=7W ;&(3L;<5o; 7Jd; :;7(4f;d; * 5V! ->4?75K:S-Rb3B75D:3672 Y ;X723XE3=7 7@3E;<3NL<;<5DC(3C]X3=7@7@3f75K(5K:`/ d3;i4 7@3 2;1X-EE3 5V':37 4DVH4 :':#^ a -mfb =:; 7@ 5;@3X87j;<3B;gE<;$XV5o;@4D7@5V-;X< $> '& > (4 :hR3 54F[F7J3C 7J4D4D(,=7@3B;+7J3C@E3=7@74-:7G;<3OL<;$C@3C G* 5K! :@>4D7@5V-h;+* 3723X@7@5o; 3Y)AH4y! :5;]3C672f;@5975K:I3=73X5V=3C 7597 ;<3f/ .e4D5o;O;4-;OK3B;1!/4-597%; j 3XF7@ 7YC43;i;4D.;jV3X-87@3873d;+* -/10 +'=-EE3YK3X! -87i3X>4-2 3C 43%4(4w8v < T'W ^ a -mdb j X4DV5VR@3CC((3XE3X87Wg*=< $9> '& > 3; 7@3h5V(4f=3;2;@5KRV33/A ;5=-.; 5H;$X-.;&;<:JX-EE3hR9$X'$8C. 3;<3E * 4e-:5K] * &4F[F723X27(4D4D(,=7@3 5o 3E3=7E=:7( $43X;@5VRK3() X4-K5VR3X57J ZE; K3CZg* 5V=3C 7597 ;<3>;<3G* 37J3C 7B/
q PFQ3TNQfIUTgM.O q K P o IO Q q cPFLNT@I'O q cML qut \:2;2 3Gg* 3723X@7 ; 7.$B=5DC(3xe
=-EE3 (3 ?4DV3 =3X874-K3;<3
M+(5KN;8* 4`7'7 5K3Z$=3B; ;4D5V3XE3X87y; h=3C 74D5V(-hR(@3Z;<3Z2$C4-K5o;@4D7@5V-;Y7J43;2;$C3;^;<3 ∑ ;<36; [F; 7 ,CE3;
7%43;A 3X[5V87@3C4f7@5V-S4e:3 ∑ / 723 7W4DV-%;%4:5;@5Ka=:EL3hK3N-NR@3ZE5V5VE4-];3I $M4DV5;4?75K:;
'$B=3B; ;4D5V@3E)=3=7@7@3;79$X5?C.C4?75K:6+J3X7dV3IE<;<5?CJ3X @<N;<3; 3; 7@5V-;5; R;@5V;<5H4D5V3;0+'7J: 74D87; Yg* 379$X5K3C=37(4f; ; $X3W;<3ZG* 37J3C 7B/
& 4 $ Z > <U20 w J] Z x @<U w&W ^;$0C((5K (3YEt[:3X3(^@;@-5K7^g* 3; 7@5V>4?73 ;4D;iR5H4D5o;^;+* 3W3; 79$X4DX3Bbj 3($C;5V4-31+:"7J3 7h$8:-5K7@5VE3XE3X87J; 7'72f;@3XE 3>G* 37J3C 7 7@5VK5o;@3+4 E-5V;ZH4h=:(4D5o;2;@4-=3E;<3
a = 1
a = 2;<:'$X3B;17(4f; ; $X3B;^ 3; 72375D:3XE3X87kbk/ 7'72f;@-;W 3Ag* 3723X@7W;@-5K7WX437(4DRV3&;+* 5V(;<5o 3X +-NR3A;<3
;$8!/4D5VKH4D.=3;7(43;2;$C3;; AV3;2 3XVV3;A5V ! :(;<36;@-S4e5o;8/ * 5V] $7J:(; 7Jd; 5K7@5-3CL3C87)>H4# 3; 7@5V- C@4e-3 j : ;;$!)Q-R;@3X23$E;<3B;I;$0!/4-5KVH4DX3;A$CK:5D::'$X3B;Z;<3
xe3=7NK3B;Z5K.= 3 j : ;Z;4-;A:D7@3E3; 7@5V>4?7@5V- 7+_5KO3; 7E7 ;
4D5o; 3ChRH4DRV31 3 xe3872$;@3X873 3i($C;<5H4D3f/ 5K(-6+MK3 a =-3; 7J:(;4D875;2 3O; * X7@3W=:;@5V;$C4-RV3XE3X87
$CK33$3/ Y4D; X3ZC43;0+MH45 3; 7@5V-`C 4e:3 j - ;_:R; 3C f$ (3 ! : 73YX-X3X87@4?75K:h;<3;$0!/4-5KVH4DX3;;4D.; 6!/4-5KRV35V87@3X2?4DVK3W;<3W73X(7; A7J3C@E3X7^;3E=-5D::3X
a3=7xe/
R & x3w W .#87 5975D:3XE3X879+ 37J3C 7-X437(4DRV3Q;<3 ;79$X5?C(3C E<;<3 =:(4"!H7 6(-hR(@3 7 7,-7($CK33$Q;<3;$8!/4D5VKH4D.=3;67(43;2; $X3;0+;<:87L4 E:5K; $::4DV3;I 7%43;/ E5K(5K >+ 5V !/4 7 a = 3
;<:'$C3;E72 ;$8C(5V^V4(XV4f; ;@3WL<;4-K3Z;+* 3I;5; 7@5KR 7@5V- =::75K 3f/
A ;I387( $B; 3C:7-;6; NV4 C: 3i/DaBiQK3B;h;<3C; 5K7 $B;m(x)
7<; 5K7@3;J; 3CK:c%4(43 3F;3 =3B;6; 7.$B=5DC.X4 j75K:;8/ %4:5;@597>5o=5
xe = 100+[βl, βr] = [1.1, 5]
+(p, q) = (1.5, 1.5)
3=7a = 2
/'-j !/4-5;4D87-?4D5K3Ca+
: 5V(;<5o 3-; ZH474-RK3-i/DaLg* $8:- 7@5V-a;<3EG* $BX4D@7 j 7 [F7J3L3=7IH4Q;<5o; ; [$X7@5K3 ^ 4 w U w 4)4bW;<3M/ g ;EX3=7@7@3
;<3C@(5D,C@3W`4-K3 ]3; 7 ! : 73 72f;@5975D:3XE3X870+e7 ;]H4d;<5o; 7@5KR 7@5V- M 7(@55K$8::5K3OK3B; ?4-K3 2; V3;N7 ;N4(4 73;;<3X/
-g X43;O;+* 4'$;@5974?75K:+3X87@3AV3;Z;79$X5?C9X4?75K:;0+dX3XVK3Y; E<;<3d4f; ; 3 37 ;jH4D2:-3A5VX3X@7@5K7 ;<3d3=7Z7J3 7X-; 7@5K7 3C #%4:5 J7(4Dj;'$0!/4 7/F\O37J3C(;4-:79+fX3%4-5 L;<:3X)
MH4E 3 3^H4]7 ;gV ;<3 4-(; x → ∞ /
\ 4J; 7.$B=5DC.X4D7@5V-; E<;<3]75D5VD$:-5V3Y;-YV3;Z?4DV3 %;;<3X
V3;Y7 ;Y4(4 7@3B;8/\O-EE3W ;OV3]72$=5o;@3X-;4 § i/ i/ +=3=7@7@36; 79$=5DC.C4?7@5V-7 $CE-5:-3I;+* 3J=-((4D5o; ;4D.=35$XV383$C3Z;<3;X-(;5975K:;A;+* 37V-5K74D7@5V-Q;<3 ∑
1 )+!#%,/- 8/8/ . #%""#%+#/) 1&(&( 2-,/17 .+.! 2 28 # &( '8/,/.#)!,/# '.+,/ .!- 1)!,
?i − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* 3=7;<3dH46X-F5=7@5V-+; 5K3C5KVK5o;2; 3CL3C870+83=7 ;O35=-.; 3C5KVK:;Z; : 7@5VK5o;4?7@5V- 3&V-%; 3&V4J=-((4D5o; ;4D.=3< $9> '& > J3B;b7Y?4DV5V;'$X3]7(4D V4- ::3X-;@3X; ;O ;<3;4D5o; :;g:Rb3=7@5-3B;8/ A5V;@5 + ;Y=:; 3C5KVV-;W7 7,-7^;3%4-5o;@5KH4N($M;<5V4-3A7(4-;'$0!/4 70+ 54f; ; 3 3A7 ;^H4D2:-3Y5KX3X@7@5K7 ;3E 3YG* 3; 7.$C4-=3f/
x
m(x
)
0 50 100 150 200 250 300
espérancemodemédiane
xe=100
M i/DaBi 3d3X;@5K7 $;m(x)
;<3WH4>;<5; 7@5VR 7@5V->4D2:-5V(4DV3M
a σ γ σ γ σ γ
!""# !$" ! % ! &! ' $' # $( & # &# # $" %$ $( $& % ' ' $& % ')$ $" $!" & ' $& "$# " # $" & $"$# ! # ! $&*$!
l,+ .i/Da - ; 7@5V>4?7@5V-.;A;<3Wg* $C4D@7 j 7 [F7J3σ
3=7&;<3WH4>;<5;2; [8($X7@5K3γ
;<3M/
& DV
q PFQ3TNQfIUTgM.OhL r O r I'OUT qut A ; !/4-5;@-.;G* [ha)$ & v$[S 4 w ; 5`4-87@3):MH4X; 7.$B=5DC.X4D7@5V-h;+* = 7K3 (xe, α)73X 3 P (X < xe) = α5V(; 5K7 3Wg* 3723X@70+(;3]!/4 =: ;@ ; j 43X3X87@3-+FX-J4"!H7^4 74-87&;<3W $M4DV5;4?75K:;Z7(43; j
; $X3B; * 5K3; 7 $=3B; ;4D5V317J ;'$0C(5VV4X72$=5o;@5K:α2e; 5
α = 0.5^ ($C;<5H4D(3Bb7+`5V84X7J3C E = 7K3 (xe,l, xe,r)73X 3 xe,l < xe < xe,r
/ -5K7 a = 2/ ; (3I ,:-V3W74f=5K7@3 x u Z5m8s > Z $ < >kv v '&UQ;<3;&;<-$X3;yC.75D:3;0+5K 3; 7
4-K:2;^4-5; $h;<3h;'$X V3X 3h($=724<;<3J;@5VL5VH4D5V@3E)>H4 ^ 4 w 0 v '&UF5V87@<; 5K7@367( ;X4(4 772 $8?4D 3X&V4E74D5VKV3aX-@3B;72-J;4D87@3f/ A5V; 5 +'; 5
α = 10%+<g* 3723X@7A3; 7=3X.;$Z4e-:5KY723X a = 10
;-'$C3;/
\O3=7@7@3I2,8::K3Z3; 7@3ZR(5K3C3C87@3XJ; 3E7(@0$M; 3A7(4DA;'$0!/4 7/ 5G* 4-(4D[;b73ZR(4e[3$B; 5V3XQ4EV4(72f;2;@5KR5VV5972$h;<3X-F-3C2;@3X^4e-3BWg* 3723X@7J^@X36 5]3; 7X:3$C'$X4DV3XE3X87AV36X4f;2b<+(H4(?4DV3 Y;<3
a 3N ;y7372f;@-;A4-; @7@ 7
72 dR 7W;<3E!/4-5K3I2$C4`::5K&g* 3723X@70+(3X7d-74-LE3C:7Y;<3h 5 !/4D5V@3E723X%=3-:5KX;4437 $X($C@5K7 $'3WK:2;2 * 5K 7(@f7Jd; 3 h;@3 5K;36 4D87@5VV3W7@37 72$X5;/ ;Y! -E3I;+* 3X%4',C@3B;+( ;7J3C; :; * h=:;@3X; ;^3B;b7&3C :3$C'$X4D4D7 7@3C5D::(4DR(K3E72 C<3X 3J?4DV3 a / \4;<5;2 ; ;@5K: 7J3 7 =7@3E$::4DV3XE3C:7Y-5K3C87 $X3A7J4DW;<3; 3; 7@5V-;Y; 7 [F723C K3A 4-:75KV3] 3: ;Z7372f;@3 5=-3; 7J:(; j 7 j 5K9) (357(@f7J: 75K:F;<3A=:72f;4D87%; ∑ 7%43;i;<3∑ 7-hR9$;3C 7(4D(3W;(4D;V3A7(43;2; $ %/
A ;3872$;@3X87@:;Z; ^H46C: 3Ai'/a K3B;;<3X;@5K7 $;m(x)
7; 5973;Y;@3XV-V4(; 7.$B=5DC.X4D7@5V-;3xe = 100X-EE3 4D875KV3;3 ;@3 5V α + 72 E;<59$C@3C87@3;?4DV3 2;L;<3
α/ %4(-5o; 5K7N5o=5
[βl, βr] = [1.1, 5]+(p, q) =
(1.5, 1.5)3=7a = 10
/
x
m(x
)
x
m(x
)
x
m(x
)
0 50 100 150 200 250
xe=100
10%
25%
33%
50%
66%
75%
90%
M i'/a d3X.; 5K7 $B;m(x)
;<3WH4>;<5o;b7@5VR 75K:>4D2:-5V(4-K3M
^g; 3CK:α
7@3X 3 P (X < xe) = αbk/
q PFQ3TNQfIUTgM.O L r OT@OU q3os I q L q Q o PdLNT T TVUBP t \O-EE3+5V(;<5o $4 § i/ i/ i/ + ;6; 7.$B=5DC(-;; $ j7(4- $CE3X87>V4U74D5VKV3 $ 5?4DV3X87@3 ai
+i = 1, 2
;<3h%4J43 3h 4D875KV3[; = 7V31+g3X7F ;>@3B=:hR5V-.;NV3;; 7.$B=5DC.X4D7@5V-;A3XS4-;37<74D87AV4F;$X>4D%%43-^@i/ bk/+\O3=7 73h3X-hR(5K(4-5;@-'$X3;2; 5K7@3J=38723XJ;4D87 (3I@3C->4 jV5;4?75K:;3;
ai 3Z ;5KV ;b7@:;^;4D.;g* 3<3X(7V3A; 5D?4D87B/
$&%G(' / Y & v (xe,1, xe,2)T u $U v|w8> < T$TDw Q +$ l + VXW &ts 434ks $ $ & 4&U-4 m8s wDT u w)( $w8>kv UuZ0 w 4)4 v|w a1 = 2
x &UUIZ w 4 $ &ts > - (Fw0> xe,1$U x Z $w U x < ] ]-w U v w0v a2 = 20 x &UUuZ w 4 $ &`s > - (dw8> xe,2
$U x Z $w U x < ] ]-w U v,W &ts > U w$ <"414ks > w 4 v ](w8>fT <F0 &UJU9< 4)48<U20 wJxfw`T u w)($'w8>kv l&U x & v < & > 2ω1a1 + 20ω2a1 ≤ 20 W < >6x Z d0<ts v l&U 0 [ & 4 vZ xfw] ]-w U v ω1 = ω2 = 1/2
l/4& v a1 ≤ 20/11 w8v a2 ≤ 200/11 Wi Uc0 [ & 4 v < T & > 4 a1 = b20/11c = 1 w8va2 = b200/11c = 18.
\)E3C=:@3-+ @< 3I5V87@3C@3:84?7@5V- 7J ;2;$C3I;<3IG* 3F723X@7A3=7 36X-(4-5;2;4DX3&73%45o 3^/-Rb3B7@5-3eb;<3∑ +g- 7J3 7>7@3C87@3X;<3 7 $B=5o; 3C# 3X 4-:75KV3Q3; 7+K3 7 ;+;5D::3[;<3h=:CJ4D.=33/ \* 3; 7#7(4D@7@5o K5,X3XE3X87
?i − L!)f &* +6 3!7*4 1 &% - 7 !)0/8%$ &* * &!G0/%'+ 4* =:; 3C5KV$>K:2;2 3EK3B;E; 3 5V;E;@-87I387 CE3;/ *Ac 4-:75KV3#; 7.$B=5DC$#) 7J3 7X-@3B;72-(;@3()QV4[;(4?7@3+;3;<3X5,X3N;$8!/4D5VKH4D.=3E;<3*b3 3B; ;@3 ^/;$-3XE5KJ4`:-3eb&;3 ∑ 3X7 " 4D875KV3) ) 3>;4?73L;+* 4D =7Z;<3 ∑4e?4D87L;$8!/4D5VKH4D.=3"^@;@-5K7 (3 X3X; 3Bbk/ Y4D;LG* 3<3X(7V3 72$8$C;<3C870+]5K>4`::5K(-; * : 4-597>;<3 ! -5o;J7( ;=:CJ4D.=3&3C+H4J; 7.$B=5DC.X4D7@5V-F;3
xe,1 3]=3CKV3d;3
xe,2/ &K:2;
ω1 = 2/33X7a1 = 2
/ .#X5V;<3CEL3C870+-F;'$0C j5K7
ω1=:EL3OV4X7-R(4-R5KV5K7 $ 3ig* 3723X@7N;@-5K7 7 ; ; 7@3iK:2;2 * 5Vf; 79$=5DC(3ig* -;<3j;<3
x1,e 3WX3X 5<;<3 x2,e
/
.#G3; 7]$::4-K3CL3C87575VL:;5V4-;3f$X5?CJ3X] 3LH4 ;79$X5?C.C4?75K:U;<3EG* 5V87@3C ?4DVV3[xe,1, xe,2]
3; 7+0 & [ Z > w U vw4e-3B&V3ZE;',XV3YRJ4e[3$;@5V3X / 7@5VV5;@3A7J =3CV4LH4J! - V3-^@i'/ b0/ -5K7
βe = log
log(1 − α2)
log(1 − α1)
log−1(xe2/xe1).
5 βe /∈ [βl, βr]+ 3723X@7@5o; 3B;W-R;@3X2?4DRV3N3X7672$C;<5o75D:3-; ::76X- 5=7 3CKV3;/ *A3 ; : 75K: 3B;b7E$5V;<3CEL3C87
;+* $XH4D2:-5VYE<;$C $CL3C87[βl, βr]
/ \O38723X(;4-870+;@5βe βr
+K3>E<;,XV3+;<3 U3X5VR VO;<3-7J4D4D(,=7@3E;<3(! -E3βe
3; 7 7%43 ; * (3 V-5W:@>4DV3e3X7[g* 3723X@73B;b7 =H4D5V@3CE3X87F7@f7 5V! ->4?7@5D! / .#W3; 7Q4-K:2;'$X3;2;@4-5K3;<3 72-J;$X3XV4 =2$C;<5VR5VK5K7 $
(α1, α2) ; * ) :R<7@3C5V
βe = β0 ∈ [βl, βr]^@7(4D3<3C7(K3UV46Et[-3X(3Bb0/
\1* 4D:-:@5K7 4E3E; 5`4-870+(;$C@53$N;<3 A3 7@: j &4`7'4;@-+'7J3C@E3X7d;<3h2$C4-K5o;@3XY! -E3XVK3CE3X87=3X7 73E7J:(;$C4D7@5V-4e-3B&V3W74 ;<3EX-F-3C ::3XX3=:; 74D87
|αk+11 − αk
1 ||αk+1
2 − αk2 |
=ω1
1 − ω1
)+G* $X7437J3k → k + 1
/&5K.; 5 +9;@5ω1 → 1
+α1
$8:- 3E723 4 = %;&;<3Lg* 4D:-:@5K7 4(L3f/J\ 4 ; [email protected]@5V-e;3NH4=:;b7 7@5V-3; 7&;<-('$X3W3X &3<3B; § i'/ '/a^@7 / a_f_fb0/
M o TVU q a3/7n M.ONL P o IUBT@M9O L q I Q o PFLNT T TVUP t
0 < α0
1 < α02 < 1 α0 = (α0
1, α02) `0 = (xe,2/xe,1)
β0 `(α) =log(1 − α2)
log(1 − α1)
δ−1 = ε+(1, 1)
ε ! #"$&% '() 0 < ρ 1 * + " -, ./ ρ ∼ 100ε 0
132&4 "& k = 0, ...K 5 &46, ‖δk−1‖ > ε
• 4 $ δk =
((1 − αk
1) log(1 − αk1) (`(αk) − `0) /`(αk)
−(1 − αk2) log(1 − αk
1) (`(αk) − `0)
) 7
• 4 $ hk = − (1 − αk1)
(1 − αk2)`(αk)
• 4 $ αk+1 =
αk
1 − ρhk(αk2 − αk
1),
αk2 − ρhk(αk
2 − αk1) ω1
1−ω1
$A%G'%,'/98 ;:<U-4β0 = 3 w0v (xe,1, xe,2) = (200, 300) W & U>=&< 40<U v (α0
1, α02) = (0.05, 0.95)
l?<Uc< < T < > 4βe ' 10.03 W Au w :@= w8>kv 4 w $U x s v6x <U20 shU w T$T 4 4 w]-w U v >k> Z%< T 4 vw W i U -A: w ω = (1/2, 1/2) Wfi U vg> <`s ew< T < > 4 α = (0.3, 0.7) WY T u <U(=<"4 w ] <$U v|w U9<U v α0
1 = 0.25lB<U vG> <ts tw βe ' 5.78 W w w T$T 4 4 w)](w U v $U x s v
w 4 vw U20C< > wvG> <C= dD< >kv,Wj tw 0 xfw 4E=&< x 4<Z5m8s T ^8> Z 45lF<U vG> <ts tw α = (0.4, 0.8) W Y T u <Uc< shU w dD< >kvw 0C<U -W<U20 wx <U-4 T <>0 > Z x ^ T v Z xfwpT < ^ < > U w $U&dZ > w s > w l<U-A: w ω = (0.95, 0.05) WjT < > 4 α = (0.26, 0.64) Whb U ew8> 4 w]-w U v l< tw 0 ω = (0.05, 0.95)
lG<UH< ^8v w U v α = (0.55, 0.93)/
& DV
O q PfU ML q PFONP o I q L TgK8UBM Jo I qutJ* 3Y!/4 X-:3$C'$X4DV31+?4 7@ g;+* 3y?4DV3 xe * 5V! j
5K70+8G* 3F723X@7Y7J3 74e:-5V 3d375V5V-F;3dH4h;<5o;2;[($=7@5V3&;<3YV4N;<5o;b7@5VR 75K:F;<3X+8;<:87OH4I2$87(4- 75975K:-7J3 7
X7@3Y437'72$=5$X37(4D (3Y($X7 4<;<3W;+* 45; 7@f:-4DEE3J^g 5_ * 3B;b7 * ; 0$M;4D'$Y;<3WV4N($=724<;<3Z;<3WV4NR5o; 3B j75K:.bk/g4D A-@7^5 bo w8v < T'W ^ a bY7372f;@3X87Y;<3; 3:-E3X873X
IR+ 3CS5V:73X2?4DVK3B;A73X(7J:@3C;Y;<5o; b:5K87%;+23X%4:5;@5o; ;4D87V3; ! -87@5,X3; 3XL;<3;N?4DV3 %; -R;@3X23$C3;2$X3CKV3XE3C:7y^/;3YX3X; 3j :.bk/ F %4(4f 3O5V87@3C ?4DVV31+G* 37J3C 7]5K(;5 3Z;@4y7(@:R(4DR5VV5972$O;<3i;$0!/4-5KVV4-=3-+7J4D3<3X(7V3W;@ ; ! -L3i;<31! 2$ 3C=3O ;3 7(4-@5G/C\ 4A2$87(4- j75975K:N3X7gK3:hR3^;<3;G;<:'$C3;NC.=7@5-3; 723 -3X87g4DV-%; =7@3Y7372f; $;]3X! :75K:L;<3^H4572$=5o;@5K:N4?7@7@3C5K87@3f/
$&%G(' / Y < v (50, 80, 170) vG> < < Tw s > xfw x Z d0< T$T <U20 w < ^ w8> Z w w8v 90shU w < TDw s > x3w 0 w Uks > wD_
x`> < v|w&Wfi U)Z ='< > wBT u w ='<:0 w xfw v|w] = x3w x Z d0< T$T <U20 w w U D1 = [0, 90[ w0v D2 = [90,∞[ W Y < w U v (n1, n2)TDw U/< ]B^0>2w x3wEx <UUIZ w -O0 v tw > w = w 0 v tw]-w U vWx <U D1w8v D2
WeY s ==&<D<U`m0s wpT u w :D= w0>kv $U x m8s w && xfw0 [ <U20 w-x3wv < ]B^%w0>Jw U ='<UU wx <U T < = Z > < x3w D1
W i U"< x <U20n1 = 2n2
WY Z =< > <U D1w U D1,a = [0, 50[
w8v D1,b = [50, 90[ W
b U x m8s<U _ T u w :@= w8>kv m0s w T < x s > Z w-x3w w#wgXw 0 v tw#w v 0C< ] = > wx <U D1WY s ==< <U< T < > Bm0s1u T $Ug
x m0s w " + xfw 0 [ <U20 w x3wv < ]B^2w8>w U =<UU w x <U D1,bW shU w m8s w v <U ] T < >2w %s > shU w =< >kv v <U xfw
D2l Tw v $U 0%< =< ^)TDwJxfw w = > <U/<U20 w8>)WY T u <Uks ==< w m0s wpT u w :D= w0>kv U w = w s v = T s > Z =<U xt> w _ shU w m8s w v <U=< >kv <U v ks > shU w <ts vG> w ks ^2x <U x <U D1
lE<UBd0< v7T u [ a =< vO[ S w m8s1u T < * s , n1 = 4 x <UUIZ w x <U D1W
j s -U< T l T < x <U20 = w8>! s n1 + n2 = 6 x <UUIZ w -O0 v tw W
- /+' /- - #-% -6!'[-\1* $=74`723h;<3N@3BX4DV5VR4D7@5V- C((4-K36X-@3B;72-J;4 7@4e?4D5V=5975 3N;<3hG* 4-(4D[;b73ZR(4e[3$B; 5V3X+5KJ;$8723X(;(4D j
E3X87F;<3S7@ 7 =::743=7F4e-3V3[ V3;>37J3C 7%;8/&3 ! :5; =-.;b7 5K7@3B;EK3B;+E<;$XV5o;@4D7@5V-;>;<3B;+3F723X@7@5o;@3;;<5o; 7J:5KR(K3B;+ 5V]3B;b7W'$B=3;2;4D5V@3h;3NV3;]=:hR5V3Cd3X7W;<3LR2-3XW;<3;d74D5VKV3;
a1, . . . , aM$8:3X87 3CKV3XE3C:7Y7@37
::4-(;<3;Y5o; j ) j 5;;<3hV4-X-(4-5;2;4DX3Y:Rb3=7@5-3I437'7J: 72$X3A7(4-AV3;A;<:'$C3;yn
/A ;A;<:-.;^3XhX36; 3C; 3C 3B;1=:; 3C5Ko;/-gCJ6+:V-%;2 3^g* 4D(4-D[; 7@3A;$B=5H;<3&;+* 75KV5;@3XiV3AE;',XV3X)Z5;2 3;1X-(79$=7@5K7@5D! B(η0, η1, β)
+ ;A;<5o;2 7-;d; 7-R,XE3h; * $B 5KV5KR(43:-3I;<3IG* 5K! :@>4D7@5V- <9= > < > ! 5V3h4 ;@ ; j E<;,CK3B;^3X87@4D873C X- @3X.=33/
t t t HJM9O K q O K r K q OU oq q q0o UK\:2;2 3Y7 ; 5V3 2;3F723X@7@5o;@3;
e1, . . . , eM;@-87j;<5o;72-(5KRV3;0+D;<3 NE<;$XV5o;@4D7@5V-; 723 -3X87 =7@3 7@5VK5o;$C3;/
\ 4Q0C< ]<^ $U9<D<Ua0C<U tw : w
π(θ) =
M∏i=1
πω′
ii (θ)
∫
Θ
M∏
i=1
πω′
i
i (θ) dθ
,
-8 ∑Mi=1 ω
′i = 1
+<E5K(5KE5o; 3ZH47J3C 73I;+* 5V! :@>4?75K:;3 T] KVR(4fo j \3C5KR(K3C
KL(π;π1, . . . , πM |β) =
M∑
i=1
ω′i
∫
IR
π(θ) logπ(θ)
πi(θ)dθ
3X72$C4DV5o; 3 0C<U w Uks>377@5VE4-Y;3;Q;<59$C@3C:73;3723X@7@5o; 3B;8/\3B;72-5H;';ω′
i
@37 $B; 3C87@3X87g* 5K(72-@74-=33XH4?75D:3;<3B;>3723X@72;/Z\O-o:3 w8v < T'W ^ a -m-mdb7+ ;<3B; A4-:75KVH4 ^ _f_3_db7+G\4-t[ FC43=5H4 ^ _3_aBb
?i − L! &* + 3!7*4 1&% !)0/ % &* * &!G0/%'+4* 7372f;@3X87&-74DEE3X87 S3X;@3XNRV3N;<3=5972,X3;^73Xo; 3hg* 379$X5K3C=3N V3672 %=3C:74`::3h;+* 3X@3 7(4f; ; $7J ::3Xh;<3B;
ω′i
/\4Q($X7 4<;<3 d3X7'45O3; 7 a3<3X(7V3#0$CD,CR@3>;<3F$X7 4<;<:Kf:-5V3N3Cc=3-;@3X;/\5V5o; R23X2:^ a 'aeb AK7 4 C4 w8v < T'W ^ _f_ b7+?3X87@3&4 7@3;0+t! ::7i4D5V; 52;<3XX3=-.; 3C; ;GV4I@37 $B; 3C874?75K:N5H;$M4DV3&;+* :- 723W;+* 37J3C 7%;8/-jG* 4`7'7V5o 4D87) π(θ) = π(µ|β)π(β)
+J:Q-R<75K3C:7^G* 437'72$X5V4-RK3A; 74-R5KV5K7 $
µ|β ∼ G(
M∑
i=1
ω′iai,
M∑
i=1
ω′ibi(ai, β)
),
β ∼ Be
(M∑
i=1
ω′ipi,
M∑
i=1
ω′iqi
)
K:2;2 3 πi(β)3B;b7 3ZV-5 X74 Be(pi, qi)
;$0CJ5K3E; [;<:>4D5V3[βl, βr]
=:L /
\O3F7 [F723;<3E<;$XV5;4?75K: * 3; 7(X38723X(;(4D87J7(4f;+N72 * k&4`:84D ^ _3_fi,+ _3_ b<+V3E3X5VVK3 (7Jd; ;@5KR(K3f/-gP3 3=70+3CKV3Q'$X3;2; 5K7@3 3V3;L3723X@72;(;@-5V3X87L5V(;$7J3C(;4-:7%;L3X87@33 + =3 5i3B;b7>;<5#=5VK3 )&3$X5DC(3XB/d4-;gG* 5K(; ;b7@5V31+F;@3 JK3XC43;O;<3d;<3 E37J3C 7%;X^V!/4DR5C4D87j3X7O37V-5K74-87A4-RJ:(;<3:f$X'$C4-K3CL3C87g3X#=35; 3C;/ j ;<3C)N;<3Y;3 +3F723X@72;0+8g* 4F[d72D724',;@3d; * 5V(;$7J3C(;4D.=3Y3; 7Z723 =2$C;<5VRV33/ * kd4`:84D-72$0!V,C@3:3$X$X4DV3XE3X87;$0CJ5K "=:;@3X; ;7(4DYK3ER5H4D5o;W;<3E;<5;2 ; ;@5K:;d3X87@3N37J3C 7%;+ 3=7W3X.; 5K7@3(70$M;<3X])FH4L<;$CK5o;4?7@5V-/d4-;dD7@3(X4f;+=3X7 7@3L($=724<;<3N5V(7V5 3E;<3(=-.;b7 5V3 $B%4(4D875KVK: C.=7@5D!C =-.; 3C; 3X Z>4-5;7J3C@E3X7;<3W@3X7@:hR23X; ^K3Z CL3Y7 [F723I;<3ZE;'$XV5;4?75K: /
t t t T cTVUBIUTgM.ONK6L q UBI'T qut2,;e4375V;3XE3X870+; 3CK: V3;e ,:-V3; 7( $B; 3C:72$X3B;S4 7J4D4e`4-870+^- 723 7a4D@5-3Ch) ;<3c::4-(;<3;[74D5VVK3B;
a1, . . . , aM;+* $B%4(4D875KVK:-C.=7@5D!Z^V72 M 37J3C 75;@3;i;<5o;72-5VRV3;%bk/# ;W7J4D4H7O5V(;<5o;723X;4DR(K3Y;<3dR2-3C
; 7.$C@5V3 3XE3C:76X3;W74D5VKV3;I3X"! -.7@5V- ;3;Z3X>4D% 3;A; 5?4D87@3B;8/ c(D7@3a =
∑Mi=1 ω
′iai
H4Q74-5KVK3+;3g* $%4J4D87@5VKV- C.=7@5D! X-@3B;72-(;(4D87Z)LH4-=-NR5V(4D5o; :;<3B;3723X@72;/
a3/ ;-5K74e-:5Ka < n
-8n
3; 7 5K(;5C4?7@3 X-@3B;72-J;4D87)cH4c74D5VKV3 ;<3 g* $%4J4D87@5VKV- 5G/ 5g/ ;+/437'7J: 74D87hH4[ XE3 4-87@5K7 $; * 5V! ->4?75K:c 3+K3B;h;<:'$C3;N-R;@3X23$C3;
yn
/ A ;E$X7 ;<5V-.;N4 §i/ H4 =-.;b7 =7@5V- ;+* P7@3Xd5KJ;<5C4?73 B/W\O3=7@7@3[R2-3 !/4D5K7; 3C;-7 5;2 * 3CKV3Q@3C(;PK3B;>2$; 974?7%;h<=< v|w0> < > ; θ′ ∈ η, λ, µ >4 b:@5K74-5K3XE3C:7G;$7J3C(;4D87%;j;<3B;g;<-('$X3B;G $C3XVK3B;8/:\3;ai
;<:5D:3X87j4DV-%;X7@3Z@3C->4DV5; $;O3X
a′i = aia
M∑i=1
ω′iai
.
\ )>3XX-31+ 7@5VK5o;@3X&H4-7(4- 75K3h3X87@5,X3h;+* [73X2$; K74D7!/4D5K75;@3X;; 5 g* - ;$;@5K3E=:;@3X2-3XA;3;&74D5VVK3B;3C:75D,C@3B;8/\O37J3C(;4D879+'=3X@74-5K(3;O37J3C 75;@3;i5;2 3X87^4DV-%;j;<3Z;<5o; 7(4D4H7@3
(a′i = 0)/ 72$0!V,C@3W;<-.
X-;@3X2-3C0 < a′i < 1
3XND74D87N 3iG* 3F723X@7ei
437'7J: 73O4-K:2; 4 74D87];+* 5V! ->4?7@5V-J * (3 ! 43=7@5V-;<3;<-$X3 W W=x3W ^@;@-5K7hG* $B 5D?4-K3C:7L;+* 3F;<:'$X3 =3X.; 2$X3eb7+ 3XcX-(73X(4-:76 * 5V1!/4 7 p > 13F723X@72;
e172 A437'72-@7@3X^G* $ 5`4-K3C87^; * (3I;<-$X35C.75D:3Z5g/ 5G/ ;+/
/ &3I4DR2-(;(4D87@3YK5K7 72$X4?7 @3W3B;b7=:;43=2$X3])[email protected]?7@5V-; ; 3 5V;<3;y 4-:75KV3;/i3C4 = 7;+* 4 j73 %;I3C"CJ4DR5VV5972$>5K(; ;b7@5V3XVV3;8* 43=:;<3C87A; IV3 ; ;@3X;Z2$X3Cj;<3B;E?4DV3 %;]7372f; $X3B;A7(4DhK3B;3F723X@72;/(43X3d4 > $B; K74D72;O379$X5KE3X874 +fX3XVK3B; j =5;@-87W=2$C;5972$X3;W7(4-jV3;i3723X@72;g3 j CL3B;O;3
! .!) .+ ) " )!,/&(,/.!)! *) 8 #% )!! 2" .+.!,/&(,/.!)!
& DV
; 3 5Ko;I;<3-X-CJ4-=3L7 ,B;E; -3C87]; 2$8?4D $;/ \O3723X ^a O b<+ .S38[-3C i-o:3X(^ a -m b7+\ 4Dt[ @C438X5V4 ^ _3_aBbW7(4-@E5;+* 4 7@3;^5V;@5; 7@3C:7X; AV3IR5H4D5o;^5V(7J: 74D87&;<3E=3B;?4DV3 2;^;<3J; 3 5KG/ 3ChRV3; 3I;$::43:-3X (3Z ,:-V3d743X5973E 5 -3 7X 3Zg* -[4f8=:;3W4 F3F723X@72;AK3B;^-;<3;i>4 <5K>4 ; 5`4-872;^V-:5KA4 ;2; 5274-RV3C4 /Da-+\ 4D(t[ X4f8=5H4 _3_aBb<+F 5 ! 5o; ;@3X87^;<3;^R2-3B;y; 79$X5V3 3; a∗i
/
) ) a∗
i & # !&! !& "# & &# '&# & !%" &
l + i/ \O-@3B;72-(;(4DX3&743=5K7@3W3F723X@7@5o;@3 j $M4DV5972$d3C:7@3W:;<3;;<3E 4-87@5VK3B;8/
i/ 5 4-E4-5;(7(@3 -3Q3; 7#!/4-5973h 3 xei
^V! - $X3 7(4-+g* 3723X@7eibE3B;b7 (3 `4-K3 +3 $X74-:7-7 ;>H4
=-((4D5o; ;4D.=3^;<3^73X(7;j;3X=3C; @3Y7 7-7Z 3A;<3X3$X5K74DR(K3B;]7@3X(7;O;<3&;$8!/4D5VKH4D.=31+-3=7j3C-7(4D@7@5o V5V3X3X X4f;Z;<3# 4D87@5VV3L37@ XE31+5Vg3B;b7I43;2; 4-:7W;+* 438X-;<3XA)G* 3F723X@7IG* $B 5D?4DV3X87I5V! ->4?75?!^;+* 3; 3 K3W;<:'$X3
(ai = 1)-:5K3X3X 5_;+* 3Z;<-$X3]=3X.; 2$X3f/ &3d7@3C;5V(;<5oX4D7@3 2;Y;@-87Oo)N4 ;2; 5_;$8C(5o;
4 § i'/ / #N;8* 4`::597&; * h%4(-5 7(4Dd;$0!/4 7&V-%; * 4 3]72$X5;@5K:* 3B;b7d;5; 7J:5VRK367J A37V5o 3XV4(; 7.$B=5DC.X4D7@5V- /
/ A3L4 7@3E7(@0$M; 3I;<3JC4DV5KR(4D7@5V-6+:Rb3=7@5-3-+3; 772f;2; 5VRV3I3X 7@5VK5o;@4-87dV36=5972,X3 X\ ;$8C(5]3=7$=7 ;<5D$L4 %4(4375K7@3 :J: 7J3 7XC 3C a = a∗
7@3XVV36 3 X\ (a∗|yn) = 1/\O3E!/4D5o;@4-:79+.;@5G* 37J3C 75;@3
3; 7]$CK:5D::'$X3L;<3;Z;-'$C3;0+ -&;* 43;2; 3J 3 π E<;$CK5o; 3 3E5V=3C 7597 ;<3($XV38f$X3-+X3 5G3B;b7 S$=74?7;<3!/4D5K7W43;2; 4-8772 Z;<3B;]7@:R,XE3;d5KJ; ; 7@5K3C;/ G-5VA=3-%4(4`7(597@3 72 I;<3;Y37V5oX4?75K:;57 ;7 $B=5o; 3B;8/
A-7@:;]3XC(( 3K3y%4:5 B1
^@0! /t7@f7Jd; 5K7@5V- fb3; 77@3X' 3G* :67 5;2; 3E<;$XV5o; 3C 4-:75KV3;+* :;@3$CK33$1+7(4DA3<3X(7V3 +7@ 7d3X[ 5]438X-;4D87 36`4-K3 &;<3h74-5KVK3]C97@5-3hR5K3CSE-5V(;<3E 3E72$8 3E;@5G* :S; 7'72f;@3- 3Eg* 3723X@7W3B;b7EX-@3B7-^ 5o=5
a = 20bk/ S7J3 7A!/4D5V@3LV3> CE3(X-EE3X874-5K3(; ZV3;Z;3
4 7@3;iE<;$XV5;4?75K:;/-gQ $B; K7@3 (3A:-4D(;<3 35VR5VK5K7 $I;(4D;^V3Y7@4e?4D5V;<3Wg* 4D(4-D[; 7@3WR(4e[f$;@5K3C /
t t t HJI T o I'UT@M9O L r cM L q B(η0, η1, β)t
\1* 3X(7V-5J; E;',XV3)W@5o; 3B;1=:(7.$X7@5K7@5D! B(η0, η1, β)'$X3;2; 5K7@36< = > < > ' 3AG* 3F723X@717 5o;2; 3^2$872-(;@3
4E>4?75D:3XE3X87#) 3 3B;b75K:P; 7 [F723 CbV3h; [F; 7 ,CE3 ∑ @5o;2 3 j 7 j 5KA;+* =7@3h;@ L5o;-)c;<3B;+;$0!/4-5KVH4DX3;4f8=5H;<3C:73XVK3B;+:3X ; ;i; 85V3X5VVK5o; ;@3XE3C:7 k/:\ * 4-(4D[;b73&R(4e[f$;@5K3C-7J3 7^4 ;2; 5 75KV5;@3XiV3;O 75Ko;j(-7J4D4D($ j7@5o 3;^5V87@<; 5K72;&4 %4(4`7(597@3 7J d437'7 [:3XXX36%4-5 +/ A-7@:;7(4-X;@5K(7V5X5972$
λ = 1/η0+µ = (1/η1)
β
3X7λ ∼ G (a′, b′)
/
723 7A; 7'72f;@3X] * S7 ,B;dR2- 3723X@7! @(5;2; 3-+24372,;5; :KV5X5974?75K:6+ 3L; $C3L;3J5V3NEt[-3C34e?4D87A;$8!/4D5VKH4DX3I43=5H;<3X87@3CKV31+<3C ; ;&;<3;^5V! :@>4?75K:;A;$!) 7@5VV5; $X3B;y7J AH4LL<;$CK5o;4?7@5V-[;<3 U3X5VR VG/& ;A;38-:;X38723X(;4-87!/4D5V@3 h%4-5 7(4-&;$8!/4 7;@5 4 [37J3C 7A(3E7J3 7d2$872-J;<@3E)-X3=7@7@36 3B;b75K: /\ 4+;<5- K7 $L3; 7Y;<:I;<3NL3X7 7@3N3X X-(7.$X7@5K7@5V-
π(λ)3=7π(µ, β)
;4D.;!/4e:-5;@3XA4-@R5K7@4D5V3XE3X87AG* e G* 4 7@33/\3A%4-5 ;<3B;Y4d[F723X27(4D4D(,X7@3;i;<-5K7A@3C(;<3WK3B;L<;,CK3B;R(4e[3$B; 5V3X.;W$B 5974DRV3;/
?i − L! &* + 3!7*4 1&% !)0/ % &* * &!G0/%'+4* ; 5V3 %;Q4 73 %; ^d4^5H; \ 4 5K7 X3C _f_3_ +j\3 X4-@5 \O-;@-(5 _3_3i + ^t:3X4?7 \O-;@-(5
_f_ +\O:; :5 w8v < T'W _f_ R.bg:877@f7Jd;$Z;<3B;Y! ->4DV5;4?75K:;Y72 A;<:3C ; 3C;XV4-5K)-X3=7 73I-7@5V-;+* $ 5K74DR(5KV5972$A^ 0C< ] =< v ^ T v Z2bk/.e4D5K ^ _3_ db;'$0C(5K7
π(λ)3=7π(µ, β)
X-EE3YX-(7(4D7@5VRK3B; ; 5<H4W;<5o;b74DX3;<3 T] VKR(4fo j \3X5VRK3C3X87@3WK3B;E4- ::5KJ4DV3;
mW (x) =
∫ ∫fW (x|µ, β)π(µ, β) dµ dβ
mE(x) =∫fE(x|λ)π(λ) dλ
3; 7gE5K5V>4DV33/-g+;+* 4 7@3B;7@3C@E3;0+3;43%4(4-87π(µ, β)
+::(; $XV3=7@5V-(3^K3B; 4F[d723X27(4-4-,X7@3;(a′, b′)∗
;<3π(λ)7@3C;y 3
(a′, b′)∗ = arg min
∫
IR
mW (x) logmW (x)
mE(x)dx
^@i/ mdb
7J I:R<7@3C5KNV4[;3X;@5972$π(λ)
H4 7 ;EX-(7(4?75KR(K3f/ \4 7(@f7Jd; 5K7@5V- 5V(;<5o 3 3 (a′, b′)∗35o; 7@3>3=7N3; 7
5o 3E; ;iV4-X-(;5975K: 3-+7J k = 1, 2+
-[Xk|B] =
∫
IR
bk/β(a, β)Γ(a− k/β)
Γ(a)Γ(1 + k/β)π(β) dβ < ∞,
=3A 53B;b7y4-5 ; 5a > 2/βl
^V72 ^7@ 7%4-5 ;<3Bbk/
EGFIH HLK&M9NMOHP / Y < v π(β) x3w ks ==&< >kv [βl, βr] ⊂]0,+∞[ W Y < v π(µ|β) vw)T$TDw m8s w =&<`s > k = 1, 2l & [Xk] <∞<\ & w v T u w =Z > <U20 w =< >> < ==&< >kv_ MW
WJjT < > T+w : vw shU shU m8s w 0C<ts = Tw (a′, b′)∗ > 0 < T s v <U x3w Q WVXW
npoq0rLsJq+--5V A(3<3; § i/ /Da(^V7 /a_'aeb
.S4D43 @3 ; 3CE3X87 =3X7 7@37372f;@5975K:L5V(; 597O;3;gV ;<3 %;g;<3XC4Do 5J@3C(;<3C:7OV4Z($X7 4<;<3723 7(4 j7@5o 31+8D74DEE3X87y; 52g* -FE4-5V3X7 ;@5K3 2;]<F= > < > N^@7(4Di3<3C7(K3dK:2;O;+* 3$X7 ;3Y;<35; 3C;@5KR5VV5972$Bbk/. CE35K(72$X5;@3;0+MK3B; 3B;b75K>4D7@5V-;;<3
(a′, b′)7@ f$X3;5KJ; 5o; 3C:7G3C6:3$C'$X4D;<3B;Y<?= > < > ;<::7V3; X4-4f7 $C@5o; 7@5o 3;
; :87Y7(@%43B;i; %4(-5 a′ = a,b′ = b(a′, β = 1)
^@i/ db
5 ( ;5; 3ChRV3hV367( ;]X34'$C@3C:7Y4e-3IV4 ;79$X5?C.C4?75K: : 3N CL3J:-4D(;3 dE4- ::5KJ4DV31+(:R<7@3C 3N4e:3 $%4(4-87@5VKV- C.75?!g;<3Z CL3I74D5VVK3-+3; 7X=:L 3I4 ;<3 E<;,CK3B;8/ # ;Y7(4D4H7^K3A7 ;;@4D7@5o; !/4-5;4D87X4-iV3;4d[F723X27(4D4D(,X7@3;Z=-.; 3C :3X87 ;@3X.;y=-(72$84(3X;@5KR(K3f/
F m/ <ts U+u < <UF='< /0 ] =<)Z T < >S c TDw = > < =<)Z =< > ! w0>kvO[ < T <U w0v < T'W Q "%$ $%&V lm8s ks ==&< wm0s w η0 > η1
< - U x3w cd< > <U v > m8s w TDw w T$T w)](w U v]w v7T <[0%<ts w = > $U20 ='< TDwxfw x Z d0< T$T <U 0 w W w 0 [ <;:&<= > < > x3w w U v < T < >
π(λ, µ, β) = π(λ|µ, β)π(µ|β)π(β)
< tw 0
λ|µ, β ∼ G(a′, b′) , λ≤µ1/β,
µ|β ∼ G(a, b(a, β)),
β ∼ Be(p, q)
& DV
w8v T < ] $U ] 8< v <U Q WV UuZ0 w v|w shU 0%< T 00s T vG>S Z0C< ( vw s:Xl T _w U20C< > w ksX0 w = v ^)TDwJxfw ] < x - w0>`TDw w U xfw [ha = w8> ='< > < ]BS8vg> w WY <UE< < >Jvw v Z T <9= > < : ] v Z xfw > Z ks T v < v 5lU/<ts ksXc&c0Z > <U ] = TDw)]-w U vYxfw ^ < > U w8>`T <x vG> ^ s v <U G(a′, b′)
0 [ < w w)T <U Q W VXW
' '-' &( - c-- '#" - - d' \1* 4DJ4D[F; 7@3WR(4e[f$;@5K3C;<:597A3CC( ;* 4f; ; 3XY 3ZV4(=:; 7@ 75K:"< = > < > _3; 72373A=3X@74-5K(3;y=:87@4D5V:73;
;<3]=f4'$X3X.=3&4e:3AV3;Z737(@5$=72$;j; E<;,XV3W;<3 U3X5VR V 3=7^V35X-87@3873&5V(; ; 7@5V3X_;4D.;OV3 3CJG* 37J3C 75;@33B;b7Y!/4D5K7@3f/ A ;5V(;<5o -;^5o=5;3 X4D43=7 $C@5o;b75 3B;j3 23B75D:3;y 3E=3C 5 j =5;<:597X%4(3X%%43Xy)3$C@5DC(3XB/
t t t O T cM LNI T@UP L q ISL TgK8U o T r UTgM.OM(X)
\1* 5VE<;4DV5972$+;<3M
3X 3-?4-K3 6; 7@5=7@3XE3C:7E72f;@5K7@5-3>3; 7 4f;723=7I-a'$:-V5D::3C4DR(K3E;<3+H4QE j;$CK5o;4?7@5V-/ -ga3 3=70+
M;<:597I3; 7@3C57(@%43>;<3EH4[;<5; 7@5VR 7@5V-a;<3 U3X5VR Vj;<3>E<;<3-;b7@5o73XE3X87E72f;@5975?!
^g; 5:e372-3C87@5V3XVK3E E;'$XV5;4D87 e4 b3 5o; ;@3XE3X87kbk/ 4DZ4D5VKV3 2;0+JG* 37J3C 7]7372f;@3 3J5o; 5V-U;<3H4+; 2$X3N;<3E5K3L;+* =:(7Jd;@4-:7 ∑ +; E5;X7@5VX5D7(4-K3CE3X87)+; 85V3X5VVK5o; ;@3XE3C:7B/ -geg* 4DR;@3X.=3h;+* 4 7@3;;@ %=3B;h;<3Q;$8!/4D5VKH4DX3(-64f8=5H;<3C:73XVK3B;+ =3X7 73Q; $C3;<3 5K3 72f;2;,M;<3F$8::5975KE3XE3C:7 75o (5 33X (3A`4-K3 A;<3W73X(7;^:Q KV3Z18V4(7(@:R(4DR5VV5972$W;3I;$0!/4-5KVV4-=3W3B;b7A>4 <5V>4DV3(^gk! /C: 3;Yi/DaBiN3X7Xi'/a bk/$X5?C(3C=3X7 73 5VE;(4DV5972$17(4D@7@5o=5723j;<3gH4^R2-3j@37 $B; 3C874?75D5K7 $g; E<;,CK3gRJ4e[3$;@5V3XW;4D; ; :W3X;@3XNRK3f/
Y4D;V3;C43;(i)
3=7(ii)
;<3WH4(7372f;@5975K: ,+<( ;^4e:-;Z=:;b74?72$YG* 5VL<;4-K5K7 $I;3M
3X 3A?4DV3 ; 7@5=7@3CL3C87 7Jd; 5K7@5-3; j7@ ;gV3;g3<3X(7V3;^/18
β * 3; 7 7(4f; X- 3=7 C$X)
1b 5H4WK3&7@3B;b7 ;+* k&4- 75D:84D
k&4D@7@5::4-&^a-m 4dbk/ (D7@3-+'72 p 1+
d5D7(Mp|B) = inf
V ∈Vsup
x|Mp(x|B) − V |
-8Mp(x)
3B;b7hH4[;<5o;b7@5VR 75K:c3X(75V@5o 3 (1/p)∑p
i=1 , Xi≤x3=7 V 3B;b7NV4 XV4f; ;@3>;<3;N;<5; 7@5VR 7@5V-.; 5 j
E<;4DV3;-^/ ($X5 3CL3C870+;<3B;Z;<5o;b7@5VR 75K:; 5D! -E3;Z VV3;E;@-87 7@5VV5; $X3B;E=:LE3>2$0!V$C@3C=3B;2b0/ 5M
3B;b7 5VL<;4-K3L3=7]; ;5=3X@74-5K(3;XX-87@4D5V87@3B;&;<3L $: H4D5K7 $1+ √p d5D7(Mp|B) → 0
3C 7-R(4-R5VK5K7 $f/ 723 7+4-5K;@5y=:;b7 5K3 7@3B;b7>;<3 7 5o;2;@4-=3 $XV383$C3S^k&4- 75D:84D a -m R.b7+ 5^5V(;<5o 3G* 5KE<;4-K5K7 $ ;@5iH4; 74D7@5o;b75 3A3B;b7W7@%43A;3_'/ &3&-NR@3 N 7@5Vo;j5V! ->4?75 3B;G-87 7(4-gH4E; 5973X$=72$&;$-3XV377.$B;X^=! 5; j;4D87 7(4-O3<3X(7V3 3XC4-;<3XE3X87; E<;<3Bb0/ Y4D.;OV3A74DRV3C4 i'/ i,+8 ;j5V(;<5o -.;W)I7@5K7@3Y;+* 5VK ; 7@4?7@5V-G* 3B;b75K>4D7@5V->;<3 d5D7
(Mp|B)72 3C 3B;1;20$XJ4D5Kd;<'= > < > =/ 7@5VK5o;@3 200
?4-K3 2;G;3X; 5V V4D7@5V-;/ 9+;43%4(4-87
p+.;@ ;&g* 4F[F7J-7 4',B; 3E 3 M ;@-5K7 5VE;(4DV31+
P (&57
< 0.0496) = 0.999/ -g =:; 74?74D87X 3Id;
?4DV3 2;];@-87W7@ b %;WV4- ::3XE3X87W5K'!V$X5K3 @3B;]) _/ _ ,+ -"=:= 7I;<-. 3 M * 3; 7]7J43;W 975KE<;4DV3^/3XVK3W3; 7&;<: 5VL<;4-K3Z3C 3A?4DV3 ; 7@5=7@3CL3C877Jd; 5K7@5-3Z;+* 4`72,;iH47@f7Jd; 5K7@5V- fb0/
(p, q) B1 B2
α = 0.1 α = 0.5 α = 0.9
$)$ # #$( & # #$( & # #$ &% # #$%" $ # #$( $ # #$" # #$ " # #$'&# )$ # #$!"! # #"$ # ##"% # #$$ # #$%"# # #"# # #$ &' # #$($
l + i/ i - ; 7@5VE4D7@5V-;h;<3EV4 ; 74?75; 7@5o 3>;<3E7@3B;b7h; * (5KE<;4DV5K7 $ &57(Mp|B)
/ "%4-5o; 5K7< = > < > β ∼B(p, q)
; [1.1, 5]+xe = 100
3=7a = 1
/
?i − L! &* + 3!7*4 1&% !)0/ % &* * &!G0/%'+4* t t t HJM oo P IUBT@M9O L q
(η, β)
\4 7(4-4-($=7@5o;@4D7@5V-θ = (η, β)
3; 7hV4h7 ;h4D5o; $XE3X87JX-(72$843C; 5VRV3#7(4-Ig* 4DJ4D[F; 7@3FR(4e[3$B; 5V3X / A3E;'$XV5;4?75K: 0C< [ Z > w U v|w 4e-3+K3L<;,CK3 7(4-4-$X7@5 3 W(θ)
3X7 (3 X-(4-5;2;4DX3&<(= > < > ! -(;'$X3 ; ;<3;&;<-$X3;^4-=5V3X(3;^ C.75D:3;5V72f;@3X4D5K7X 3ZH4-=-2$XH4?7@5V-S^ 3X7&;<-A 3ZH4-=t?4-@5H4DX3BbO; = 7V3(η, β)
3; 7I'$8:84?7@5-3f/ -gc3 3=79+V3;Z3B;b75K>4D7@5V-;5! 2$ 3C:75; 7@3B;I;<3η
3=7β$-- 3C:7A:f$X'$C4-K3CL3C87Z;3(!/4 =:
4D8743:-:5; 7@3(^V:-5V^K3E%4J4`75K7@3 3=7 _3_ b0/A3I4-5K3I;<3E=2$C;5KR5VV5972$E=34$X3X87@3Z4e-3B W(θ)73X(;Q;<:+
5V;'$C4DV3XE3C:79+ 3]! :@E3]7F%4(3Y;<3EX3XVK3Z;<3ZH4JC: @3Ai'/a /<\4N74-RK3Ai/ 72$;@3X87@3WV3;3B;b75K>4?75K:;^; =38! jC.=5V3X87A;<3EX- $CV4D7@5V-72 3Xo 3;y;20$XJ4D5Kd;]< = > < > =/
eta
beta
0 20 40 60 80 100
020
4060
8010
012
014
0
M 9i'/a -gF-3XV377J3A=:F-3<3W;+* 7@5V43:-3(η, β)
;3E=-2$XH4?7@5V-F'$8:84?7@5-3f/
a & B2
B3
) B1
# #"# # ! # ! # # " #)$( # #&' # #$ #)$( # #$( # "' # & # % # # !"! # #" #)$&$ # $# # #" # ( # " # # # # # #"! # ' #
l + di'/ g4DV3 %;G;<3H4A=-2$XH4?7@5V--\Ot(η, β)/(σησβ)
+372 β ∼ U[1.1,5]+f; 3CK:%4(4f 3y%4-5 N;<3E<;$XV5 j
;@4D7@5V- /D\3; ?4DV3 2;'$::4D7@5-3;]5V(;<5o 3X87gV3; %4:5 < = > < > K3B; 7 ;1=f4'$X3X87G4e-3BjV3^E;',XV3y7(4D4D($=7@5o 3;<3 3C5KR KG/
4D(;S-e4 :-E3X873hG* 5V! ->4?75K:h; β ++=3B;X?4-K3 2;d;<3JX- $CV4D7@5V-S;<5VE5K 3C87d4`75H;<3XE3C:7B/ -F79$ j@5VE3X874-K3CE3X870+ V3;I;<5o; 7@5KR 7@5V-;#< = > < > =:F-3X2:-3C87-3C2;W;<3B;WK:5;W(-E4-K3B;W;$X- $CD$C3;/+\O3X7W3<3X(7V37J3C@E3X7F;<3 3$X5DC(3X- 31+O;4D; C4-;<3Q;+* $X7 ;3 7 7D7X-E P3C C24DR5VK5K7 $-+gV3;EE<;$XV5;4?75K:;>;<3[H47372f;@5975K: h3; 72373X87^;+* 35!/4 =::f$X'$C4-K3XX3=7@7@3W ,:-V3d;<3AX34'$C@3C=3f/-g 7(4- 75 K5V3X9+V4; 7.$B=5DC.X4D7@5V-;<3A 4D87@5VV3;^;+* -;<3!/4-5KR(K3-^ ≤ 50%
bO3X7&; L<;<3Z5V(; 5;@3X87AV3;E<;$XV5;4?75K:;V3;Y7 ;X34'$C@3C:73;/
; >L? "!$#%'&)(*+&)(
\1* 7@5VK5o;4?7@5V-N;<3;7@3B%45o 3; R(4e[3$B; 5V3X(3; 723XE3=7;+* 4-($XV5K:@3CH472$X5;@5K:]! $B 3X87@5o; 7@33/`\1* 5V':3$C5K3 ;3;-,(@3X7 $&5V(; ;b7@5V3XVK3Y3; 7^4D5V; 597(4- 75 K5,X3XE3C:7i5K872$X3;2;$7(4Dig* 3; 7@5V>4?75K:;<3] 4D87@5K7 $B;g73XVK3B;Y 3dH4N; $C3;<3]5K3Z H4(; 25K3f/.\O3; 7@5V;A 5+723XE3=7@7@3X87d;357(@3C(;<3 3I;$X5;@5V-Q;<3WE4-5K873X(4-=3-+;<[email protected]@5V- ;<3d@3C(7V4f=3CL3C87j; =:(7Jd;@4-:7 ∑ /\4EX-;@5; 74-=3^3X7OH4E=:F-3X2:-3C=3^;<3dV4h;<5; 7@5VR 7@5V- <F=< v|w0> < > ;<3E;<3X;@5K7 $
π(θ|yn);@-87W;3; ; [email protected]@5V-;Y5V72-@74-87@3;Y;<3Lg* 43=37<74D7@5V- ;3;Y $B; 974?72;d5V!V$X3X87@5V3Xo;/ 4-
< =< v|w0> < >
4-5KVK3 %;+DE3B; 3XGH4E; 74-R5KV5K7 $&;<3=3; 4D87@5K7 $B;15; j ) j 85o;G;<3723=75973;OE;5?C.C4?75K:;X<E= > < > 7J3C@E3=7Y)Wg* 4DJ4 j[;b73jR(4e[f$;@5K3Ch;<3;$0C((5K;<3;]E<;$XV5o;@4D7@5V-; 7(4D;$8!/4 79+MV-%;2 * 5Vf4$;@5973O3C:7@3i;<3 6;79$X5?C.C4?75K:;Z<?= > < > =/
Y4D;X3=7@7@36;@3=7@5V-6+( ;Yf$X5?C(:;^;<:A 3 π(θ|yn)4-;<f7<7@3 X-(7J: 73XE3X87A4D5o; :(4-RK3YV-%; 3
n:-4D(;<5K7/ -g ;@3X-(; K5V3 +4`72,;W4e:-5VY;$8C(5gV3;]7(@5V=57(4-K3B;5 4-87@5K7 $;]7( $M;<5=7@5-3B; 7@5VK3B;Z4 CJ4DR(5KV5; 7@3-+
;17(@f7Jd; :; (3X$=7 ;<3(^@; ;<3;i;<-('$X3B;- bjE::74-87W 3&V3X:84D5V+R(4e[3$B; 5V3X>3C+5V! ->4?7@5V-+8;4-;V3;=-J;<5975K:;&;3JX4-K5VR4?75K: X-;@3X5VVD$C3;&4 7(4-4e?4D879+7J3C@E3=7W;+* 4?7 73X5V(;<3 (3672$X5;@5K: ; #;4D87@367J 7372f;@3X 3I;'$=5o;@5K:5V(; ; 7@5V3XVK3-^ (-[4?7@7@3X5V87@3]7(4-^K3B;$X7 4<;<3B;A;+* 3; 7@5V>4?7@5V- =H43;2;@5 3B;2b0/ -jC(6+ 3$X7 ;<3I;3];@3X.; 5VR5VK5K7 $E; A;<3B;Y$%4(4-:75KVV-;y;@5K $;Y5K3C87yXK:@3WH4(; 3B7@5V-3=7XX3A%4(4`75K7@33/
A XF 1 X- - j" -6, --\3E%4(-5 ;<3h;3X;@5972$;
π; Rb3B7@5-3B;@3B 5K3C 7AH4([email protected]?7@5V-Q; R2- =:(7J: 73XE3X87&;<3B;A;<5o;b7@5VR 75K:;
< =&< v|w8> < > =/A ;y3$C@5DC(-.;^;<-]72 3C5KR K 3=7AV3ZE;',XV3E)E@5o; 3B;y=:79$=75975?!@;=3C 74D5V3;Y737(@5$=72$;;<35=:; 5o; 74D.=3A3X7i;<3]=:d:3X2:-3C=3^;<3]=3B;j;<5o;b7@5VR 75K:;0+d 5_4 7@-5o; 3C:7Og* 3X(7V-5 ;<3;i;<3C; 5K7 $B;
π7@f7Jd;$C3;
4 7(4D4e?4D87/M\1* 4D@7@5o=V3i;3^n54(f;4D+^ a :4fb3=7gV3Z%4(4375K7@3a^; K5@3;<3 A4-E4-E8: 7245 n54d;4 ^ _3_3idb;@-875o=5Jf; 7@5VX5D7(4-K3B;g2$0!V$C@3C=3B;8/\O37(4D4`::437'4335=-.;b7597 3X38723X(;4-87W * +437'723X +d 5.7(4D@7@5o=57J3Y;+* 3$X7 ;<3ZRJ4e[3$;@5V3X3]=:(7D,X7@3-+<E4-5;y 53; 7:3$X$X4DV3XE3X87Y7(43;2;$];@ ;y;@5KV3X.=3Z;4D.;^K3B;^4D@7@5o=V3;4`77K5o $;/
t Lt t HJM9O KBT@KUI'ONQ qA(3+;<5; 7@5VR 7@5V- < =&< v|w8> < > 3; 76=:; 5o; 74D873(;@5 +4 ! I3X76)E3B; 3- 3>g* -a4`:: ,:-3E;<3;h;<-$X3;0+
3CKV3X-F-3C ::3Z:3X%;gH4ZE3; 3d;$:3$X$X2$X3δθ0
-8θ0
3; 7jH4E?4DV3 j2$X3XVV3d; 7J4D4D(,=7@3f/d\OV4-5K3XE3C:79+:5V;8* 4`::597;+* 3 X-(;<5K7@5V- $U w m8s< U <UU; * 4f8=37<74D7@5V-P; 2$; K74D7E5V!V$C@3C:75K3CA^ V3;E3; 7@5V>4?73 %;LR(4e[3$B; 5V3X;J$=74D87;<:5::4-4-:75;Z=:;@5; 74-:7%;2b0/
pEqr M$PeM9NMOHLP / Y < vXn = X1, . . . , Xn ∼ M(θ) w0v θ0 ∈ Θ W < x vG> ^ s v <U < =< v|w0> < > w vYx vw 0C<U#g
v <U vwJw U θ0 =<ts >Av <ts v <$U9<)c w U x3w θ0 l∫
U
π(θ|Xn) dθM(θ0)−−−−→
p.s.1.
\4=:;@5; 74-=3(<F=&< v|w8> < > 2'$X3;2; 5K7@3d7@2,;Z723 :-5V@3Y4 354F[d72D724',;@3; π /\3d7 4$X-2,XE3Y;<3 d-R^gk! /F?4D;3X g4-4- 75a:m,+d7 /a fbG5V(;<5o 3] 3dH4J; 5K7@3W;<3;;<5o; 7@5KR 7@5V-;A< =< v|w0> < > (3; 7Y7@3B; 357(4- 7 7X-;@5; 74-87@3+V-%; 3 n 4 :-E3X87@3-+ 3C ; 7'72f;4D87;@5V7(K3CL3C87( 3K3E<;,XV3 M(θ)
3B;b7E5H;<3X87@5DCJ4-RK33=7 3IV3;d3; 7(4f=3;
Θ3=7χ
(5;d;<3hV3 %;&7@5KR ;&R2-2$XV5K3C3; ∗ @3B;723=7@5-3;; :87&3 XK5H;<5V3X;/G-5V&V3hV3XEE3aX5 j ;<3;2;@ ;7J ZG* 437'7V5C4?7@5V-a4 SL<;,CK3B;Z;<3 3C5KR Kj3=76)@5o;2 3;]=:79$=75975?!@;/\3N724'$X: ,CE3 af/ i/ ;<3A4->4DE-@7 45 n54d;4#^ _3_fifb23X! :2X31=3X7 7@3 7@f75D$X7 $1+C3XZ5V(;<5o 4D87H4=:;@5; 74-=3G3C θ0 7(4- 75 K5V3XB/# !/4 7(; 77Jd; 3C6 3+V3+E;',XV3 M(θ)
@3B;723=7@3+V3;J=:(;<5K7@5V-;I;3 a4DH; ^ a dbY3=7 3 θ0 3; 7X-87@3C ;4-;^V36; 7'72-@7d;<3
π(θ)^/4-5K.; 5 +(:S;<-5K7&V3:-5o 3XE3C:7&4e:-5V
β0 ∈ [βl, βr];4D;&D7@3EC43;%bk/ :R<7@5V3X87&H4
X-;@5; 74-=3W; .S\ -θn+;@-5K70+ ∀ε > 0
+
Pθ0
(|θn − θ0| ≥ ε
)n→∞−−−−→ 0.
\O3B;X-(;<5K7@5V-.;X;@-87f$X5?C$B;72 YV3hE<;,XV3N;<3U3X5VR Vg/_\4 =:; 5o; 74D.=3h; .S\ - 7J dV3;dE<;,XV3;Y;<3U3X5VR V3=7)e@5o;2 3;=-(79$=75975?!@;L3; 7J7(@ 3$C3+@3B;723=7@5-3CL3C87L;4-; 4f%4(4c^ a fbW3X7 i3X@7 4(-V- w8v < T'W
%1 ! , 2"
?i − L! * + * 1&% / % &* * /%'+4*
^ _3_ dbk/ A-7@-.;y 3dV-%; 3 n 3B;b7y723=7@5K70+βn
E3C(43X3Y;+* =7@3WV-5V;3β0/&5K;@5 +<K35!/4D5K7A;+* 4e--5V
βn /∈ [βl, βr] * 3; 7J7(4f; ::43:-3F;+* 5V=:; 5o; 74D.=33/ 5 n 4 :-E3X87@3-+5VZ!/4 7-=37J3C(;4D873$X5DC(3X 3FH4 ; 5973;3;(βn)n=:d:3X2:-35)NG* 5K87 $C@5V3 A;<3
[βl, βr]/
# 5af/<ts > (µ, β) ∈ IR+ × IR+ l Twf] < xXS)TDwExfw )w ^ s T$T w v xfw U v -Z< ^)TDw&W <`s > (λ, µ, β) ∈ IR+ × IR+ ×
IR+/1 l Tw ] < x3STw B(λ, µ, β) w v x3w U v -W< ^Tw&W
npoq0rLsJqut :597 fW3=7fB
K3B;Z;<3X.; 5K7 $B;d3; 72375D:3;W;<3B;Z;<3 L<;,CK3B;8/ F 7'7Jd; :;A 3(7J Y7@ 7 x ≥ 0+
fW (x|µ, β) = fW (x|µ′, β′)/ &K:2;
µβ
µ′β′ xβ−β′
= exp(−µxβ′
+ µxβ).
\* 3B;b7A3C 7(4D@7@5o K5V3Xy4D5+72 x = 0/ $B=3B; ;4D5V@3CE3X870+
β = β′ 7 5o; µ = µ′ / F 77Jd; :;^>4D5V87@3XJ4D87 37J 7@ 7 x ≥ 0
+fB(x|λ, µ, β) = fB(x|λ′, µ′, β′)
/ AV-%;
λ+ µβxβ−1
λ′ + µ′β′xβ′−1= exp
(−(λ′ − λ)x − µ′xβ′
+ µxβ).
-gx = 0
+3C ; 7'72f;4D87β′ 6= 1
+β 6= 1
+<-Q:R<7@5V3X87λ = λ′
+'7 5;^:Q@385V3X87A4 C43;Y72$8$C;<3C:7B/ $B=3B; ;4D5 j@3CE3X870+
(λ, µ, β) = (λ′, µ′, β′)/ -g"; 77Jd;@4-:7 β′ = 1
3=7β 6= 1
+ - -R<75K3C:7λ + µβxβ−1 = λ exp(µxβ)=3] 5 3X87@4|3
fB(x|λ, µ, β) = λ exp(−λx) ^/E;',XV3Y37J:3X87@5V3X=bk/\O3B=5 3; 7YX-87@4-;5=7@-5V3d4e-3Bµ <∞ /
]5K(4-K3CE3X870+<:4E;<:β = β′ = 1
/\3YE<;,CK3W3; 7A4DV-%;j37J:3X87@5V3X_3X7A- j 5H;<3X875?CJ4-RV37 5;2 3 λ = λ′ λ = µ′ ^ 3; 7 / µ = µ′ µ = λ′bk/
t t t HJM.O suq0o uq O Q q\-%;2 3 n → ∞ +J3=7 3ZV3IE<;,XV3ZR(4e[f$;@5K3C3B;b7X%4(-5o; 5; #;4DE3X87A2$8: K5V3X9+<V4F;<5; 7@5VR 7@5V-c<9=&< vw g
> < > 723 7Y; -3C87 X7@3&4`77@<5V($X3Y7(4D 3&;<5o; 7@5KR 7@5V-E->4DV33/-\3^7 4$X-2,XE3A;<3 O3C@.;b73X5V j :- .[5o; 3B;5KJ;<5 3WH4JX-F-3C ::3XX31+84 ; 3C;;<3WV4L;<5o;b74DX3Y3C?4D5H4?7@5V-F7@-74DV31+<;<3WH4N;5; 7@5KR 7@5V- < =< vw8> < > _;<3WH4?4D5V4-RK3i2$C; 5K7@3 √n(θ− θn)
-3X%;]V4W;<5o; 7@5KR 7@5V-L->4DV3YX3X87@2$X3;<3^;<3C; 5K7 $N(0,Iθ0
)(x)+-;<3=t?4D5H4DX3jH4
>4?7@5o=3Z;<3 ]5; 43XIθ0
/ ;@5V3 %;Y! - V4D7@5V-;;<3A=3Y724'$X: ,CL3A;@-87A;<5o;72-5VRV3;A^n54f; 4(4D a -4fb0/ A ;;<-(-;L3C &3<3B; § i/ /Da 3 f$X5?C.C4?75K: ;<3B;-=-J;<5975K:;7372f; $X3;7(4- A4D>4DE-@7 4(5 n54f; 4^ _3_3i +F7 /i bi;4D;V3AX4f; 3C5KR KG/
l qHLF+m Yi/on! q0o ONK8U q TgO s M9O T@K q Ker W3Y < vXn = X1, . . . , Xn
i.i.d.∼ W(θ) w8v θnTDw R & 0C< >k> w =&<U x <U vW
< v <U θ0 T < B> < w < TDw s > x s>='< > < ]BS8vg> w WY < v π shU w xfw U v Z < = > < > B=&< v tw w0v 0C<U v $U.s w w U θ0 Wpi Uks ==< w m8s w Y s == (π(β)) = [βl, βr] ⊂]0,+∞[lm8s w β0 ∈ [βl, βr]
w0v m8s1u T1w : v|w p ∈ IN∗ vw)T m0s w ∀n ≥ pl
βn ∈ [βl, βr]WLY < v h =
√n(θ − θn) WLjT < > 5lm8s<U x n→ ∞ l
∫ ∣∣∣π(s|Xn) −N(0,Iθ0)(s)∣∣∣ ds W(θ0)−−−−→ 0.
npoq0rLsJqut+:-5V &33B; § i'/ '/a^@7 / aB_ b0/
AD7-;+3XC( 3 7J =3[E<;,CK3-+ 7 5o;2 3 ∫Θ|θ|π(θ) dθ < ∞ 18
θ = (µ, β)+i- 4,+g3C D74D87
< =< v|w0> < >
θ∗n =-[θ|Xn]
+√n(θ∗n − θn)
W(θ0)−−−−→ 0,√n(θ∗n − θ0)
L−→ N (0, I−1θ0
).
A $#@2 '7( ,B;d4e:-5VA;$0C((5]V3;X75KX5D7J4DV3;X 4-87@5K7 $;d5V:72$X3;2;@4-87AK36CJ4DR5VV5; 7@3N5K(; ;b7@5V3X +2 ;7372f;@-; 3
E3; 3Y; d0<:0 v|w s >6x u w8>k> w s > ρ ;<3A%4(4f 3]72 A;5 $X3X872;i7 [F723W; * 3B;b75K>4?75K:6+)J7(4D@7@5VA;<3Z;<:'$X3B; -W/ 437'723XVK3A 3W ;A4e-:;O;'$0C(5 =3]!/43=7@3 A;+* 3X@3 y; 3E 4D87@5K7 $]?4D5H4DRV3 A 7(4D
ρ(A) =
√
max
(q50%q5%
,q95%q50%
)
-8qα
3; 7V3E 4D875KV3Z;+* :;@3α
;<3WH4>;<5o;b7@5VR 75K:;<3A+; :597^H4`4-K3 7@3CKV3E 3
P (A < qα) = α.
58V3; $X7 4<;<3B; XV4f; ;@5o 3;_3W7J3C@E3X7 7@3C87N: ,X3j;+* 4?7@7@3X5V(;<3gH472$X5;@5V-Z5V(; ; 7@5K3CKV3O;'$;@5K2$X3(ρ < 2.5)
+XV3;E<;$XV5o;@4D7@5V-;R(4e[3$B; 5V3X(3; 7( $B; 3C:72$X3B;;4-; X3W%4(4375K7@31+ 7@5VK5o;@4-87;<3B;74D5VKV3;
a7 [d7(5 3B; ;<3B;7(@0$M; 3; ;<3
C4DV5KR4?75K:67372f; $X3B;+B7(4-KV5K3C87 =3=7@7@3^;$8C.=5V3XX33/-\ * 437'723X4 I7@3B%45o 3;R(4e[f$;@5K3C3; 3; 7j;<:W! =7 3 +/
t Lt t r UBT K(LNPFQ3T@KBT@M9ONO q K 437'7J3CKV3] 3Xo 3;iD74?75K:;8/ -5K7
yn
G* $B%4(4D87@5VVK:;<3Z;<:'$X3B;8/ K3A; 7'72f;@35X-87@3C5V ;@ ; j$B%4(4D87@5VVK:
xr
;<3r
;<:'$X3B;W5g/ 5G/ ;+/ ;<3 3C5KR KO3=7 a3X;@3XNRV3cn−r
;<3n − r
;<-('$X3B;Z;<3#=3C; @3JC<3^g);<-5K7@3ebk/ :597 B H4FE<;$XV5o;@4D7@5V-e3=73X 37J b = b(β)
/ j =< vw8> < > 2+ 4tC(a;<3(73X(;<3 3>;$X5;@5V-;+* 37V-5K74D7@5V-6+<V3;ZC24DR5VK5o; 7@3; ; 4-R5VK5o;b73;y%43C2%4(3X87)NE3B; 3XV3; 4-87@5K7 $;y; 5?4D87@3B;8/
af/^\4 x s > Z wx3w w ] < a3w UU w ;<3 ∑
-[X |B,yn] =
∫
IR
Γ(1 + 1/β)
b(a, β) +
n∑
i=1
yβi
1/βΓ(a+ r − 1/β)
Γ(a+ r)π(β|B,yn) dβ
5j3; 7L;'$0C(5V3;@5a + r > β−1
l
/ -g 74D7@5o 3#=3X7 7@3 =:(;<5K7@5V-a3; 7h7 ,B;A7J3 3; 7@5=7@5-31+7 5o; * 3X:3$X$X4D
r ≥ 4^V72 ^V3;A;-'$C3; - b3=7
βl@3B;b7357(@%43W;<3-a3/
/^\4 ks > w 4 73X(7;t0
S(t0|B,yn) = P (X > t0|B,yn) ,
=
∫
IR
1 +
tβ0b(a, β) +
n∑i=1
yβi
−(a+r)
π(β|B,yn) dβ.
i/^\4 x s > Z wx3w wJ> Z x s w)T$TDw 4 73X(7;t0
^ ]5Ko:3Xo;b73X5V _3_fb.
(t0|B,yn) =-
[X − t0|X > t0, B,yn] ,
=1
S(t0|B,yn)
∫ ∞
t0
S(x|B,yn) dx.
?i − * + * 1&% / % &* * /%'+4*
7@3B;Y=3;Y 4-:75972$;y;@-87^;<-X!/43X5KV3XE3C:7^3B;b75K>4DR(K3B;W7J4D3<3X(7V3]7(4D^;<3B;O7@3%4(5 3B;^;<3 .[-87@3 j \i4-@V; 5]G* : 723 7W-R<73X5V S75K4`::3
(β1, . . . , βM )+ 18
M3; 7Z4 ;2; 5N:-4DJ; 3J72f;2; 5VRV31+ 5 ; 5-3L43; [77@D75 j
3XE3X87^H4>;<3X;@5K7 $π(β|yn)
/-gQ3 23X70+
π(β|B,yn) ∝ βr
r∏
j=1
xj
β
(β − βl)p−1(βr − β)q−1ba(a, β)
b(a, β) +
n∑
i=1
yβi
−(a+r)
, [βl,βr](β)
3; 7E=-( 3) &!/43=7@3 A7 ,B;Y3=7Z3(7J3 7 =7@3>;<5V373XE3X87E;@5K D$C31+ 3'X387<72$ @<E;<3B;Y7@3%4(5 3B;]7 7-7= ,<7@3 ; 3B;I;+* 43=38774?75K: j @3!b3=7^ A-R23X@7 \i43;@3XVV4 _3_ bk/\3%4(4`7(597@3 3B;b7L;$C;<5$ )QH4 72$;@3X874?7@5V-a3X7V4-X-(7(4-4-5;@-;<3I($=724<;<3;^ ($C@5o 3;A4:;$ 4D7@3B;8/\ * 5V=3C 7597 ;<36; XX3;^3B;b75K>4D7@5V-;^3B;b7Y4DV-%;L3B; j $C37(4-jG* $BX4D@7 j 7 [F7J3&3X(75V@5o 3C4Do $X; OV3A75K4`::3
(β1, . . . , βM )/Y4D.;jG*:3<3X(7V3; 5D?4-:79+-:#72$0!V,X3X4
5KJ;<5 3CGK3kd0<:0 v|w s >Xx u w0>k>2w s > ;$0CJ5(4 %4(437597@3 ^=! :@ K3A^gi'/a_fb b7+t7 ; 7J4DV4-:7 72 GV3;5V':3$C5V3 %;-Z/
t t t q qA ;G $ 7@5VV5;@-;gg* $%4(4-87@5VKV- ! " + ^|74DRV3C4 / i]7/ db7+D;<:87jK3B;g;<:'$X3B;G:(X3X; 2$X3B; 7@t5V3X(3X87
; E<;,XV3&;<3 U3X5VR V2;37(4D4D(,X7@3(η0, β0) = (8760, 2.15)
/ (72$;@3X87@3; jV374-RV3C4 i/ G* 3; 7@5VE4D7@5V-; !/4f7@3 d; * 3C@3 ρ ;<3
S(t0)3=7*.
(t0); 3CK:[K3B;A($=724<;<3;=H43;2; 5o 3;&;<3 &3 7@: j A4`7'4.; :"^ Xb
3=7 - . +'7 5o;A;4-;^-7@3AX4-;@3WR(4e[f$;@5K3C / A4`7'723XV-.;y * 5V3; 7&;$8C(5 +'7J (3A`4-@5H4DR(K3W;<3I;<5o;b7@5VR 75K:A+'7J4D
ρ(A) =
√
max
(q50%q5%
,q95%q50%
)
18qα
@37 $B; 3C87@3eV3 4D87@5VV3 ;+* -;<3α
;<3aH4 ;<5o; 7@5KR 7@5V- A/ d4-;K3"X4:;<@3 R(4e[3$B; 5V3X+O ; C<-;
β ∼ U[1,5]+ ;EX-;@5V;'$X-; (3>$M;<5H4D3>>4D2:-5V(4-K3
xe = 85003=7N ;A!/4D5o; :;]$8:- 3XZH474-5KVK3+;3;
;<-('$X3B;ZC.75D:3;a/(\ 4-; 25K3W3=7AH4>; 2$X3I;3A85V3Z2$;@5V; 3XVK3A;@-87A3; 7@5V($X3;A3X
t0 = 9000/\3B;i7@5V43:-3B;y; ::7
@3B;723=7@5-3CL3C877<; 5K72;]7(4-dR2D72; 7@4`7 ^@C4-;<3E! 2$ 3C:75; 7@3ebd3=7 .&\ ^gX4:;<@3LR(4e[f$;@5K3C6+90! / %4(4`7(597@3 bk/d4-;V3;A;<3 C43;0+<K3B;yX4Do ;y;@-87^3 3=7 $B;; 103 ?4DV3 2;y;@5K $X3;/
S(t0) (t0) * '& !)$
& %"
a
) $ !&' " $" ' "!! $" !" $ $")$ $"
l + 9i'/ .S3; @3B;^; !/43=7@3 A;+* 3C@3 Y; ^V3;y::4-(;<3 2; CJ4-R5KV5o;b73;/
j;<3B; ?4DV3 %;g;<374D5VKV3a
387( $B; 3C:74?75D:3;;<3;17@8$C; 3;G;3XX4-K5VR4?7@5V-( 3^ ;g4e-:; 7372f; $X3B;;4D.;V4-;@3=7@5V- 7 $B0$M;<3X87@3-+- -:597XXV4-5K3XE3C:7 3ZV3]!/43=7@3 A;+* 3C@3 ρ ;<3;A;3 :-4D(;<3 %; C24DR5VK5o; 7@3;4?7@7@3X5V87^V4J72$=5o;@5K:FE5K(5K>4DV35- 3(^ ρ < 2.5
b<+' 3dV3;($=724<;<3;W! 2$ 3C:75; 7@3B;Y7J3C@E3X7 7@3C87y)J723X5V3-^ 7(43;%b;+* -R<73X5VB/:\4IL<;$CK5o;4?7@5V->;<3 3C5KR K 3A ; 7372f;@-;17J3C@E3=7O3X> 7@3A;<3AE3; 3XgV3A:hR3;<3W;<-$X3; C.=7@5-3;O'$B=3;2;4D5V@3B;172 ^4?7@7@3C5K(;@3-++<F=< v|w0> < > 2+ 3572$X5;@5K:+5V(; ; 7@5K3CKV3; 7@:R,XE3
< =< v|w0> < >
;<:'$f/ #723 7Z7(4-O3<3X(7V3&3X;$C; 5K3&V3&(-hR(@3Y;+* 37J3C 7%;j'$B=3B; ;4D5V@3y72 OG* $X7 ;33/ F y=3X7O33C(7K3-+ ;<3 F3F723X@72;X; #;@3X87/
A ,( - - [-XF/+5 !\1* 5V 3C=3;<3jE<;<5DC.X4D7@5V-;Y< = > < > f; ]K3B;2$; K74D72;Y<B=&< v|w8> < > -3B;b7 723X 3 @<jV4&E3; @3i;<3ig* 5K.=3X j
7597 ;<3N;3;::4-(;<3 2;yCJ4DR5VV5; 7@3B;&;$8C(5V3;7( $B0$C;3XEE3X87/A ;XX-;@5V;'$X-;V3;dE;5?C.C4?75K:;X; 5`4-87@3;0+V3;Y7 ;y7(@:R(4DRV3;iV-%;;<3Zg* $=7 ;<3I;<3ZG* 4D(4-D[; 7@3Y5K(; ;b7@5V3X:
af/ 3Z$8:3,C@35?4D5V4D7@5V-Q;<3;R2-3B;(βl, βr) / 3A4'$B; 5K74D7@5V-3X87@3WK3B;y;79$X5?C.C4?75K:;3; 7.$C4-=3`($C;5V4-3?L<;<3
i/ 3ZE<;<5?C9X4?75K:;3Zg* -;<3
α; * 4-:75KV3W>4D2:-5V(4DG/
yCJ+;<3AEt[-3C3XGV3;g $B; K74D72;0+D- 75KV5o; 3C4(aB_3_5$%4(4-87@5VKV-;yn
;3^74-5KVK3n = 10
+`;@5K $;1;@3XV-EH4WK:5;<3 3C5KR K_;<37(4D4D(,X7@3;
(η0, β0) = (110, 3)/ A ;OK3B;WX3X; -; 7(4Di;<3;;<:'$X3B;G75K2$X3B; (5?! :@($XE3C:7
;@3XV-[110, 130]
;<3]!/4=- )L-R<73X5V 74 ;3]X3X; 3WLt[:3XQ;<3 _ #/ C<3t0 = 130
/
t Lt t q PFQ3TNQfIUTgM.ONK q K P o I'ONQ q cPFL TgI'O q cM L qA ;XC<-;&5X5
xe = 1003=7a = 2
a = 4/& ;=:7J4D-;^V3;&E;'$XV5;4?75K:;
Bi+i = 1, 2, 3
^gk! /7372f;@5K7@5V- dbd;(4D;YV3(C43;Y18
B1; 79$0C.X5K3
xe=:EL3 3E($C;5V4-33/ S7Jd; 3
β ∼ Be(p, q); [βl, βr]
/ 6;'$=5H;<37@ 7E; * 4-RJ:;
βl ∼ U [1 − εl, 1 + εl]3=7E- C<3
βr = 5/ E3; @34DV-%;IG* $BX4- 7 j 7 [F723;<3B;
3B;b75K>4?75K:;I;<3B;A:-4DJ;<3 %;5CJ4-R5KV5o;b73;I3C"! -=7@5V-c;<3+G* 4 ::L3C874?75K:c;<3εl/ \3;Z2$; K74?7%;E;@-87E! @5o;
;4-;iK3W74DRV3C4 i/ /&3 <5D,CL3CE3X870+:4:;<377@3YV4E;$CE4-2%4(3d5VF-3X%;@3d3C;$B=5H;4D87βr ∼ U [5− εr, 5 + εr]3X7&3X C <4-87
βl = 1/ [@37@<; 597d;(4D;&K3h74DR(K3M4 i'/ X3;A CL3B;X$BX4- 7 j 7 [F723;^3C ! -=7@5V-S;<3
εr/ d4-;
V3;^;<3 C43;0+ (p, q) ; ::7^E<;<5DC$;A;3]!/4 =: )(X-;@3X2-3C -[β] = 3
3=7g4-[β] = 1
/
εl
[X] S(t0) (t0)
B1 B2 B3 B1 B2 B3 B1 B2 B3
a = 2# #" &# " # #&!"' # ## # #&# " & & #)$ $ !$ !&! # #" $ # #" &! # #" ' " ' !" ' !&## &' %$ '&' # #" # #" " # #" " ' !"# ' !" ' !" # &! ' #"# ' # #&"' # #"&# # #"" % !"! % % !&%
a = 4# #" ' &# # ## # #! # #& !"! !& !" #)$ &! '"% ' # #" &# # #" &! # #" " & ' # %"' #"' *$ # #&$ # #& # #&& &' '& '"# ! ' # #& # #&& # #&" ' #"' '*$! ')$&$
l + i/ -W;b75K>4?75K:;^;<3B;Y$X4- 7 j 7 [F723;W5; j ) j 5o;O;3YE<;<5DC.X4D7@5V-;Y; βl+72 %4J43 3]%4-5 F;<3YL<;$ j
V5;4?75K: /
\4 E<;$XV5;4?75K:B1
^/($C;<5H4D3eb4`7'7(4-4V7 =:LE3UV4 7 ; ;b74DRV31+^4 ;@3X;18 K3B;7@-5o;::4-(;<3 2;CJ4-R5VK5o;b73;N 5<3XJ$X>4D3C87 ;@-870+`;+* 3Z!/4 X-E:f$X'$C4-K3-+CE:5K; ;@3X;@5VRK3B;4 A?4D5V4D7@5V-;];<3B; R2-3B;
(βl, βr)/
\O3W2$; 974?7=:! -@7@3YV3]%4(-5 7(4D^;$8!/4 7&;3B1
7372f; $Y4 § i/ i/ ^V7/ O bgV-%; 3YG* 3F723X@7@5o;@3Y3C `4-K3 X3X87@4DV3+3; 7>;$XV5C4?73 ) ; 79$=5DC(3C/ 4DE4D5VKV3 %;0+ V3;NE;5?C.C4?75K:;N;3
βl7t- 3X870+7J %4J43 3;<3B;
E<;$XV5o;@4D7@5V-;0+J$8:f,X3XE3X87YL:5K.;d;<3723X@7 R(4D7@5V-;-< =< v|w0> < > 3E;<3;YE<;<5?C9X4?75K:;$ 5`4-K3C87@3;]; βr/d\* 3; 7 L2$; K74D7 7( 7D7gR9$X'$8C. 3 3CN3 23X70+3X3=7@7@3^R2-3y; 79$X5V3 33; 73C(:3$X$X4DE-5V; ; ; X387<75KRV3
;+* =7@3WE<;<5DC$X3E 3 βl+' 53; 7AK5$X3A)LH4X-J4D5o; ;4DX3Y; 5V3X5VKV5o; ;@3XE3X87B/
?i −
εr
[X] S(t0) (t0)
B1 B2 B3 B1 B2 B3 B1 B2 B3
a = 2# #" # & " # # & # # ( # # ! & ! !#)$ !" !"' # # #& # #&' # #& " ' '&!# %&% )$! ' # #$ # #&! # ## ' !"' ' $ ' "# &! ' $ ' &! # #""! # #" # #"" %)$' % &! % $
a = 4# #" !$ # #"' # # & # # ' # # ' " % #)$ &# )$( # # &' # #""# # #"&# '"% #"! *$## )$ !"! " # #"" # #" # #"" $ &# !" " ' # #" # #"' # #& ' " ' !& ' !&!
l + i'/ - ;b75K>4D7@5V-;;<3B;W$BX4- 7 j 7 [F723;W85o; j ) j 5o;j;<3YE;5?C.C4?75K:;W; βr+7J Z%4J43 3%4(-5 >;3YE<;$ j
K5o;4?7@5V-/
t t t q PFQ3T Q3I'UT@M9O L q r I'OUBT q K J; 7'72f;@3j5o=5 3jG* 37J3C 7 ;79$X5?CJ3 J 4D87@5VK3 xe = 100
; * :;<3α/ J%4:5;@597
a = 2/ 3Xo 3;
?4DV3 %;j;<3α+8( ;ZX4- K:;gG* $X4- 7 j 7 [F723
σ;3;O3; 7@5VE4D7@5V-; CJ4-R5KV5o;b73;i;<3
E[X ]3=7S(t0)
3X-! :75K:F;<3;?4D5V4D7@5V-;
εl3X7εr
3; 72375D:3; ; βl3=7βr+D;4-;K3B;G XE3; X-(;5975K:; 3y72$0$M;<3XEE3X87B/f\O3;2$; 974?7%;
; :87Y7( $B; 3C:72$;Y; ^K3W74DRV3C4 i/ m/
$# " &# " %"# %" εl#)$ # !
εr
#)$ # ! l + 9i'/ m -W;b75K>4?75K:;^;<3B;Y$X4- 7 j 7 [F723;WCJ4-R5KV5o;b73;Y5; j ) j 85o;i;<3ZE;5?C.C4?75K:;Y; βl
^@4(4 7%bi3=7βr
^/R(43;%bk/Ah; RV3=7 =:@3; 72-(;])AG* $X4- 7 j 7 [F723
σ; (E[X ], S(t0))
/`\3;7J 2X3X8743:-3B; 5%, . . . , 95%=:@3; 72-(;<3C87
)Lg* -;<3α
; 4-87@5VK3 xe = 100/
A ;5X-; 74D7@:;X 3Lg* 3; 7@5V>4?75K: < =&< vw8> < > ;<3LV4; 2$X3N;3J5V31++)#7(4- 75KW;+* 3(;79$X5?C9X4?75K:h7(4- 4D87@5VV3;]37@ XE3B;+`3; 7GV47 ; ;@3X;@5KR(K3^4 IL<;<5DC.C4?7@5V-.;];3;R2-3;/3\O3i2$; K74D7]3B;b7g;<: L4- : E3X873X!/4e-3 i;<3WG* $X7437J3W;<3W@3BX4-K5VR4?7@5V-# 5+7(@f7Jd; 3Y;<35X-@5:-3CjG* :;@3Y;<3;Y 4D875KV3;W! @5o;Z7(4DiG* 3F723X@70+3XK3B;^E<; H4D87A4 7 A;<3B;Y75KX5D7J4DV3;Y?4DV3 %; (20%, 25%, 33%, 66%, 75%, 80%)
/
-gC(6+(D7-;Y 3WV4(;@3X;@5VR5KV5K7 $Z;<3Wg* 5V(;<5oX4?73 ^;<3A; 285V3W4 ::L3C87@3WK:2;2 3 α → 1:<H4(;79$X5?C9X4?75K:
;<3α > 0.5
3; 77 ;y! 4`::5KV3A 3E=3CKV3h;<3α ≤ 0.5
V-%;2 * 5V ;8* 4`::597&;3A7 $M;<5V@3 hX-(72-@7@3XE3C:7Y! 7 B/\O3 $B; 974?7Z3X87@4|3>V3#=:LE3C:74D5V@3-; 5`4-87 :V-%; 3EG* 3F723X@7A! @5K7 c5V87@3X2?4DVK3>;<3 =2$C;<5VR5VK5K7 $-+5Kj3; 75K872$X3;2;@4-:7;<3Y72$=5o;@3XN7 ; 7(4D@7@5o V5,X3XE3X87V4]; 7.$B=5DC.X4D7@5V-N; 4-:75KV3Y; 79$X5V3 9+?K:2;2 3jG* -Rb3B7@5D!2;3g* 5V':3$C5K3 3B;b7];<3 72$C;5K3GK3Z=:72-@7@3CE3X87 ! 7 ];3 ∑ /M\4 3B;b75K:GCX3=7@7@3 ?4-K3 ; 79$X5K3 @31X-@3B;72-J; j7 j 3XVK3E) 3h;$8!/4D5VKH4DX3Z-R;@3X23$C31+') (3h;4?73I;+* 4D X7ALt[:3X(31+< 3B;b7 j 3CKV3Z37437J:D$C3])(7J4D@7@5V&;<3h;4D7@3B;;+* 4D X7&4D87 $C@5V3 3; 7J3C@E3X7Y;<3 :-3Xd; ;$C@5V3 ;<36X3=7@7@36; 7.$B=5DC.X4D7@5V-[3X7d;+* 4DE-%=3C [email protected]@5V-;<3>G* :;<3E; 4D87@5VK3-++7 5o;I;<3>H474D5VVK3-$B 5D?4DV3X8731+_3X"73XJ4D87Z3X"=-(7<73>K3B;Z2,8:-V3;57372f; $X3B;W4 §i'/ i'/ / /
VV 2
; '/i L?/ " ? ? @? 7? 0 ?& ]I7?
Y4D;1X3=7@7@3X;@ ; j 7(4D@7@5V31+D;4-; #C4-;<3Y7 ;1:f$X'$C4-' 3^K3^E<;,XV3&;<3 U3X5VR V +: ;G ;G5V87 $X3;2;@-;4 C4Do ];<3IH4E74-5KVV3
n;3N;<:'$X3B; W W=xXW
xn ∼ M(θ)4`77J: 74D87AV4+ CE3I5K'! -E4D7@5V-h 3IV3;d;<-('$X3B;
:R; 3C f$X3B;yn
+3 5.;@-87 X3X; 2$X3; )Z;<@:59733/8\4]?4DV3 O;<3n
3; 7j3XE3 3=7 >5V(;<5oX4?73 7J3C@E3=7@74-:7i;<3AV5KE5 j73XjH4E?4-K3 O;<3dg* 4F[F7J3C 7(4-4-(,=7@3
a+4 ; 3C5KF;3AV467F8$C; @3&;3&3X4-K5VR4?75K:J7372f; $X3d4 § i/ i/ / /
723 7$8::4-K3CE3X87^3; 7@5VL3C7(4-^K3I4`77J: 7a/n
G* 5K(72-@74D.=3Z4`7'72-@7 $C3E)EV4-X-J4D5o; ;4DX3(<9= > < > +5; j ) j 5o;;<3WG* 5V! :@>4?75K:F:Rb3=7@5-31+ C<3C 3]`4-K3 ^;<3
a=:LE3 K7@57V3W 35! 4f75K:;<3
n;(4D;;<3B;
$X7 ;<3B;^;<3E;@3X;@5KR(5KV5972$#^V:-5VZ7J4D^3<3C7(K36\O3XV3 w8v < T W _3_-4fb0/
A ;E;$0CJ5;2;@-;h7@ 7E; * 4-RJ:;cH4[74D5VKV33 375D:3;<3B;E;<-$X3;=:EL3 =3CKV3; * $%4(4-87@5VKV- W W=x3W4e[84D87 <>=&< v|w8> < > ^G* 5V 3C=3[H4"7 ;-7%43[;<3h=3CKV3S;<3B;>;<-$X3;+ $C3XVK3B;
yn
/Z\O3=7@7@3S;$8C(5K7@5V- ;;@3XNRK3Z;<:W$8:-5K7@5VE3Z;4D;AK3EX-87@3873WR(4e[3$B; 5V3X;<3ZD7@3]$X7 ;<3f/ J43=3Z4 ;<5# K7 $B;&;<3AX4- ;<3EX3=7@7@374D5VKV3W3 375D:31+< ;Y7372f;@-; 5V(;<5oX4?73 A437'7<5K>4D87
n3X X-;@5V;'$X4D87Y 3WV3;^;<3 $B%4(4D87@5VVK:;
437'7J: 73X87dH4 CE3( 4-87@5K7 $E;+* 5V! :@>4?75K: ; WK3(7(4-4-,X7@3θ/\ 4723X@7@5V3X.=3E;<3-=3=7W5V(;<5oX4?73 W * 3; 7
7(4f;O;5;2 72$X3W5o=5+; 5V:4 7@4e-3C2;O;+* 3]=-(7(4-4-5;@-4e-3&V4h74-5KVV3d3 23B7@5-3-+F7(4DV3dR5H4D5o;;+* 3<3X(7V3;y; V3;jL<;,CK3B;j37J:3X87@5V3XJ3=7;<3 U3X5VR Vg/ * (D7@3&5H;$C31+-5V=-.;b7597 3 7J:5K87i;3d;$7(4D@7 72D7@3C87@5V3X97J O;<3B;3%43C2%4(3;W7 ;^3X 7`! -J;<3 B/
* /--- '#" - - - :597 πJ < = > < > 2(-5K! :@>4D7@5D!+;<-87K3W3= @3CE3X87]< =&< v|w8> < > 97Jd; ; ,C;<3WH4hE3C5KVK3 35?4-K5H;<5K7 $
! 2$ 3C:75; 7@3(^@;@-5K7A;+* -;<@3Y>4 <5VE4-+; 3CK:H4N;'$0C(5K7@5V-"aW; § i'/ /aBb 7J $B%4(4D875KVK: W W=x3W &3]!/4=-;@5KE5VV4-5K31+?:L;'$0C(5K7jG*o< = > < > πJ
c
7J Gg* $%4J4D87@5VKV-L-R.; 3C f$yn
+`7J-7@3C:75K3CKV3XE3C:7 X3X; 2$3/&5K;@5πJ
c
* 3; 77(4f; ! :28$XE3X87 $::4D.)
πJ 7(4Dg3<3X(7V31+;4D;jK35X4f;G372-3C:75K3C +
πJc
3B;b7ig*=<E= > < > 7@f7Jd;$X7(4D &3 4D87@5o;w8v < T'W ^ _f_'aebk/ G-5V § i'/ / /a72 y7 ;^;<3A72$=5o;@5K:;8/
Q;$0C((597&H4N74-5KVK3W3 23B7@5-3I;3;^;<:'$X3B;-R;@3X23$C3;Z7(4-
n∗ = arg minn
-X∗
nDJc,J(yn,xn)
^@i/DaB_fb
4e:3
DJc,J(yn,xn) = KLπJ
c (.|yn) || πJ (.|xn).
^@i/DafaBb
-gU;+* 4 7@3B;A73XL3B;+2G* $B%4(4D87@5VVK:xn
E5V5KE5o; 3LH4F;<5-3C ::3XX3h;<3 T] KVR(4fo j \3C5KR(K3C DJc,J(yn,xn)3=7W-
:R<7@5V3X87WV4F74-5KVK3L3 3=7@5-3L3XUEt[-3X((4D875; Yg* 3X.; 3ChRV3L;3;xn
72f;2;@5KRV3;/ 723X =:597Z;<-.xn
=-EE3G* $B%4(4D875KVK: W W=xXW 5V 7 3X]74-;72-@74-87 3E5V! ->4?75K:e;<-87ZG* 5V(7(43=7-< =< v|w0> < > 3B;b7ZV3J7( ;]7(@%43;<3Ng* 5V! ->4?7@5V-Q74-; 7J: 72$X3E7(4-
yn
++X36 5N!/4D5K75;@3X;Y;4D.;AV3JX4:;<3IR(4e[3$B; 5V3XS;3h-7@3E$=7 ;<33/_\ * 4- 75XK3V3E7 ;d $B=3X87; YK3; b3=7d3B;b7]=3C 5;<3h\5K w0v < T'W ^ _3_fbk/J\1* 3<5; 7@3C=3N3=7Yg* 5o=5K7 $L;<3
n∗ 3; 7:84D4D87@5 7J4DH47@f7Jd; 5K7@5V- ; 5D?4D8733/
EFuH H K)M9NMOHLPWm'/ Q $U w8v < T'W "%$ $%&VXWY <`s TDw E0C<U x v <U xfw ) < TDx Q ) < TDx V w0v <ts > Z w8>tw m8s w<TDw xfw U v Z <'=&< vw8> < > x <U`Q W V < w U v = > <C= > w 5l T u Z0 [ <U v T$T <U
x∗nw : v|w < ew 0G= > < ^ < ^ T v Z w0v w v shU m8s w&W
\1* 3; 7@5V>4?75K:h;<3n
3B;b7N:3$X$X4DV3XE3X877 ,B; X ,<73 ;@33/ -gVK3W7J3 7 X7@3WX-;@5V;'$X4DRV3XE3X87 ;@5V7(K5DC$X3OK:2;2 3V3dE<;,CK3 M(θ)
4:;<E3=7^;<3;y; 74D7@5o;b75 3B;O34(4 ; 7@5-3B;8/ :597 T (3]; 74D7@5o;b75 3Y34(4 ; 7@5-3YL5V5V>4DV33/ j:3X870+ DJc,J(yn,xn)
3; 7 (3! -.7@5V- =-F:3<3&;<3n
3=7i;3t(xn)
+3X(;(4D87OG* 3B;79$X4D.=3&;4-;5^@i'/a_dbG5V 7@5VK3f/
?i −
G-5V+4D5V;@5^g* 3<3X(7V3[m'/.S4-5;E3C :3$X$X4D +;3;E($=724<;<3;+; * $B%4(4D87@5VVK:(4`::372-(;'$X2$"^@-:5K#%4(4375K7@3 b= 7$X3;y)E;<3B;($=7 4(;3;A;<3A::4:;<5V3X87Y;@-87^'$B=3;2;4D5V@3B;+3=7^3X(;3X87AK3AC4Do X-;@5V;'$X4DRV3XE3X87A4D; /
$A% '%m'/fR < x3STww :D=&<U w U v wT W &; 7'72f;@3 3LG* $B%4(4D87@5VVK:
yn
=-875K3C:7n − r
X3X; 3;W;<3N7 [d723 73::@3B; ;@5?!c1, . . . , cn−r
/ J* 4372,; §i'/ / /a1+H4L;3X;@5972$< =< v|w0> < > 97(@t:3X(4-:7^; %4:5 F;<3:3 38[;Y=3X.; 2$W3; 7
πJc (λ|yn) ∝ λr−1
(n−
n−r∑
i=1
exp(−λci))1/2
exp
−λn∑
j=1
yj
53; 77373]7J ^7 7 r ≤ n 4-(; n ≥ 1
/ A-7@-.;y = 1/λn
-8λn
3; 7&V3 .S\ - 3X7x
V4EEt[:3X3Z;<3xn
/ &K:2;
DJc,J(yn,xn) = −n log nx+ log Γ(n) − (n− 1)-
n[logλ] + nx-
n[λ] +-
n[logπJc (λ|yn)]
18g* 3; 79$X4DX3d3B;b7A;$8C(5V357J4D^4`7'72-@7Y)πJ
c (λ|yn)/9\O3XH4N,C3E)NV4(;@- 7@5V- 5o 3 x∗ = 1/
-n[λ]
3X7n∗
7@3CKV3E 3
ϕ(n∗) = log n∗ − Ψ(n∗) = log-
n[λ] − -n[logλ]
^ ≥ 0;+* 4372,;G* 5V'$8:84DV5972$Z;<3-3C;@3X.b<+
ϕ$X74D87 3! :75K:UR(5 b3B7@5-3-; IR+ / A(3L($X7 4<;<3+;<3 &3 7@: j A4`7'4.; : 3B;b7A7(4-!/4-5973XE3X87I4-;437<7 $C3
)Eg* 3; 7@5V>4?75K:S;<3n∗ 4e-3 (3hR2-367 $B=5o; 5V-/ A-7@-.; 36;@5 ;X%4:5;@5o; ;@-; πJ
c = πJ ^ G*=<9= > < > ;3-3 238[;Y; 74DJ;4D;b7+<( ;:R<7@3C-;
n∗ = r+=3E 5L5V:@3Yg* 5V! ->4?7@5V-Q4`77J: 72$X3]7(4D^V3;^;-'$C3;/
F K3d74-RV3C4 i'/ IV3;W?4DV3 %;O;3n∗ ;@-87YC4Do $X3B;j3X7iEt[:3X'$C3; 7J K3B;W$B%4(4D875KVK:; ;@5V $;Y;@3XV-
3NK:5];<3J7(4D4D(,X7@3λ0 = 1/100
/\3;Y;<-$X3;5=3C; $C3;; ::75%4-5o; 5V3; 5D! -($XE3X87Y4 4(4f;@4-;[;4-;[80, 130]
/-\3;W7J 2X3X8743:-3B;g;<3X3X; 3&3X7OV3;O $B; K74D72;Z; :87OEt[-3C'$; ; _6$%4(4-87@5VKV-;^V72 i-R7@3X(5K;<3;1$X4- 7 j 7 [F723; 7 ; 723=7597%; 3_/ bk/ #24`7'7(4-4V7 72 X3; $B%4(4D875KVK:; 3^H4A=3X.; 3A4 E5V7J437OK5VE5972$; ^g* 5K'! -E4D7@5V-6+7 5;2 3 n∗ 3; 7@3]7F%4(3W;3
r/
n
$# )$ )$( " $& ' ! $" !$
l + i'/ g4-K3 2;^;3ZV4N74-5KVV3A85V@7 3CKV3 n∗ /
)9-( )' %' 7J4DVK5V3XK3B;F;<5# K7 $B;; X4- Y;3
n∗ +O ;#7(@f7Jd; :;>G* 5V(;<5oX4D7@3 4`77@<5V>4?7@5D!I;$0CJ5Y;3V4Q>4-5D,C@3#; 5?4D87@3f/ c=:; 5H;,C@3- 3+V3;A$B%4(4D875KVK:;
yn
3=7xn
4`7'72-@7@3C:7IH4[ XE3>5V! ->4?75K:a;3]5; 43X5^ 3C; 7'7Jd;@4-87W * 3CKV3&3B;b7j7@ b %;j;$8C(5V3Bb ; iV37(4D4D(,X7@3
θ/8\ * 4:;;<5K7@55972$d;<35=3=7@7@3d5K! :@>4D7@5V-
7J3C@E3X7i;<3Y;$X@5V@3; 5V(7K3CE3X87OG* 5V 3XX3d;<3B;O;<:'$C3;Z=3X.; 2$X3B;1; OV3X7J4D4D(,=7@3θ/ F 7'7Jd; :; 3 yn
VV 2
X-87@5V3X(3n− r
?4-K3 2;O;<3XX3X; 3; )h;<@:5973cn−r = (c1, . . . , cn−r)
/ 4DJ;EK3B;r
;<-('$X3B;g:#=3X.; 2$X3B;3; 74-:73;Y;@-87 W W=xXW +2g* 5K'! -E4D7@5V-Q;<3 ]5; 43CA4`7'72-@7 $C357J4D
yn
; θ 3B;b7
Iθ(yn) = HD(θ) +HC(θ)
-8
HD(θ) = − - θ
[n∑
k=1
∂2 log f(yk|θ)∂θi∂θj
]-
[r|θ] ,
HC(θ) = − - θ
[n∑
k=1
∂2 logS(yk|θ)∂θi∂θj
]-
[n− r|θ] ,
4e:3
-[r|θ] =
-[
n∑
k=1
δk|θ]
G* 3B;79$X4D.=3>; -NR3E;3+;<-$X3;I- =3C; $C3;W;4-;ZG* $B%4(4D87@5VVK: / .S4D43 @3 ;@3XE3X870+ ;Z;38-:;4e:-5V 3]=:(4D5o;2;@4-=37( $B=5o; 3Y; 7=3B; ; ;i;<3X3X; 3X72 yX4- V3XY=3=7@7@3Y3; 7.$C4-=3f/-gG* 4-R; 3C=3Y;<3X3=7 73JX-(4-5;2;4DX3 j X3J 5]3; 7];@ :3X87&V3JC43;0+(5V(; ; 7@5K3CKV3XE3C:757(4DH4D87 j ;7372f;@-;&;+* 7@5VK5o; 3CdH4;<5o; 7@5KR 7@5V-E3X(75V@5o 3&;<3B;1=3C; @3B;G-R.; 3C f$X3;Y^@--5V %4(4375K7@3 b ;@5
n−r 3; 7 :-4D(;6+D ( #C43;g;<3=3X.; 3;<3h7 [F7J3 73::@3B; ;@5?! /A ;]; 77Jd; :;A'$B=3B; ;4D5V@3CE3X87f$X5?C$C3;&K3B;5X-(;5975K:;XV4f; ;@5o 3;&; * 35o; 7@3X.=3L3=7;<3W2$8: V4-@5K7 $W;<3
Iθ(yn)^V-:5Ky7J4D^3<3C7(K36\OH4Do-3 4D-ha 3_dbk/<\1* 5V! ->4?7@5V-Q;<3 ]5o;43C;b74D(;(4D;
Iθ = − - θ
[∂2 log f(x|θ)∂θi∂θj
].
3B;b7hH4h 4-:75972$;+* 5V! ->4?7@5V-cEt[:3X3+437'72-@7 $X37(4D 3;-'$C3 W W=xXW ; θ / A ;h;'$0C(5o;2; :;h4-K:2;G* 5V(;<5oX4D7@3 A;<3W74-5KVV3Y3 23B75D:3A7(4DY7(4-
n(θ) =∣∣Iθ(yn)I−1
θ
∣∣1/2.
d5;@5KR(K3CL3C870+n(θ)
3B;b7Z5VF`4-@5H4D873J7(4-W@37(4D4D($=7@5o;4?7@5V-/ 4-(;yn
* 3; 7A7(43;AX3X; 2$1+ n(θ) = n72
7 7 θ / 7@3XE3C:79+ ;^4e-:;O7@ b %; r ≤ n(θ) ≤ n18
r3;b7AV3W-NR3W;3I;<-$X3;(- =3X.; 2$X3B;8/
\4F;<5 # 972$6$85H;<3C:73E72f; $X3E7(4-X3=7 73N;$8C(5K7@5V-[3B;b7YV4+;'$8723X(;4-=3h3Xθ
;3n(θ)
/ \1* 5V! ->4?7@5V-S;3]5o;43C3; 7]G* 5V! :@>4?75K: 75KV5o;$C3 ; 3CKV3XE3X87g;4-; JC4-;<3 ! 2$ 3C87@5o;b731+`18NK3Z7(4D4D(,X7@3
θ3; 7 C$f/yC(
;<3]=:; 7@ 5K3 5V(;<5oX4D7@3 i:Rb3=7@5D!];<3n+ ;W7(@f7Jd; :;O;3YG* 3; 7@5VL3C^4 >4 5V ;<3]84D5o;@3XhR(V4-=3
θn/ Y4D.;gd;G3<3X(7V3;0+-G* 5K(;5C4?7@3 ! @(57(4- =3X7 7@3&$X7 4<;<3d3; 7 7%43A;<3AH4Y74-5KVK3&3 375D:3
n∗ ;'$0C(5V34 7(4D4e?4D870+83=7&472 A4e?4-:74`::3Y;+* =7@3WR23C4 = 7 7( ;y!/43=5VV3])-C4Do V3XB/
$&% ' /GR < xXS)TDw-w :D=<U w U v w)T'W =:;@5V;,C@3 $B%4(4D875KVK: 37J:3X87@5V3X
yn
=-873X(4-:7r
;<:'$X3B; W W=xXW 4`7'72-@74-87EG* 5K! :@>4D7@5V- ;<3]5o;43C
Iλ = λ−1 +3=7n − r
X3X; 3; c1, . . . , cnr
;3E7 [d723 7(@f:-3;2; 5D!O4`77J: 74D87E%4(43 3Eg* 5V! ->4?7@5V-Iλ(c) = λ−1(1 − exp(−λc)) /AV-%;
n(λ) = n−n−r∑
i=1
exp(−λci
).
?i −
n
$# )$% )$' & $& " ! " $" !&
l + 9i/DaB_ g4-K3 2;;<3ZG* 5K(;5C4?7@3 n(λn)/
-g 2$ 75KV5o;@4-:7LK3B;6$B%4(4D875KVK:;E;@5K $;L;4D;hg* 3<3C7(K3m,+]-c72$;@3X873; hV3+74-RV3C4 i'/a_QV3;J`4-K3 %;Et[-3X('$X3B;O;<3
n(λn)/-gVK3B;^4`7'7<5KE3C:7Y=:@3=7@3XE3C:7]^ 3CV3;y; 3; 7@5VE4-87i$8:f,X3XE3X87%bjV3;O74-5KVK3B;i3 3 j
7@5-3B;n∗ 72$;@3X87 $C3;;4D.;g* 3<3X(7V3Im'/
$A% '%5a_/GR < xXS)TDw(x3w )w ^ s T$T WA ;g;$C@5-:;5X5 (3&4`7'7(@<5VE4D7@5V-E;<3
n(θ)7J gV3AE<;,XV3&;<3U3X5VR V + 75KV5;4DR(K3&;(4D;GV3; 7F8$C; @3B;
;$X@5K7@3B;O4 § i'/ i'/ / /'\O-;@5V;'$X-;jV467(4D4D($=7@5o;4?7@5V-θ = (µ, β)
/ -5K7 n− r?4-K3 2;O;<3]=3X.; 3)L;<-5K7@3
(c1, . . . , cn−r)+-;37 [d723 7(@f:-3;2; 5D! / J* 4`72,;V37 4'$C-2,XE3 3X7gV457372f;@5K7@5V-Ji +? ;-R<73X:;+?3CED74D87
κ = 1 + π2/6+
n(µ, β) = n− k1(µ, β)k2(µ) − 1
nk21(µ, β)(k2
2(µ) − 2κ) + o(n−1)
18
k1(µ, β) =
1
2κ
n−r∑
i=1
exp(−µcβi
),
k2(µ) = κ+ (γ − 1)2 − 2(1− γ) logµ+ log2 µ.
F NK3E74DRV3C4 i'/a3aE ;I5V(;<5o -;hK3B;Z3; 7@5V>4?7@5V-.;h; E5V5KE5o; 3 N;3 T] VKR(4fo n∗ 4-(; πJc
3B;b7hG*=<= > < > ;<3 :3 38[;=3C; $N;$0C((5]4 § i'/ / /aZ3=7
πJ 3; 7&V3 > w d w8> w U 0 w = > < > ;b74D(;4-; /-gh7(4-4-K,XV31+J ;;<-(-;H4?4DV3 A;<3WG* 5V(;<5oX4D7@3 &4`77@<5V>4?7@5D!
n(θn) = n(µn, βn).
\O3;Q2$; K74?7%;h;@-87[Lt[:3X$; ; _ $%4J4D87@5VKV-.;[;<3a74-5KVK3 ?4-@5H4DRV31+;@5V $X3B;h; 3CK: V3;h7(4D4D(,=7@3B;η0 = (1/µ0)
1/β = 1003X7β0 = 2
3X7 X3X; 2$;+;<3 V4 CE3 !/4 X- 3 ;(4D;Fg* 3<3X(7V3em/O\3B;F;<3 5KJ;<5C4?73 %;Y; ::7Z5;@5KR(K3CL3C87i7 ,B; 7(@%43B;+ XE3]; 5
n∗ 3; 7@3Y)N3XX-3&$8:f,X3XE3X87W7( ;Z!/4-5KR(K3f/\3AX4Do ;<3n∗ 3; 7&;<5#=5VK3Z3X737@ XE3X87X= ,<7@3 +/
n
n∗ n(θn) n∗ n(θn) n∗ n(θn) n∗ n(θn)$ $# $# ! %" ' # #"# )$ $" ' $" %&#$# & ' *$# )$ ! &# ! " $" #"' $")$! ! "# ! ' $" &' $" !
l + i'/a3a g4-K3 2;W;<3LH4+74D5VKV3C.75D:3n∗ 3=7I;<3EG* 5V(;<5oX4D7@3 n(θn)
/\ * 5V(;<5oX4D7@3 n(θn)+ 7@2,;];@5K(7V3-)
X4- K3C0+<437'75VE3dH4L74-5KVV3C97@5-3n∗ 3CQV4(; @3B;b75K>4-:7^$8:f,X3XE3X87/
; > @?2A ?
4E-(A" - -6'(X-' 5' X
^6R < x3STDw-w :D=&<U w U v wT bk/ S; 77Jd; 3L4e--5V
yn = (y1, . . . , yn)?4DV3 2;d-R;@3X23$C3;7(4DE5 V3;2 3XVV3; r ?4DV3 %;&(-&X3X; 2$X3B;
xr =
(x1, . . . , xr);<3Q;<3X.; 5K7 $Q372-3C87@5V3XVK3
f(x|λ) 3X7+;<3! :75K: ;<32$87(4- 75975K:F (x|λ) / D73
cn−r =
(c1, . . . , cn−r)V3;N?4DV3 %;;<3y=3X.; 33/ F 7'7Jd; :;N7J -:5K ; 7.$B=5DC.=3C 3^;<3X;@5K7 $Z7@:R(4DR(5KV5; 7@3;<3B; 5V; 74-:7%;
;<3AX3X; 3 fc :-7@3I4-K:2;
δ(λ) =
∫
IR
fc(x) F (x|λ) dx.&K:2;0+-Q-R<75K3C87^;(4D;V3;A;5 $X3X872;yC43;^;<3EX3X; 35; 5?4D872;0+
af/I0 w Uks > wJx3w6v a = wfb = > <c > w d : δ(λ) = n−∑n−ri=1 exp(−λci) /I0 w Uks > w shU dD< >#](w ks > [0, c∞]
:δ(λ) = 1 − 1 − exp(−λc∞)(λC∞)−1
i/I0 w Uks > wJx3w6v a = wfb-b : δ(λ) = r/n/
npoq rsuqut\3B;^ $B; 974?72;&;<3;C43;A;36=3C; @3I;<3I7 [d723 Z73:-3;2;@5?!]3=7 (5?! :@E3E7t5K3C3C:7A;<3hg* 4D@7@5o=V3I;<3d3 4D87@5o;+.[: 73X4Z3=7d4D;<5N^ _f_'aBb0/ Y4D.; X4f;O;3]X3X; 3d;<3d7 [F7J37+<G* 5o 3?4DV3 ;35X3X; 3&3; 7H4(?4DV3 &;3IV4+V4-; 74D7@5o;b75 3h; * :;<3
r;<3IH4(7(4D@7@5V3Z:h=3X.; 2$X3I;3Ig* $%4J4D87@5VKV-
yn
/J\3Z(-hR(@3r
3; 7X-;@5V;'$X2$5X-EE3Y5V(;$7J3C(;4-:7&;3;^:R; 3C ?4D7@5V-;i;<3
yn
/ &K:2;
δ(λ) = P(F (x|λ) ≤ r
n
)=
r
n.
! ^6R < x3STDw-xfw )(w ^ s T$T b0/&D7@:;
fW (x) = βµxβ−1 exp(−µxβ)H4Q;<3X;@5K7 $+;<3 U3X5VR Vj3X7
FW;4 ! -=7@5V-a;<3> $7(4D@7@5K7@5V-/ -5K7 γH4h=:; 74D873F;+* - V3X ^
γ ' 0.57722bk/ -5V3X87 γ1 = π2/6 + γ2 − 2γ > 0
+γ2 = −2(1 − γ) < 0
/ -g ; 37H4 C4D87^;4D;V3; CL3B;y=:(;<5K7@5V-;y 3]72 ^H47@f7Jd; 5K7@5V- +<: ; 7'72f;@3A=-((4H7@3 3Z;<3X.; 5K7 $
fc(x);<3B;Y?4DV3 %;^;<3A=3C; @3f/ (D7@3Z4-K:2;
δ(µ, β) =
∫
IR
fc(x)FW (x|µ, β) dx.
PD73 7(4D4D5VKV3 %; δ(µ, β) = δ2(µ, β) + [δ(µ, β) − 1](γ1 + γ2 logµ+ log2 µ
)+ π2/6 − 1
/ AV-%;EH4E3; 3Z;<3 -3 238[;Y72 ^H47(4D4D($=7@5o;4?7@5V-
(µ, β); L<;,CK3I;3U3X5VR V X3X; 2$5)>;@:5973Y3B;b7
πJc (µ, β) ∝ (µβ)−1
√δ(µ, β).
npoq rsuqut]\ 4;<3C;@5972$L;+* 3(?4D5H4DRV3 X ;3 U3X5VR V1=3C; $C3);<@:5973J7J4DYV4 ?4D5V4-RK3C
^/5K(;'$8723X(;4-87@3;<3XbO3B;b7
g(X) = [fW (X)]IX<C [1 − FW (X)]
IX≥C =[βµ Xβ−1 exp
(−µXβ
)]IX<C[exp
(−µXβ
)]IX≥C.
?i −
\4L>4?7@5o=3Z;<3A;2=-3Z43;2; X5D$C3Y4 7(4-4-,X7@3; (µ, β)3; 7&4DV-%;
S(µ,β)(X) =
[ 1µ IX<C −Xβ
(1β + logX
)IX<C − µXβ logX
]
\4L>4?7@5o=3]43B; ;@5K3C3A; (µ, β);8* $X@5K7^3X; 5K7@3
H(µ,β)(X) =
[− 1
µ2 IX<C −Xβ logX
−Xβ logX − 1β2 IX<C − µXβ log2X
].
\4L>4?7@5o=3Z;<3 ]5; 43CA-R<73X 3W3; 7
I(µ,β) =
[1
µ2 P (X < C)-[Xβ logX ]
E[Xβ logX ] 1β2 P (X < C) +
-[µXβ log2X ]
]
5 X W(µ, β)4DV-%;
µXβ = Y E(1)/d-.-
[Xβ logX ] = 1βµ
-Y [Y (log Y −logµ)]
/3\O3=7@7@3O3; 7.$C4-=33; 7^R5K3C[;$0C((5K3EC4D
g(y) = y exp(−y) log y3; 7^5K87 $:-4DRV3]; IR+ +7J µ > 0
+<3X7∫
IR+
ye−y log y dy = Ψ(2) = 1 − γ
18Ψ
3B;b7AH4J! -=7@5V-[;<5::4DE>4 ^/;$C@53$WV3::4-@5K7 4(L5o 3Z;<3WV4! :75K: :84DE>4fbO3=7γ
H4(=-.;b74D87@3Z;+* - V3X^γ ' 0.57722
b0/ ;E;<3 7 $B=5o; 5V-.;6;@-876! @(5K3B;67(4-h33C(7K3Q;4-; &R4DE$^597 738: ^ a +7/O m j O dbk/ AV-%;
-[Xβ logX ] =
1
βµ(1 − γ − logµ).
d3A7 ;0+ - [µXβ log2X ] =1
β2
-Y [Y log2(Y/µ)],
=1
β2
-Y [Y log2 Y ] − 2 logµ
β2
∫
IR+
ye−y log y dy +log2 µ
β2
∫
IR+
ye−y dy
=
g4-[Z] + γ2 − 2γ
β2− logµ
β22(1− γ) − logµ ;+* 4`72,; F "^a b<+
18Z
3B;b7 3A?4D5V4-RK3I4-D$M4?7-5V@3A; 5D?4-:7AH4>;<3C; 5K7 $N;<3hn hR23Xk(x) = exp(−x) exp(− exp(−x)) ;$0C((5K3
; IR +(;35?4-@5H4DX3 π2
6
/&5K.; 5 +3XQ-74-:7γ3 = (1 − γ)2
+
-[µXβ log2X ] =
1
β2
[γ1 + γ2 logµ+ log2 µ
]
7 5; det I(µ,β) =
(1
µβ
)2 [P 2(X < C) + P (X < C)
(γ1 + γ2 logµ+ log2 µ
)]
−(
1
µβ
)2 (γ3 + γ2 logµ+ log2 µ
).
3C[;$C; 597X;4D;^;5 # 972$Zg* 373;2; 5V-;<3δ(µ, β)
/
^6R < x3STDw-xfw )(w ^ s T$T b0/-gQ2$ 75KV5o;@4-:7^V3;-74D7@5V-;^; 724'$X: ,CL3 +<--R<75K3C:79+;@3XV-K3Y7 [F7J3I;3E=3X.; 31+
af/I0 w Uks > wJx3w6v a = wfb = > <c > w d : δ(µ, β) = n−∑n−ri=1 exp(−µcβi )
/I0 w Uks > w shU dD< >#]-w ks > [0, c∞]:δ(µ, β) = 1−(βµ1/β)−11−Γ(β−1, µcβ∞) 18
Γ(x, α)3; 7 H4y! :75K:
::4DE>4L5K.=-(7,=73
i/I0 w Uks > wJx3w6v a = wfb-b : δ(µ, β) = r/n3B;b7A5V(;$7J3C(;4D87&;<3
(µ, β)/
npoq rsuqutY\3SC43;7 [F7J3 #73::@3B; ;@5?!EX-@3B;72-(; ) ;<3B;Q;<-$X3;;<3"=3C; @3&C<3;C1, . . . , Cn−r
/ &:P (X < C) = r+
∑n−ri=1 F (Ci|µ, β) = n−∑n−r
i=1 exp(−µCβi )/\3]X4f;O;35X3X; 3 (5?! :@E3Y3; 7i5VE$M;<5H4?7
3C#=-.; 5H;$X4D87GH4A7(4D4D($X7@5;4?75K:U = µXβ / d4-;gV3XC43;G7 [F723 7+8;<3AH4Z XE3y!/4=-# 3d;4D;gV4A7(@3 -3
;<3WH47@f7Jd; 5K7@5V- +<::R<7@5V3X87X;@5K(7V3XE3C:7X 3 δ(µ, β) = r/n/
/ :597 (µ, β)V4 7(4-4-$X7@5;4?75K:S;+* aL<;,CK3>;<3 U3X5VR Vg/ -5K7 xe ∈ IR+∗ + a > 0
3X7α ∈]0, 1]
/ (D7@3
b1(a, β) =((1 − α)−1/a − 1
)−1
xβe ,
b2(a, β) =
(Γ(a)
Γ(1 + 1/β)Γ(a− 1/β)
)β
xβe ,
b3(a, β) =aβ + 1
β − 1xβ
e .
[4`7'723XVV3J 3 B ;$;@5:-3NV4>E<;$CK5o;@4D7@5V-S5V(; 597367(4-dV3J%4:5 [;<3b(a, β)
/&K:2;0+72 &7 7]%4-5 Q;3π(β)
;<3E; 7'7J: 7 [βl, βr] ⊂]0,+∞[+3X %4:5;@5o; ;4D87E< = > < >
µ|β ∼ G(a, b),
:4
(i) P (X ≤ xe|B1) = α,
(ii)-
[X |B2] = xe;@5a > β−1
l
+
(iii).
d[X |B3] = xe;@5βl > 1
/
Y4D;dV3C43;(iii)
+ V3hE<;<3xe
;<3hH4;5; 7@5KR 7@5V-S>4- ::5K(4-K3M
3; 7 (5 3L3=7A;b7@5o73XE3X8757Jd; 5K7@5D! / d4-;V3;^4 7@3;yC43;0+F;@5 βl ≥ 1
+M
7Jd; ; ,C;<3Z4 E-5V; QE<;<3E; 7@5=7@3XE3C:7y72f;@5975?! /
npoq rsuqut 4`7'723XVV3] 3 FW;$;@5D::3WV4J! :75K:Q;3Y2$87J4D@7@5K7@5V-Q;<3ZV4LV-5;<3 U3X5VR VG/ 4>4-K:2;
P (X ≤ xe|B1) = M(xe|B1),
=
∫
IR
∫
IR
FW (xe|µ, β)π(µ|B1, β)π(β) dµdβ,
= 1 −∫
IR
1
1 + xβe
b1(β)
a
π(β) dβ,
= 1 −∫
IR
(1 − α)π(β) dβ = α.
?i −
\O3h2$; K74?7d3; 7W4 ; ;@5?4-V4-RK3N XE3J;@5π(β)
* 3B;b7]7(43;d;<3(; 77J: 7X-(7(4f7/ 4DW4D5VKV3 2;0+(3C ; 7'7Jd;@4-87a > β−1
l
+
E[X |B2] =
∫
IR
∫
IR
µ−1/βΓ(1 + 1/β)π(µ|B2, β)π(β) dµdβ,
=
∫
IR
Γ(1 + 1/β)b1/β2 (β)
Γ(a − 1/β)
Γ(a)π(β) dβ = xe.
\O3+2$; 974?7L3; 7h3XX-3#?4DH4DRV3;@5 7'7 (π(β))3B;b7J;@3 K3CL3C87NR2-'$)h:84 %4(3-7(4D βl
/ -gC(+ E;3$8:3X87 3C;<3
M3B;b7;@597 $Z3X 72-5V87
x 6= 07@3CKV3E 3WV4E;$C@53$C3
m′(x); :597^ VK3f/ 4
m(x) =
∫
IR
aba
(b+ xβ)a+1 βx
β−1π(β) dβ.
:3 F 77 (π(β))X-(7(4f70+<H4E;$X5`4D7@5V- ; ;G* 5V87 $8::4-K3Y3; 7;@4-;^;<5# K7 $Z3X7
m′(x) = −∫
IR
abaxβ−2
(b+ xβ)a+2
[xβ(aβ + 1) − b(β − 1)
]π(β) dβ.
^gi'/a b
\:2;2 3 a > β−1l
+(K3A%4:5 ;3b = b3
723XL3X7y85o;@5KRV3XE3C:7&; * 4- K3C m′(xe)/ 4-A4D5VKV3 2;0+ ∀x > 0
+<-[4
m′(x|B3) =
∫
IR
aba3xβ−2
(b3 + xβ)a+2 (aβ + 1)π(β)
xβ
e − xβdβ
5j3B;b7N; CE3; 5:-(3# 3+V4e;<5 $X3X.=3xe − x
/ #O * 3<5; 7@3;<-- * ;@3 8$C@[:c O;<3m′(x)
/F 7'72f;@-.;X 3 βl ≥ 1
/ YK3B;%4:5 (i)
3=7(ii)
;<3b+J3CS $ 7@5VV5;4D87dG* 373;2; 5V-c^gi'/a b<+(5V 3B;b7Z4D5o; $h;<3
E-87@3X * 5K3<5; 7@3x0 < xe
7@3C 3A72 0 < x < x0+m′(x) > 0
/ QD73E * 5V3<5o;b73E; 3ChRH4DR(K3CL3C87x′0 > xe
73XZ 3 ∀x > x′0+m′(x) < 0
/N\O3CV4S5K(7V5 37(4D6X-87@5V 5K7 $# 3 m(x)7Jd; ; ,C;3+4 E-5V;
E;3]X-(75;^;(4D;[x0, x
′0]/
/
\3+73XE3δk
3; 7LV3F4`7'72-@7N;<3V4h! -=7@5V-`(αk) − `0
)e4- K3CN3Cαk ; ;@-c::4:;<5V3X87/ \O3XVK3 j X5
;8* 4D( K3A 4DJ; βe = β0+4e-3B
βe = log
log(1 − α2)
log(1 − α1)
log−1(xe2/xe1).
\1* 4D:-:@5K7 4E3W4LV4J! -E3 ; 3XVK3
αk+1 = αk − ρkδk.
\3y7(4f;j4:;4`7<74?75?!ρk
3B;b7 %4(-5o; 5(;3y!/4 X--)I43;2; @3C 0 < αk+11 < αk+1
2 < 1)Wg* 5972$X4?75K:
k → k+1/ E-7@3
3 δk[1] < 03=7δk[2] > 0
/ 4DZ4-5KVK3 %;+; 5 `k > 1^@X3( 5G3B;b7Y7@ b 2;YV3(C43;W;4-;Yf;W4`77K5oX4D7@5V-;%b7+
−1 < hk = δk[1]/δk[2] < 0/AV-%;+
αk+11 < αk+1
2 ⇔ ρk[2] < ρk[1]δk[1]
δk[2]+αk
2 − αk1
δk[2]
^@i'/aidb
A3&\ ;<3c^@i/DaBifbN3; 7F;<-.;3h%4-5o; 5Vρk[2] = −ρk[1] + (αk
2 − αk1)/δk[2]
/ ;$;@5K3ρk > 0
72 4 ::E3X87@3C
α13=7&;5KE5V 3X α2
^ ⇒ 0 < αk+11 < αk+1
2 < 1bk/.\* 3; 7X%4f;@3]7Jd; ;@5KR(K3W3X %4:5;@5;2;4D87
ρk[1] = ρ(αk2 − αk
1)/δk[2].
-80 < ρ < 1
/ %4:5;@597A4-K:2;
ρk[2] = γαk
2 − αk1
δk[2]
(ρδk[1]
δk[2]+ 1
)
-80 < γ < 1
+X3A 5 7J3C@E3X7^;3Z@3B;723=7@3XE^gi'/aidbk/\1* $=74`723Z;<3A3 7@- j A437'4;@- ;8* $X@5K7&4DV-%;
αk+11 = αk
1 − ρ(αk2 − αk
1)δk[1]
δk[2],
αk+12 = αk
2 − γ(αk2 − αk
1)
(ρδk[1]
δk[2]+ 1
).
Q;$;@5V@3W-R7@3X(5K Q4`7'72-@7^;<3E=:d:3X2:-3C=35=-.;b74D87A4 = 2;;<3B;^5972$X4?75K:; :
|αk+11 − αk
1 ||αk+1
2 − αk2 |
=ω1
1 − ω1.
-R<75K3C87&4DV-%;
γ = −ρ δk[1]
(ρδk[1] + δk[2])
ω1
(1 − ω1)
7 5;
αk+11 = αk
1 − ρ(αk2 − αk
1)δk[1]
δk[2],
αk+12 = αk
2 + ρ(αk2 − αk
1)δk[1]
δk[2]
ω1
(1 − ω1).
-R<75K3C87A3XC(
αk+12 − αk+1
1 = (αk2 − αk
1)
1 + ρ
δk[1]
δk[2](1 − ω1)
.
;^V3Y7@3C@E3ρ
3; 7723=7@5K7&;<3`4-87Ea1+7 ;^G* 4-D::-59724E35;@3X4J7 $B=5o;A^ >4D5o;^V3X87%b0/
N/& ;4`7'723XV-.;K3B;i7@3XE3;^;<3ZH47372f;@5975K: / -5K7 π(β)
;<3E; 7'7J: 7[βl, βr] ⊂]0,+∞[
/ :597 π(µ|β)73XVK3
3 7J k = 1, 2+-
[Xk] < ∞ -8- 3B;b7EG* 3B;79$X4DX3#7(4-N4`7'72-@7) H4e;<5o;b7@5VR 75K: MW;<3Q;<3X;@5K7 $
mw(x)/ -5K7 λ V3]7(4D4D(,X7@3W; * QE<;,XV3Z372-3C87@5V3X;<3I;3X;@5972$
f(x|λ) / 7H43X3E; λ 3h;<3C;@5972$<= > < > π(λ)
;<3ZV-5 G(a, b)3X7A-D73
mE(x) =
∫ ∞
0
f(x|λ) π(λ) dλ.
&K:2;i5V3<5o;b73 5 3EX 7(K3 (a, b)∗ > 07@3X 3
(a, b)∗ = arg min
∫
IR
mW (x) logmW (x)
mE(x)dx.
npoq rsuqut:3λ ∼ G (a, b)
+(a, b) > 0
-[4
mE(x) =aba
(b+ x)a+1.
9 ?i −
7 7X%4:5 F;3π(µ|β)
3=7π(β)
;<3A; 7'72-@7[βl, βr] ⊂]0,+∞[
+:Q;'$0C(5K7
ϕ(a, b) =
∫
IR
mW (x) logmW (x)
mE(x)dx,
= −HW − a log b− log a+ (a+ 1)-[log(b+X)] <∞.
18HW
3; 7]G* 3X87@375V3;<3V4Y;<5o;b7@5VR 75K: MW3=7'- 3B;b7G* 3B;79$X4DX3W7J4D]437'7J: 7 )
mW (x)/&K:2;
ϕ(a, b)3B;b7
=::75K 3Y3=7&;3 #! :5;^;'$X5D?4DR(K3]7(4-i4`77J: 7y)(%4(43 35?4D5V4-RK3-+7 5o; 3 - [(b+X)−k] < b−k <∞ 72 k = 1, 2
/ 5 (a, b)∗3B;b7 37@ X ;<3
ϕ+4DV-%;W3* 3B;b7 L5V5V /\ 4JX-F-35K7 $Y;<3
ϕ; IR+∗ × IR+∗
5K(72f;@3A * 5V3; 7 5o 33/-j[3 3=70+
∂ϕ(a, b)
∂a= 0 ⇒ a
b2= (a+ 1)
- [ 1
b(b+X)
]
3=7
∂2ϕ(a, b)
∂a2= 1/a2 > 0,
∂2ϕ(a, b)
∂b2=
a
b2− (a+ 1)
- [ 1
(b+X)2
],
= (a+ 1)- [ 1
(b+X)
(1
b− 1
b+X
)]> 0
7J a = a∗/&D7-;3C; 5973E 3 b∗ +;* 5K3<5; 7@3-+<3; 7X; : 75K:;3YG* $ 4?75K:
g(b) =- [ 1
b+X
]1 − log b+
-[log(b+X)] − 1/b = 0.
4g(b) → −∞ 4D(; b → 0
+b > 0
/ ;WV4 =:(;<5K7@5V-" 3 - [Xk] < ∞ 72 k = 1, 2+ -a4 7J
b→ ∞
g(b) =
-[X2] − - 2[X ]
b2+ o(b−2) =
g4D[X ]
b2+ o(b−2).
$B=3;2;4D5V@3CL3C870+F7(4DX-87@5V 5K7 $Z;<3g; IR+∗ +(5K3<5; 7@3
b∗ > 07@3X 3 g(b∗) = 0
/&K:2;
a∗ = (-[log(b∗ +X)] − log b∗)−1 > 0.
U4[4-K:2;57 3$Lg* 3<5o;b73XX3E; * 5 3-X 7V3 (a∗, b∗) > 0 5gE5K5VE5;@3>H4;5; 74-=3+;<3 T] VVR(43o
3X87@3WK3B;>4D2:-5V(4DV3;/
/\3B; X-(;<5K7@5V-.; 7(@3CL5,X3;; 7 4'$C-2,XE3 ;<3 i3X%; 7@3X5V j -: .[5;@3; ;@-87H4 X-;@5o;b74DX3 ; .S\ - 3=7[;<3;=:(;<5K7@5V-;=H43;2; 5o 3;N;<3F2$8: H4D5972$F;<3V4eKf: j 84D5o;@3XhR(V4-=3 L(Xn; θ)
^G\OH4Do-3 4D- a3_dbk/ -jKV3;; :87 ; 3XVV3XE3X87f$X5?C.$X3;X7J YK3B;&E<;,XV3;d2$8: K5V3X%;XX-EE3 U3X5VR Vg/& ;dK3B;&4`77J3CK:;AR5,8-3CE3X87=5 j ;3;2; ;8/ :597 f(x|θ) V4E;<3C;@5972$E=-.; 5H;$X2$X3f/
a3/G* 3C;@3XhR(K3 x : f(x|θ) > 0 3; 7AK3W XE3A72 7 7 θ ∈ Θ /L(θ, x) = log f(x|θ) 3; 7>7@-5o;! -5o;>;<5 $X3X875V4-RK3 7(4->437'72-@7-)
θ;4-; -:5;@5KJ4`:-3 Dθ0(δ) =
]θ0 − δ, θ0 + δ[/ D73
L+L
3=7 ///L
K3B;^;<59$C@3C:75K3CKV3;^; * :;<3;]a-+ 3=7XiL3; 7J3B75D:3XE3X87/&K:2;-
θ0 [L(θ0, X)] = 0,-
θ0 [L(θ0, X)] < ∞,-
θ0 [L(θ0, X)] = − - θ0
[L(θ0, X)
]< 0
3=7A5V3<5o;b73 3]! -=7@5V- M(x)7@3XVV3] 3-
θ0 [M(X)] <∞ 3=7sup
θ∈Dθ0(δ)
|///L(θ)| < M(x)
/
4DG4-5KVK3 %;+ π(θ);<-5K7 =7@3y=:87@5V 3i3X717Jd; 5K7@5-33C
θ0/-\3y; 7'72-@7g;<3
π(β)$X74-:71X-(7(43=70+?V3;1X-(;5975K:;
β0 ∈ [βl, βr]3X7βn ∈ [βl, βr]
7J ^7 7 n 43;2;@3 ]:-4D(; ;@-87A'$X3;2;@4-5K3;y)EH4-=:;@5; 74-=3Z;<3π(θ|yn)
/ C(J4D +( ;d;38-:;X;@3 K3CL3C87d;$XE::7@3C&V3hV3XEE3J; 5`4-870+ 5 =:; 7@5K7 3NV4 =-J;<5975K: 75KX5D7J4DV3N;<3IH4:3X%; 5V-; 724'$X: ,CE3W;3O3C2; 7@3C5K j :- .S5;@3;Y7372f; $X357(4- A4D>4DE-@7 4(5 n54d;4"^ _3_3i +F7 /i bk/ / Y < v Qn(θ) = 1
nLn(θ) − Ln(θ0)<\
Ln(θ) =∑
i=1 log f(Xi|θ) WjT < > l ∀δ > 0l ∃ε > 0 v|w)T
m8s w
Pθ0
(sup
|θ−θ0|>δ
Qn(θ) < −ε)
→ 1.
npoq rsuqut d; :;δ = (δ1, δ2) > 0
3=7θ = (µ, β) ∈ IR+ × [βl, βr]
/ 4 +3X>4 <5KE5o;@4-87y; µ +
sup|θ−θ0|>δ
Qn(θ) ≤ sup|β−β0|>δ2
Q∗n(β)
-8Q∗
n(β) = logβ
β0µ0+ (β − β0)
n∑
i=1
logXi
n− 1 + µ0
n∑
i=1
Xβ0
i
n.
[4>4DV-%;0+ ∀β ∈ [βl, βr]+(3Xh; 7'72f;4D87 3 β0 ∈ [βl, βr]
+(3=7X;@ ;^V4-X-(;5975K: 3ZV3;X1, . . . , Xn
;@-875G/ 5G/ ;+/ +
Q∗n(β)
W(θ0)−−−−→p.s.
Q
(β
β0
)= log
β
β0µ0+ (β − β0)
-θ0 [logX ]− 1 + µ0
-θ0
[Xβ0
]− log
-θ0
[Xβ].
-R<75K3C876^ 3X 75KV5;4D87AV3; XE3B;7 [F7J3B;^;<3EX4- 3ZV473 -3Z; Y4'$X: ,CL3 7 /fifb
Q (x) = logx+ Ψ(1)x− log Γ(1 + x) − Ψ(1),
! :75K: 5 3; 7A KV3I3Cx = 1
3=7; 7@5=7@3CL3C87A'$8:84?7@5-3W3CQ7@ 7d4 7@3A72-5V87x > 0
/ ^7 7 δ > 0+(5K
35o; 7@3Z;<-.βδ ∈ [βl, βr]
+βδ 6= β0
+3X7ε′ > 0
7@3C 31+ ∀β ∈ [β0 − δ, β0 + δ]⋂
[βl, βr]+
Q∗n(β) ≤ Q∗
n(βδ)W(θ0)−−−−→
p.s.−ε′.
3X[;$M; 5K76^ H4X-F-3C ::3XX3X73;2 3A;-,(@3W5V(; 5o;@4-87H4-=:F-3X2:-3C=3d3XK:5ob
Pθ0
(sup
|θ−θ0|>δ
Qn(θ) < 0
)≥ Pθ0 (Q∗
n(βδ) < 0) → 1
;+* 18+<'$X3;2;@4-5K3XE3X879+8G* $X:0$Z; V3XEE33/
EFuH H K)M9NMOHLP '/ O q o M o TgPfUBPhL q K M9O Q3UT@M9ONK I cI t ∀β > 0l ∀n > 0
l ∀r > n/βl ∀p < 1
l
limr→∞
rp
Γ (r − n/β)
Γ(r) r−n/β− 1
= 0.
?i −
npoq0rLsJqut 7@5VV5;@3ZV4J! :@ K3W3<4f73W;3 7@5V@V5V': ^ A@7@5V6+ a +F7 / b0/
Γ (r − n/β)
Γ(r) r−n/β=
√2π(r − n
β
)r−n/β−1/2
exp (−(r − n/β) + µ (r − n/β))√
2π rr−n/β−1/2 exp (−(r) + µ(r))
4e-3Bµ(r) =
α
12r
18α ∈]0, 1[
+
=
(1 − n
rβ
)r−n/β−1/2
exp (n/β + µ(r(rβ − n)/n)) .
d-.Qβ,n(r) =
Γ (r − n/β)
Γ(r) r−n/β− 1 = r−1 n
β
[1/2 +
n
3β
]+ o(r−1).
c3X6;'$C; 5K7NH4hX-F-3C ::3XX3:3X%;E_[V-%; * -a 975D7(K5V3 Qβ,n(r)7(4-
rp 4e-3p < 1
/n
3X7βC$B;+
g* 4`77@<5V>4?7@5V-Qβ,n(r) ' 1
3; 7Z?4DH4DR(K3d4375V;3XE3X875^@-:5K F3<3X(7V3]5: 3i/Da fb7+f7J β ≥ 1+n ≥ 13=7X)7(4- 75K^;<3
r ≥ 3/
0 5 10 15 20
12
34
5
r
(r^1
/bet
a)(g
amm
a(r−
1/be
ta)/
gam
ma(
r))
M 9i'/a.+f1:- 7@5V-;^; 7(@<; 5K7 r1/β Γ(r−1/β)Γ(r)
3X ! :75K:;<3r+.7J β = 2
/
# ( - -% -6!'[- X' ! &( -d4-;=3X7 736; ; j 7(4- 75K3W4D(3<31+( ;X=-.; 5H;$X-.;OV36C43;^;+* 3I37J3C 75;@3A 4D87@5K74D7@5-3E 5;* 37@5VE3
;<5V@3B7@3CE3X87; ^V3A7(4-4-,X7@3η
;<3 3C5KR K 3=7A: 7( ;AE4- ::5KJ4DV3XE3X87y; X /
t t t q0o UBT@K q cT@ONT cI qut\3EX4f;^;+* 4D87@5VK3-+;+* 3Z3; 79$X4DX3d ; * QE<;<3W3; 7A $:-$7J4D^H4J7(@f7Jd; 5K7@5V-Sa_/
EGFIH HLK&M9NMOHPaB_'/ Y < v ηe > 0 w8v 0 < α < 1 W < v <Ub4(a, β) = ηβ
e χ22a(1 − α)/2,
b5(a, β) = (ηeΓ(a)/Γ(a− 1/β))β =&<`s > a > β−1
l
lb6(a, β) = β−1(aβ + 1)ηβ
e .
jT < >
(i) P (η < ηe|B4) = α,
(ii) & [η|B5] = ηe,
(iii)R
d[η|B6] = ηeπ(β) w vyx3w %s ==&< >kv [βl, βr] ⊂]0,∞[ W
npoq rsuqut d4-;^V3EX43;(i)+:Q4
P (η < ηe) = α ⇒ P (η < ηe|β) = α/ 0+';@5
µ|β ∼ G(a, b)+2bµ|β ∼ χ2
2a
/ D73
ηe(b) = (2b/χ22a(1 − α))1/β /AV-%;
P (η ≤ ηe(b)|β) = α/ Q3XQ;$C; 597Y4D5o;$CL3C87G* $X:0$
(i)/
Y4D;gK3XC43;(ii)
+-H4]73 :3^3; 7j5KE($C;5V4D7@3&3C#=:; 5H;$C4-:71 3 - [η] =-[-[η|β]]
3X7j3X 7@5VK5o;@4-87jV4A! - V3;<3B;^E-E3X87%;^;<3IH4L;'$0C(5K7@5V- 7 / af/d4-;^V3EX43;
(iii)+<5KN; E7&;<3EX-; 74D7@3C 3 π(η) =
-[π(η|β)]
3B;b7;$C@5?4DRV3-^V7 5;2 3 π(β)
3B;b7&;<3A; 7'72-@7[βl, βr]
bi3=7&;+* $X@5V3
π′(η) =
∫
IR
ba
Γ(a)
exp(−b/xβ)
η(a+1)β+2
(bβ − ηβ(aβ + 1)
)π(β) dβ.
d3ZV4E CL3E!/4 =: 3A72 AV4(7(@f7Jd; 5K7@5V- (7 / O +<5V3B;b7&4-5; $Z;<3I-7@3X 3ZV3IE<;<3ηe > 0
3; 7 5o 372 ZH4E;'$XV5;4?75K:
B6+3X76 3EK3B;ZE;'$XV5;4?75K:;
B43X7B5
72f;2;,M;<3>4 E-5V; aE<;<3-;b7@5o73XE3X8772f;@5975?! / 4-G4D5VKV3 %;0+?-LD73Y 3^H4W;<5o;b7@5VR 75K: M ′ ;<3
η7J3 7 =7@3 (5KE<;4DV33X 3Y?4DV3 1;b7@5o73XE3X87
72f;@5975D:31=37J3C(;4D87 3gH4^;<5o;b7@5VR 75K: M ;3X
* 47(43;;3gE;3g;$0C((5.^V7J4D3<3X(7V31 4D(; a < β−1l
b0/
t t t 0OU q0o s I q L q Q o PdLNT T TVUBP tY4D;YV3C43;d18 5V87@3X2?4DVK3L;<3(=2$C;<5VR5VK5K7 $
[η1, η2];+* -;<3
(α1, α2)3; 7Z;5; 7J:5VRK3-+ ;5; :3:f$X-;
;<3%4-5o; 5V b = b(a, β) 5 35=:LE3^H4E;@- 7@5V->3F7(K5o=5K7@3&; * -7(@:RD,CE3&;<3AE-5V(;<3; X4D2$;y^V7(@f7J j;@5975K:ca3aeb7+ 5 E5V5VL5o;@3NH4F;<5o; 74D.=3h3C87@36 4D87@5VV3;5; 4(4D5K7 $B;&3X75 4-:75KV3;d3 3=7@5D!@;8/ &5K;@5g* 373;2;@5K:;<3
b2$C4-K5o; 3 X-;@3X; ;J74?7@5o 3; * 75KV5o;@4D7@5V-6;<3B;E;<3 cX-(4-5;2;4DX3;/ \1* 4 b 7L; * (3 72-(;'$X4?7@5V-
7585V$8:-5H4D87^G* [ g* 4 7@3I;<3; 4D87@5VK3B;^3; 7&o)E3XX-3A7Jd; ;@5VRK3f/A(3I7@3CKV3E7372f;@5K7@5V- * 3B;b77(43;!/43=5VV3 jE3X87g4-;(4`7<74DRV3^4 X43;]-8LV3; 4-87@5VK3B; ; ::7>4D2:-5V(4 I3X
X+`K3B;1 4D875KV3;;<3
M * $X74-:7 7(4f;37V5X5973;/
EFuH H K)M9NMOHLPa3a3/ <`s > i = 1, 2lU < v <U ηi(b) = (2b/χ2
2a(1 − αi))1/β w0v bi = ηβ
i χ22a(1 − αi)/2
WLY < v2Tw= > < ^)T'S)](w(x3w ] <$U x`> w h0%< >k> Z
b∗ = arg minb∈[b1,b2]
(1 − η1(b)
η1
)2
+
(1 − η2(b)
η2
)2.
jT < > b∗ w : v|w l w v shU m0s ww8v 'u Z0 > v b∗ =
(b−1/β1 + b
−1/β2
b−2/β1 + b
−2/β2
)β
W
npoq rsuqut 4f%4(4D87 β +7J i ∈ 1, 2 +((D7@:; δi = ηi(b)/b1/β 3=7W3 37 -.;dV3-%4(4D:-3XE3C:7W;<3`4-@5H4DR(K3
x = b1/β /(\ 4J! -=7@5V- )EL5V5VE5;@3X; ^V3I;<->4D5V3[b
1/β1 , b
1/β2 ]
3; 7
f(x) = η−21 (η1 − δ1x)
2 + η−22 (η2 − δ2x)
2.
&K:2; ∂f∂x (x)
3; 7;b7@5o73XE3X87y=-5o; ;4D87@35; [b1/β1 , b
1/β2 ]
C4D ∂2f
∂x2(x) = 2
(δ21/η
21 + δ22/η
22
)> 0
/-j3X>4D j 4-:7 3 b1/β
i = ηi/δi+: X-; 74D7@3I4-K:2;Y 3 ∂f
∂x (b1/β1 )∂f
∂x (b1/β2 ) < 0
/\4! -.7@5V- ∂f∂x
7Jd; ; ,C;<3Z;<: $X (5 3E; [b
1/β1 , b
1/β2 ]
5;$8C(5K7AK3ZE5V5V 5o 3I;<3f
;4-;G* 5V87@3X2?4DVK3f/
Chapitre 4 t d #ll % L # &
)" *& ) " &&) & )) s uI|.~wZ|.~ π ) & yn
) * * " " & && & &)" & * ) ! *"& & * " #* ) & & *& & ) " )) %$ )* )" )& ) ) & ' s)(g~+* ~Px/yz " ))" )-, *)& /. & ) *#0 *&1& ) * 2 & ) 3 zys*54 #"#" ( ) ) & &), & ) ) * ") *& & &
& * )" & suI|.~wZ|.~ & . 6 & ") 7 ) 8 & )& & ) & "9 " * . : )) <; = ?> sZyXs@sAZ|/z/zBzCSyDg|~Pyz|.~w"C "E& & & ) ) * ) ) )) * *& "GF "%"/&#&# <H & " ) " ) I" ) )" )& & * )" Dg|%J > ~K(* z )" ) ) )
6Y ( 9- 'n(143 KL;I8CBD\SRS8CBT;E6 111F11F111F11F111F11F111F1F1F11F111F11F111F11F111F1 3ML9Hn(1DJ Nv:<]O\VBT;E>VQPOF;I:/Q2Qt1F11F111F11F111F11F111F1F1F11F111F11F111F11F111F1 3ML9n(1D0 RTSV]VUpXW @A:hBT8g]O:hBT;96$1F111F11F111F11F111F1F1F11F111F11F111F11F111F1 3I3I3n(1on Y 2R9Yde(NT]OQ;IW BP=(]RIN'@ARIQ7]OQ)1F11F111F11F111F1F1F11F111F11F111F11F111F1 3I3 nn(1DG pq6)B46987:hB46(QCBP@kRI=VRIe(87RS8CBT;E6;[WZUpXW 1F11F111F1F1F11F111F11F111F11F111F1 3I3Sn(1D_ Nv:hBT;I:@ARINPBTm(:<R[8CBT;96[O BD8\S]UpXW 111F11F111F1F1F11F111F11F111F11F111F1 3OJI_n(1T p e ;IQCQCBPmNT]#B4Yde(:<;O\9]SY#]S6E8 11F111F11F111F1F1F11F111F11F111F11F111F1 3OJIn(1DH W ;E6(@AN4>(QCBT;E6 111F11F111F11F111F11F111F1F1F11F111F11F111F11F111F1 3O023n(1D pe(e ]S6(=(BT 1F111F11F111F11F111F11F111F1F1F11F111F11F111F11F111F1 3O023
a_
−
Y
AE-:Z7 43Y4D@7@5o=V3;G:#; Rb3=7@5-3 4e[-3B; 5H4DE5V! 3X3X.=3y! :j3; 7@5V>4?75K':I7 437(4D4DE3=73Xθ ∈ Θ
`!4I;<3B=5o; 5V- j>4Do5K:LE<;<3C M(θ)
+J>4DF[>`!]7 43C E3X875K:7243W5KJ;<3;@5K4DR(K3WR2384(4e5V-`!]7 43A72f; 7@3C@5V-<;;'; y4(3X7 4375K:Y5V! :@>4?75K:e5o;WR(4-;D[ X-(;5975K:3C; / Z[[7245o;Y7@3X i3E3XX-(7(43;2;W4 ^@:':5V! :@>4?75K: ::5D:3XURd[3723X@7f75V5K:;E^/-A4DF[+-7 43C; %=3Y`! ; Rb375D:3Z5K'! [email protected](;Q4J7(@5V-A;<3C;@597 [
π(θ)!/4DY! - 7 43
! @3B 3X87@5o; 7YX-CJ;3XX3d38::5K:F- 7(4D4DE3X7@3Xθ
R@ :3487RF[>-R;@3X2-3M;+;4?74yn = (y1, . . . , yn)
/ Y4(3d:Rb3 j7@5-3Yo$^K3M;:-3d`!7243W@3M4DV597 [-[5V3XH;<3C;FRF[E7 43WV5Ko:3XV5D4<; L(yn; θ)
X4-+7 4 ;iR23WK5K7 73X3C;Rd[E7243A%4-5o=3d`!4^@:':67(@5V-4D(;F7 4(3 4e[-3B; 5H4D4D(4-D[; 7ZX4-F74Do:3d4J7Jd;b73X5K:i;<3B=5o; 5V- ^59724 i3Xo=:E35X-;@3 3C=3B;8/ #Ef; 7WX4f; 3B;+-7 43B; 3&4- 75XK3B;j4-@3X3X87@3C@3M;E-+4Z3X5V! -%=3CE3X87j`! 7243X72f; 7@3C@5V-g-R ;b73;2;/ kA$i38:3X9+:4]CJ2; 7;<5- K7 [5o;' ;b7]%43o5V':>7 4J4?7&7243N4e?4-5KH4DRV3Zo$^V3C;::3Z5;dD75; 3C?! j =::74:;<5o7@: [-+R230! -3Z7243h5V! 3X3X.=33/A-7@3W7 4J4?7A5V C43;@3W`! 45D:f4[=3C; :@3M;;(4?74E4-(; ; >4DV ;4D(7V3E; 5 X3B;+ y43C7 434e[-3B; 5H4D5K'! -E4D7@5-3Z4`7 j7:43%4F5;Z-3C [E3X-EE3X(;<3M; ^ ^-R23X@7 _3_aBb7+:7 4(354(3=7@3C@f:-3C3X5K7 [N3!7243Y;(4?74N-^4J45:34FV38:3X `!N=3X.; :@5V':X4-4J4e-37 4(3X;@4-E3A3 3=7O43;g4 ^@:':A75K:2^59724=:@3=7G;(4?74/ 75K:1X4D>R23 i3XV.%4d; 3CER 7g7243X72f; 7@3 j@5V- 7 4(@3M4?7@3C;7@YK3M4-;Z7dRJ4-;I;<3B=5o; 5V-.;8/ #(; ; 7@5V4-34D(7V3;NX4-hR23j74Do-3Ch5V i-(385H4DVK3 i5KVD[#^ _f_ fb0/
Y43X6+.75K:^o8($^K3M;:-3Z4DJ;3C4-_o$^K3M;:-3AC4D 4(4e-3A; [LE3X7@5W-V3;5V[4-=- 5=7/y\ @5V ; [1+7245;7Jd; ;@5VRK3=- 5=7Z;@3X3C(;O-7j7@J4(4e-3&RJ3C3X+ %4; 7 ;<5V3C;F4DK7 4 :34>5K7YX-; 7@5K7 7@3;0+f! @: 72-5V87O3! 85V3 +4D+5V(7J: 74D87W7(@3CK5VE5K(4- [L7 4e[-3B; 5H4D-; Rb3=7@5-3d5K! 3C@3C=3f/AD75X34($O3-3Xg7 4(4D7i5V+72435; 7@5o7O:R;@3X2`4 j7@5V-+3!_7 43 4e[-3;@5H4D(7(4D4-;5D::=+:7 4(3X75K:O;<5o; 7@5KR 7@5V-F4-(;E7 4(3;4D(7V5K': ^V7(4-4-E3=7@5obE<;<3X=:; 7@5K7 734 y4:K3I;<3B=5o; 5V-[E<;<3C 4DJ;Q597]; 3C3X-;A-7&3XV38?4D87^7@;<3=73XE5K3N;<38::@3C3;^3!G;5;2=387(4-0[RJ3X7O3C3XQ7243X! @: 4 7 43C-3=7@5oX4-Y7J:5K87>`!585V3 A/*Y43 75K>4D2[c=:87@37E`!]; %4 4S;b7 ;[ 5o;L7 43 7(4f7@5oX4-O5V(7(43=7>`!4e[-3;@5H4D+3B;b75K>4?75K:RF[3X'::5K3C3X%; y43X7 4(3I;4?74L4-@35728:@[+5V! ->4?75D:33/
E3X ;NX-(373C; O:@o;];<3C4-4D7 43C ^5K7 4L-R ; 7@(3;2; 7(@:RK3C-;8/ [LE3X7@5C4DVD[-+e! 75K:(4D'; 7@ j7 3;&`!G7243J7(@5V-&4-(;Q7243;4D(7V5V':+E<;<3XN4(4e-3hRJ3C3X ; 7 ;5K3M;[7@-R<74D5V[38::K5:-5VRV3672f; 7@3C@5V-A5V 3XX3y43X 472f;2; 5VRV3N;<5o;2=387(4-0[5;d5V=3C4f; 5V': +2o$^5V':F:3I7243h7i;@ %=3B;A3!j5K'! -E4D7@5V-h4J43;&7@FR23L@3 jb3=7@3C; / .S4D5VQ@38! 3X3XX3;^4-@3 d3 ]5K3X7 75 ^ a 'aeb7+Y4^5V;"^a-m b<+kA5VKZ^ a bj\ X4f;A^ a3idbO4-(;3; 7J3B=5H4DVD[ * kd4`:84D ^ a +Ya-m:m,+Za3_,+ _f_3idb7+ A(;<4-;3 * kd4`:84D ^ _3_fbW4DJ; A'::3X%;^ _3_f_fb0/ -E3F4 7 4(-%;43;LnY3XVE4- w0v < ^ a db]=-.; 5H;<3Xh7 43 ; 5V 974D3C ;Z3 23B7%;h3!i7243 ;4D(7V5V':SE<;<3X4D(;U72437(@5V-N-4 72f;2;@5KRV3Q;<5;2=387J4D8[ ^5K7 4 7 4(3Q;(4?74/ ]5V2; 70+- `4-; .[d;4(-t ^ _3_ 4fbE4(4e-33 #=5V3X87D[ 4D5o;@3+7 4(37-RV3X>:A4 7@5V>4D2[5o;2; 3N5;d7 %43BoF7243:-<;<3;2;d`!G7 4(3(;4D(7V5V':FE<;<3X M(θ)
^5K7 4 @3B;723=7&7@yn
/7'7(@f7@5H4?73'i4e[:3;@5V4-hE3=724<;'; 4(4e:3OR23X3C+;<38:3XV3723C;N5K 4e[:4-@5 i3X2:-3Cy^ _f_3_db5VL7 45o;24e[3/ 3X-(;D[-+5?!OF385H;<3C=3N`!j3C@-A5o;Y-R<74D5V3C;6+ i364J4e-3Z7 %43BoQ47Jd; ;@5VRK3J75K: j ;4?74#=: 5o7B/# Y5KJ; ; 7@5V4-=::7370+D7 43;4D(7V5K:YE<;<3C5; ; 4DVD[ i3XV j 7@5V3C;L4D(; 4e[-3B; 5H4Dh7@3%4(5;G4D3 ; 3M;I7Zt-3C2X-E3 73X5;@5V-K5VE597%;^`! 7 43]! 3 3C87@5o;b7&437'7:43%4/
X; i3;4D5H;+! 3 O:@o 4(4f;YRJ3C3Xa;<-3L-e7 43-; Rb3B7/ -1?4D; .Sf; 4-t ^ _3_ 4 + _3_ R.bA4-@3N7 4(3>4D5VL30! 3C@3C=3B;g;<3M4DV5K':]73X5;@3X[ ^5K7 4E7245o;G7@f75o`/ Y438[=-(7 7@3^7243
P− ?4D 3^`!_4DE-R.; 3C :3C;; #X5K3C:7;b74?75; 7@5o ^5K7 4+3; 7237O7@N597%;O>4- ::5K(4-^V7(@3M;<5=7@5-3ebg;<5o;b7@5VR 75K: / #F5V(; ; 7@5H4D.; 7 ;<5K3B;+b;$C3 w8v <^ _3_aBb4(4e-35; :3:-3B;b73C;+7@ ; 3Z4 ]5; 43Xi73; 7R23=7i3X3C 7(@5V-4D(;3X(75V5dE3C43; 3;i`! =3C 74D5V87 [+4D(;4D < x <C04`7'7(@843%4 4(43;&RJ3C3X&7@f7Jd; 3M;QRF[ ; 3C4 ^ _3_aBb ^5K7 4 7]; 74D7@5o;b75C4D ;b75?C9X4?75K: /# 3=7@5V- / +CJ2; 74D(;F7 4(5K;Q4`7'7:4f%43;i4D3Z4D(4-D[ C3C;+/
X;A4-4DK7@3C@J4?7@5-3W4`77@843%46+ i37(@f7Jd; 3d7 43]! :KV$^5V':X@5K7@3C@5V-F7@>;<3X7@3=7&4672f;2;@5KRV3A=: 5o7/W\O: j
7 7@3W7 4(3Z4D7@5V
X\ J (π|yn) =KL
(πJ (.|yn) || π
)
KL (πJ (.|yn) || πJ )
y43C@3KL(π1||π2)
5o;7 4(3 T] KVR(43o j \3C5KRV3X^;<5-3C ::3XX3dR23=7i3X3C;5; 7@5KR 7@5V-; π14DJ;
π2
KL(π1||π2) =
∫
Θ
π1(θ) logπ1(θ)
π2(θ)dθ
4-(;πJ 5;N5V;3C4DV[ 4S:5K'! -E4D7@5-3#7(@5V-B/ X\ X4- RJ3Q38:84D;<3C;a43;E4 x < v << >2w2w-w .v 0 > v|w0> < / !
X\ J(π|yn) ≤ 1++7(@5V-W4D(; ;4?74 j :-5-3C =:CJ;<3C=3L@3:-5V-;X! -
θ4D3J=Vf;@3L3X :34U4D(;S7 43(75K:7(@ j
72f;4Dπ
5o;5V43:-3X3XE3C:7 ^5K7 47 43yn
/'-j;@3Z4X- 5=7i5o;^;<3=73=7@3C; / Y45;^;30C(5K7@5V-Q5o;3C4f; 5VD[E74-; 7Jd;@4-RK37W4-F[E! =7@5V-E`! 5V:73X3; 7
g(θ)/Y43CN7243=5K7@3X5V-(;4 H;NR23=:(7 7@3M; ^59724E7 43y75V-%;-
g(θ)/ X;@5K:
7243 T \ j ;<5-3X2:-3C=3d3X; 3;O7 4(3Z5K(;38723X(;<3C=3Z3! 7243Z3; 977+4DF[>R5 b375D:3Z@37(4D4DE3X7@5 M4?7@5V-/
# 3=7@5V- / i,+ i3>;<3=74D5VG7 43>4- : E3X872; y45o%4 V3C4:;S77 43#%4:5X3N3!O7245o;A=5973X5K: 4DJ; ::5D:3(;@-E373723X@7@5V3; ! -j5K72; ;@3y43CE7 4375K:j5o;145V3X4D%%45oX4DV[J; 7J3B=5DC(3C;>:g5o;O;<38C(3M;+43;O4E=:hR5V(4?75K:>`! 75V-%;8/Y43E;<5# K7 [S4D5o; 3M;[RF[S597%;W;<30C((5975K: 5o;W597%;W4`7'7V5oX4?75K:S7 ::3X3C4- C43;@3;/ -E3L5V;3C4D1C43;@3;Y4D3EC(%; 7@[7@3M4?7@3M;I5V 375K: / / #L4dE:@3Z:-3C3X4Df! 4-E3 i-o +i3O(3X3C;h7@Z4:; ; 7 X\"; 5VX3
πJ X4-NR235K(73723XB/
\5Vo5V':E ^;<5# K7@5V3;7(7@:RV3X-;i3XX 87@3C@3M;F5V i4e[:3;@5V4-+E<;<3X;@3XV3=7@5V-6+ i3Z;<30C((3Z5K 3=7@5V- / 4D .vg> /0A! - 3!i7243#=5K7@3X5V- / X; 4DG7-o;Z`!E;3XW;@3XV3=7@5V-c4D3 ;@3C;+43; < vg> < 8< = w Z4D(; =< v|w0> < > = > < > #^i3X2:-3C 3C@5%4(5ya,+ _3_ + $C@3 a:mfb0/ #a7 45o;6;@3=7@5V-6+ i3>E4-o-3;@-E3E=:(7(4D5;@-.;RJ3X7O3C3XQ7 4(3I5V;3C4D]4D(;7243h4:; ; 7@3C; ! :@ `!72436=5K7@3X5V- /'Y43E5o; 5V-Q3! X\ 4f;A472f;2; 5VRV3h7-N! -Y7 4(3C4DV5KR4?75K:e3!W; Rb3=7@5-37@5V-%;0+97(@f7Jd; 5V':F:Rb3=7@5-3J`4-@5H4D.=3LRJ (;';Y-Z;<38!/4 K7?4D 3B;+J5;&7 43; Rb3=7d`! 3B75K: § / /7d5o;&:3I`!7243N>4-5KS;<5 3X3XX3; ^5K7 4e7 43J7FX3C; @3A7372f;@3C;RF[ -1?4D; .[d;4:t ^ _f_ 4fbk/ ]5K(4-K[1+ i3X-;@5V;3XW;@-E3d5;2; 3;j; 3&7I7243X! 3 3C87O3C3C;+3!4-;437<7@5V':X\ 4D(; 7372f;@3]; :L3W3;@3C4-2%4X 3;Y! :Y! 7 30i-o9/
i3;@5V;3;0+D4DV-':&7 43Y; 3B75K:;+?34D(7V3; ^5KVF7 7G5K875X-(7(4D5o; :W7 433; 97%;:R<74-5K(3C;IRF[W7243X5VL3X7 4<;4-(; X\/'-W;723X5V4-K[1+j7 43[3F72-(3X87@5H4D^4DJ; U3X5VR VAE;3Xo; ^5KV^RJ3Q !/4e--5K7@3F34D(7V3;(; 5VX37243;<3X7@3B7@5V-e`! =: 5o7&R23=7i3X3Ch75K:d4-(;S;4D74>5o;&3!17(4D@7@5o V4-&5V:73X3; 7&5VS@3CK5H4DR(5KV597 [f/ 5=::7@5VR 75K:;4-(;[4-:-V-RJ4D X-(7(4-@5o; : ^5K7 4S7 43
P− ?4- 367@X3C; 3h4-@3E; EE3C; 7e5K[7243JX-X ; 5V- ;@375K: /U37245K(oQ7 4(4D7WR2D724 E3X7 4<;';W4D3(=-(7V3XE3C:74D2[[4D(; ; 38! ; 5VX3h72438[h::5D:31+9! -Y7243 i4e[:3;@5V4-S4D(4-D[; 70+_473X5;@3Y5K3 3!J7243y75K:G5K! :@>4D7@5V-L5KE73X(;3!2VX4D7@5V->4D(; .=3X@74-5K87 [I:h72437(4-4-E3=7@3C ;7(4f=3(^5K7 43; 7J3B7g7I7243d;4D74Z5V! :@>4?75K: / Y4 ;G7 4(3 4e[-3B; 5H4DE4DJ4D[F; 70+85V#7(4D@7@5o V4-y43X ; Rb375D:375K:2;14(4e-3R23X3C3CK5o=5K7@3M;+C4D %43o+4-(; 7Jd; ;@5KR(D[+E<;<5D!V[+7243A75K:/
O 72-5V872;Z3!Z5V3 4J4e-3ERJ3C3X =:;@5V;<3C@3M;e5VU7 43F@3B=3X87A[-3M4D%;8/Y43(CJ2; 7I-3E5o;E=:E5K': ! - 7 43>3C j::5K3C3X5K:h=-E 5K7 [a4-(;c; :3::3; 72;W7 ;@34 ; 5V(7K3 K3`!^7 4 NR y45VK3F7 43 ;@3=:(;c4`77@843%4"4(4f;h4; 74f;b75; 7@5oX4-.! (;4D7@5V- / ]5V2; 7@[';@-E3W-74?75K:;^4D3W5K87@<; X3C;+/
:n ∈ IN∗ gV3=7
Xn = X1, . . . , Xn ∼ M(θ)RJ35V(;<38723XJ;<3X87@[64-(; 5H;<3X875C4DVD[c;<5o;b7@5VR 73C;6@3M4D j
: j -3B7@: j ?4D 3C;4D(;- ?4D5V4-RV3;d5K 7 43-;4D(7V3; 7(4f=3χn ^5K7 4a4 7@:R(4DR(5KV597 [S;3X;@597 [ ! 75K: ^@72;F!%b
f(x|θ) 4-(;e4 ; 25D?4-W^ 3XV5H4DR5VK5K7 ['by! 75K: S(x|θ) /Y45o;572;F!O5;W;<38C(3C; ^59724e3; 7J3B7d7@4;<:L5V(4D7@5V':
−
E3C43; 3 y45o%4S5o; ; 4DVD[S\3XR23; : 3f/Wk&$O3-3C; :E3h`!O Y34D(7V3;^5VK ;@3L;5;2=3=73NE3C4f; @3B;8/W\3X7θ ∈ Θ
4-(;F;<3CD73d = dim Θ
/G\3X7yn = (y1, . . . , yn)
R23Z4D+:R; 3C :3C;#;4D(7V3&3!n
;4D74/ ; 4DVD[ i3^5VK=:; 5H;<3CA7 4J4?7
yn
=-874D5V;r .=3X.; :@3M;5G/ 5g/ ;+/W;4?74
xr = (x1, . . . , xr)! :KV$^5K: M(θ)
4D(;n − r
C<3C;^|7 [F723 j b&5:3487 j =3C; :@3M; ?4D 3;;<3CD73C;
cn−r = (c1, . . . , cn−r)/ Y4 ;7 43E-R.; 3C :3C;[V5Ko:3XV5D4(8<;&X4DUR23
^@5K7 73X[43;
L(yn; θ) =
r∏
i=1
f(xi|θ)n−r∏
j=1
S(ci|θ).
Y X; 3C4 ^ _f_'aBb ; :f:-3B;b7%;4DN5K87 5975D:3 5o; 5V-I3!<7 43i5o; ; 3`3B;723X5V4-K[W5Kh7 43YX4f; 3y4(3X3 M(θ)
5o; 4&KX4D7@5V- j; C4DV3EL<;<3Cg/W[F75C4DVD[ M(θ)
5;W5VU7 43>(4D7 4DG37J:3X87@5H4D !/4-E5K[ ^ - bk/]kA3+;<38C(3;h4 X-F-: 7@5V-7; 7^R23=7i3X3X7243A75K:4-(;F7 43ZV5Ko:3XV5D4(8<;
δπ,L(ω) =
∫ ∞
−∞π(θ − ω)L(yn; θ) dθ,
=
∫ ∞
−∞π(θ)L(yn; θ − ω) dθ
y4f;@3Z72436?4D 3I5VX@3M43;@3;(^5K7 4[7243J; 4(437J36;@5VL5VH4D597 [`!7 4(f;@3A! =7@5V-;d`!θ/ >4 <5VE4-?4D 3 δm
π,L5;
4?7@74D5V3M;N5Vω∗ / 4(3X>R2D7 4(! 75K:;j4-@3 (5KE<;4D :7 45o;g>4 <5VE5 M4?7@5V-E5o;g3C4-K5 X3M; y43CE7 43A7OZE<;<3;
4D3>3 4-K5 X3M;+/ Y43X 4 ;4D74 j 43:-3X3XE3C:7J723X%=3X874`:-3>5o;L;<3C; =3C;c4f;δπ,L(0)/δm
π,L/&\OK3M4DD[ 7245o;JX@5K7@3 j
@5V- 4(43;i-; 74D7@5o;b75C4D E3C4D(5K':(;@5KX3W597=:;@5V;<3C2;i5V(7K5o=5K7@3CD[4D(; ^@:':-[L7 4(4D77 43WV5Ko:3XV5D4<;! 75K:θ 7→ L(yn, θ)
5o;G4W;<3X.; 5K7 [I-θ/ .[-3Xt:3X ?4@;4?74 j 4`::@3C3XE3X87 723X%=3C874`::3O;<3B;D7 7t5V;<34]X4-K5VR4?7@3M;
=5973X5K: /
W "!AW -1?4D;Z4-(; .[f; 4:t ^ _f_$#-4fbX7@t5H;<3E4 ; 74D7@5o;b75C4D7243X: [e4-(;S; 3-3X4D3<4-(7K3B;5! -W7243+;<3=73=7@5V-U`!;4?74 j 75K: =:&%(5o7/ Y43[-=-.; 5H;<3Xj7 43Y>4D2:-5V(4D.75K:O;<5o;b7@5VR 75K: MT
`!4 < %s% 0 w .v v < v v /0T^5K7 4h;4D(7V5K:L;<3C;@597 [
fT (t|θ) / Y45o;^;<5; 7@5VR 7@5V- 4(4f;^;<3X.; 5K7 [
mT (t) =
∫
Θ
π(θ)fT (t|θ) dθ.
! 7243Z-R.; 3C :3C;; 74D7@5o;b75to = t(yn)
5o;A4(; 7(@5o; 5V':?4D 35! - MT(4DE3CD[ y4(3X7 43Z>4- ::5K(4-
P− ?4- 3
FT (to) = MT mT (t) ≤ mT (to)5;i387@3CE387 43C4=:&%(5o75;;<3X7@3B7@3M;+/ Y45o;O7 [d7(5C4DVD[>8 %; y43C FT (to) ≤ 0.05
:FT (to) ≥ 0.95
/ #7 45o;1C43;@3'b7 43d;4?747(@t5H;<3K5K7 7K3A:GE; 7'72-@7G7@W7 4(f;@3y`4- 3;`!
θy43C@37243y75K: 7V4f=3B;597%; ; 7'72-@7k/
#Q37^34D(7V3 O3I;3X5D:3FT (to)
! :A4;@5V7(K3Z3F72-(3X87@5H4D_E<;<3X y43C;(4?74E4-@3A=3C;@-3C;+/
() +* ,.- a3af/ & :D=< 9w v @< < x3w 0/21 77J.13/5476#498;:$/
yn = (xr, cn−r)y4/=<7/
xr = (x1, . . . , xr)8><7/ ? =/ ? 16@<7/'ACBG/ Bg/ A+/A8>478 DBE4 4FA/ ? 1BG4 [
f(x|λ) = λ exp(−λx) !H6@< x ≥ 08 ? A cn−r
BI181/J46`!n− r
=/ ? 16@<7/'AKA98L478/NMOB ? BEMK8>P 1 #JBI/ ? 413478>4BQ147BBI1t(yn) = r−1
∑ni=1 yi
/ 9<6$M F? A&R/=<2: ^ _3_'aeb /26$R&478@B ? T ∼ G(p, qλ)y49/=<7/
p = q = rB ? =8$13/26`!
4 [F7/ j J/ ? 19</h^ 6@< B ? 4/ ? J/ ? 16@<7/'A=8$13/eb 8 ? A p = rq = r + 1/2
B ? '8@1/K64 [ / j =/ ? 1<7/h^G\6? 6 ? j 8@1 [.M&476>47B 8!<76&BEMK8>4BI6 ? b#" /$PI8%J/ 6 ? λ 8&=6 ? ':$8L47/ G(a, b)
<BI6@<A&BQ147<BIR!&47BE6 ? (/'</a, b > 0
")/ ?
FT (to) =Γ(p+ a)
Γ(p)Γ(a)
∫ qtob
0
up−1
(1 + u)p+a du.
*6$/=:$/=< L49/ 1/ ? 1/+@BI:@/ ? 468 ? /,=/'1718@< [4<7/'16@PQA-J6%!PI/6P− :L8>P / 1 BI1 ? 6>4)JPI/'8@<," .F[ 16/9PIA /
66.13/(5%, 95%) 0 98 ? 47BEPI/'1 <78>4/'< 498 ? (2%, 98%) 21 /'1BIA/'1>49BI13/,42BI18 /, 47/'AORF[28 ? JBIPIPI8@<BE4 [@1B ? J/
45/MK8><5+@B ? 8@P+A&BQ147<BIR!&47BE6 ? 45</ 8L47/ ? 1 46</=%9/,J4 MO6@<7/ 45/2R/6 8;:.BI6@< 6 45/ 178>MPIB ? + MO6&A&/=PfT (.|θ) 4 8 ?45/39<BI6@<A&BQ147<BIR!&47BE6 ? " / ? 6>4B ? +
U8 ? 8 ? =BEPIPQ8><2[ 13478>4BQ147B ? 8>MO/'PD[ 8 14 8L47BI134B7 (96$1/ 49/A&BQ134<7BER!4BI6 ? A6./ 1
? 6@4A&/, / ? AK6 ? θ ^98:6.1 w8v<; _3_.#3bL4/ 8&456@< 1=9<6% 6.13/476&J6$M!&4/4/P− :L8@P>/ 6 49/J6 ? A&BE4BI6 ? 8@P
MK8><5+@B ? 8@P ABI134<7BER?&4BI6 ?
mT (t|U) =
∫
Θ
π(θ)fT (t|θ, U) dθ
476 <7/=MO6L:@/545/OB ? %?/ ? J/ 6 49/O178>MPIB ? + MO6&A&/=PB ? 45/@9/,4A"CBPE/ 8><7P>D 945BI1$/E4 6$<4&1 /=PIPH6@< -MO6&A&/=PQ18 ? A F (t0)
BQ1 6/ 4/ ? /FPEB7JBE4 (9/ ? J6 ? '!+.8L4/<BI6@< 158><7/G91/'AH"2* 6$/':@/'< 0/'BER!9PEPA&BQ147<BIR!&47BE6 ? 1I H6@<5B ? 14 8 ? =/EJ 8><7/<76@RPI/=MK8L47B 13B ? =/ 8 ? D MOB ? BIMK8>P1 GJBI/ ? 413478>4BQ147B BQ1 4/ (6$PE/K178>MPI//")/ ? 45/? MO/=<7B7=8>PK=6$1346/+45/J6$M!&478>4BI6 ? 6
FT (to)=8 ? R/B>+%H"
L MON QPSRUT V W
X3 Y[Z \ ^]`_ba@c edFd \ / ? 6@4/ πJ 8 ? 6 ? B ? H6@<7MO8>4BI:@/f<BI6@<@gDBE4h+@66&AiH<7/ 0 / ? 47BI134=6L:@/'<78/+@/K6 4/ 1 8D@/ 1C6$134/'<BI6@< B ? 47/=<7:;8@PI1,jπJ BQ1 ? 6 ? B ? H6$<MK8L47BE:$/ 8 ? A BG4 1C6$134/'<BI6@<A/ ? 1BG4'D πJ (.|yn)
DBE/'PIA1C 6.147/=<7BE6$<&=6 ?Fk A/ ? =/ </,+@BI6 ? 1gDBG45"8 ?8%6#9<78>4/H<7/ 0 / ? 4BQ145:L8>PIBIABG4'D%" ^ 8l<7BE6$< BI15178>BQAF8imC< on#p5; n -;oq m 3= p < p I B . Jr" )BI1 ? 6>4BI6 ? 6 ? 6 ?!s B ? H6@<7MK8L47BE:$/ ? / 11 8@1O8L44< 8/ 47/'A 8>4347/ ? 4BI6 ? H<6$M ? 9M /'<6%91O8&456@< 1," 6>47/4598L4 49/=<7//6tBQ134@<7BI6@<A&BQ134<7BER!4BI6 ? 1+@BI:B ? +6/&47BEMK8@P+/ 147BEMK8L476@< 1 8%6J6$<7AB ? + 46fH</ 0 9/ ? 4BQ14$J<7BG47/=<7BI8 I 8@1 MOB ? BEMK8tBE4'D!J R!&451?F8<7/'134<7B J4BI6 ? 9198>PIP DCA6./ 1 ? 6>4 PI/'8$A 46f ? B7JBE4'Dlg(9/ ? 1?u<7BI6@< 1D/#t&BQ14 /6t!J/64g(/ ? B ? :L8><7BQ8 ? =/147<^ 49</8@</ </ 0 BE<7/'A
B ? 49BI1'8@1/ 45/ <B +/.4 s * 8@8><MO/'8$1<7/ BI145/ 8!9<6%<BQ8L47/6$B =//" /'/ 6@R/=<4 Ivw%w!x !B398/H"
vzy| J$H6$<2MO6$</ /6tFPI8 ? 8L47BE6 ? 1," )/fH6@<7MK8>PA&/ k9? BG47BE6 ? 6$B~ <7BE6$<7128 ? A </'PI8>4/'A </6H/=<7/ ? =/'1C'8 ? R/H6% ? A B ? 8@171 y 8$11/=<7MK8 ? I'xE// J 8>43478 I'xE// J 8 ? A8:6.1 8>P Ix,%/ JH6$< <7/6+/9PI8@<8 ? A ? 6 ?Fs </,+/PQ8><'8@1/'1," /5</'MOB ? A45BI1 A&/ k9? BG47BE6 ? /'</ 8o 47/=<E"
-!H7K #F"& n#q θn(α) n~q!n =< qn6p < p α− 6 ; ^q n <' θ ; n5 < @q!n n < n6p;q < Xn = (X1, . . . , Xn) n P (θ ≤ θn(α)|Xn) = α
(= p < p ; -q < n; q < p/n6p mC< on#p; n -;oq m 9= p < p-f D< p θ
Pθ (θ ≤ θn(α)) = P (θ ≤ θn(α)|Xn) + O(n−i/2),
?n#pnCq!n n q !; /n q!n pn 6 n ^q q = p < ; q ?n#p5n θ b : n5; θn(α) p; A <
A&/ 8>PIP>D 66$1B ? +πJ 8@1 8uB~ gDBE4049/-B>+%/'13458;:;8@BEPQ8>R9PE/ 6@< A&/=< BQ15A&/'1BE< 8>R9PE/%" 4=8 ? R/@+$/ ? /'<78@PEP D
A&/ k9? / A8@1O45/016@P &4BI6 ? 6/5898><4BQ8>P AB /=<7/ ? 4BQ8>P / 0 98>4BI6 ? "¢¡./ ? d = dim Θ = 1(¡ /=P7 y /'/=< 1
Ix,/%£ J&<76L:@/ A"4598L44/[¤@/6<7/6D&1-<7BI6@< BQ18 1/,=6 ? A s 6$<7A&/'<KMK8L459B ? +¥<BI6@<8 ? A¦BPQ8><54@/ Ix,/% J&+@BI:@/ 1 8
−
147<78@B>+%.4H6@<5g8>< A A&/6H/ ? =/F6/ 45BQ1l96@B7J//" ? MCPE4B 98>< 8>MO/=4/=< 1G9<6$RPE/'MK145/ 1 /=<5+@/'< s 1 /=< ? 8>< A&6 I'xE// x,%%v J pn n#p5n m np p 8><7/@g/=PIP=4 ? 6og ? 46 g6@<54 R/J4347/=<E"1/,JBQ8>PIP D+4/ ; m <;op </6H/=<7/ ? =/-<BI6@< 1 8><7/9198>PIP>DFMO8>45B ? +2g(/ ? B ? 4/'</ 14 8 ? A ? BQ178 ? =/@ 8>< 8>MO/J47/=< 1 8@</O6$<3456/+$6 ? 8@P I B ? 49/ 13/ ? 1/ 6/45/ BI1/'<B ? H6@<7MO8>4BI6 ? J#"
*/=<7/g/=6 ? 1BIA/=<BEMPIB =BG47/=P DO498>4πJ (θ)
=8 ? R/ A&/ k9? / A 1? 4598L4πJ(θ|yn)
BQ1~<76//=<6@<498>445/=<7/8><7/ / ? 6/!+%F8;:L8>BIPI8@RPE/ A8L4 8
yn
" ? A/=/'A 4/'</2/#t&BI134 M 6&A&/'PI1eH6@<:g(B7 4598L451^πJ A6./ 1 ? 6>4 /6t&BI134AH6$<
8 ? Dn
"<B6@M /=4B ? + M 6&A&/'PI18><7/ 6/ 4/ ? B ? 49BI1('8@1/ I 1 /'<+$/=< y ? xE/%£ Jr")/ ? Fg/ 1!+%+@/ 1446Cg6@<54gDBG45/'8% 1RMO6&A&/=P 8@171MOB ? + 4598L4<7BI6@< 18@</ B ? A&/6/ ? A&/ ? 4,"
41/=/'MO1</ 8@16 ? 8@RPI/498>4 πJ &8$18 ? 6 ? B ? H6@<7MK8L47BE:$/<7BE6$<.16%PQAKR /$J6 ? 1BIA&/'</ AO8@1B ? 8/+@<7/=/'M / ? 4<gDBG45yn
" 1 / 13BQA&/'1 πJ (θ|yn)MOBIM B7=1 4/KPIB 4@/=PIB 66A L(yn; θ)
" 1 /'< ? 8@<7A6 Ix,% J B ? H6@<7MK8>PIP>D[<6% 6.13/ A 4762913/4BQ1A&/ ? 1BG4'D 8$1D8 R / ? MK8><54G<7BE6$<~H6@<49/ 1459AFD 6/ 1R/E 4BI:@/ 8$11/'171M / ? 471,"
? 8 MO6@<7/H6@<7MO8@P :BE/,g( 6$B ? 4 g/-<76/6$1/O46z=6 ? 1BIA/=< πJ (θ|yn)8$1 8 A8>478 s A&/, / ? A&/ ? 4<7BI6@< g(B7
k t&/'1 8 PIBEMOBE4 46C45/ 8%6J/,&478@RPI/ B ? H6$<MK8L47BE6 ? D.BI/=PQA&/ A0RD 8 ? /#tF/=<4 I B ? A&/6/ ? A&/ ? 4H<76@M 4/ A98L478 Jr"9<BIA/'8OBQ1D4613/'/
πJ (θ|yn)8@1D4/&<BI6@< A&/ ? 1BE4'D6 ? θ 6/ 83m q q o n !n6prq?n#p n m q i; pnn-n ^q) q yn
")/ ? g/5A&/ k9? / 8 ? p -;oq nCp5n pn6q R /=4'g/'/ ? 49BI1DBQA&/ 8>PK<7BE6$<
πJ (.|yn)8 ? A45/ 8@1713/ 11/'A<7BE6$<
πRD
KL(πJ(.|yn) || π
).
I " x J
? A&/'/'A KL(π1||π2)13478>4/ 1 45/ <7/6+$</=4AF/ 46 45/$6@B7J/6/
π2g(/ ? 4/ 4<5/ A&BQ134<7BER!4BI6 ? BQ1 π1
" 6>4BI6 ? 16 </,+@<7/J4)gDBG45<7/'1 /E 446OB ? H6@<7MK8L47BE6 ?Fs 45/=6$</=4B74766@PQ18$14/ Hs A&BI:@/'<+$/ ? =/'8 ? R /:H6/ ? A B ? B6L:@/'<8 ? A)6@MK8$1 I'xE/?x JD8 ? A g/=/=4B ? + 8L434 8 8 ? A[8:6.1 I9vw%w #%Jr"
πBI1 1^ 498>4 I " x JBI1 PI8@<+$/
π(θ)gDBEPIPR /
J6 ? 13BQA&/'</ A 8@14766& 8@<)H<6$M 45/ A8L4 8 B ? H6$<MK8>4BI6 ? 6 ? θ "
6og BE4 BQ1 ? /EJ/ 1178><5D 476f66$1/ 8=6 ? 13478 ? 4 C 1^ 498>4 B>KL
(πJ (.|yn) || π
)> C
πBI1 A&/EJPQ8><7/'ACB ?
J6 ? %9B7 4$gDBG45045/KA8L4 8!" )/2< 8L47BE6 ? 8>PI/&H6@<66$1B ? +CBQ158@1H6$PEPI6og16j
πBQ151?! 6.13/ A 476 R/OB ? H6@<7MO8>4BI:@/%"
KL(πJ(.|yn) || π) > KL(πJ(.|yn) || πJ)
BG4'8 ?? 6>4 MO/'8 ? 498>4 π BI1PE/ 11 B ? H6$<MK8>4BI:@/ 4 8 ? πJ g(B7BI1K4/ PI/'171KB ? H6@<7MK8L47BE:$/ <BI6@<E" )9/=<7/#H6@<7/CBG4M / 8 ? 1 498>4 π BI1- 8><-H<76@M
πJ (.|yn)(g(9B BIM@9PEBI/'1 498>4
4/0B ? H6@<7MO8>4BI6 ? 1<76L:.BQA&/ ARDπ8 ? A yn
8><7/ ? 6>4 B ? 8/6=6@< A8 ? J//" B6 ? 1/ 0 / ? 47P>D DBG4CPI/'8@A91 476 66$1/C = KL
(πJ(.|yn) || πJ
) "b¡0/56$R&478@B ?lk9? 8>PIP>DK45/ ? 6$<MK8>PIB! =/'AfJ<7BG47/=<7BE6 ?
DACJ (π|yn) =KL
(πJ (.|yn) || π
)
KL (πJ (.|yn) || πJ )
I " v J
8 ? A 4/&H6@PIPE6ogDB ? + <5PI//jπBQ1 18@BIA¥m #" m q &gDBE4
yn
B DACJ (π|yn) > 1
")/O1! /'<715J<7B &4J<7/J%9/E 471
4/'8><7/#PH96@B7J/ 6πJ "$ ? 4/$H6$PEPI6ogDB ? + Fg/ 6$MOBG4 16@MO/J47BEMO/ 14BQ1 ? 6@478L47BE6 ? H6$<13BIMPIB =BG4'D%"
BPI/'8@<P D / k9? BG47BE6 ? " v BI1 ? 6@45</ 147<B7 47/'A 476 45/ Ks A&BI:@/=<5+@/ ? J/28 ? A08 ? 6@4/'<$96@B7J/B ? 4/&%PIB s BIPI:@/6DJPQ8@1716B ? H6@<7MO8>4BI6 ?Fs 49/=6@<7/J47B MO/'8$19</ 1 I %PIB+8 ? A BIPE:$/6D x,/% J~=8 ? R / MK8$A&//"~*6og/=:@/'<&8$118@BIA R/#H6$</ 8 :@/'<D2BEM6@<478 ? 4bH/ 8L49</D6/45/ Hs A&BI:@/'<+$/ ? =/BQ1 BE471 B ? :L8><7BI8 ? J/4765</,98>< 8>MO/=4<7BI178L47BE6 ? "¤%9147B k =8L47BE6 ? 1 H6$<91B ? + BE4~<7/#H/'</ ? 4BQ8>PIP>D-=8 ? R/eH6% ? AB ? * 8@<347B>+.8 ? Ix,/% J8 ? A B ? 8 ? 6L:B(' 8 ? A¤$6/ ? 16 ? I9vw%w/£ J I H<76@M R6>45J6$M@?&478>4BI6 ? 8@P8 ? A 8 ? 8>P D$47B :BE/,g( 6$B ? 4 15J#") 6>47B =/f6og/=:$/=<54598L4O6/9< A&/ k9? BE4BI6 ? BI12<7/'134<7B J4/'A 46 k9? BE4/ sA&BIM / ? 1BE6 ? 8>PM 6&A&/'PI1," ¡./ ? 4/ MO6A/=P~98@1 8 ? B ?Fk9? BG47/ ? M2R /'<26/98>< 8>MO/J47/=< 1 45/ Hs ABE:$/=<5+@/ ? J/OBI1 ? R 6% ? A&/ AC8 ? Al=8 ?? 6@4R / 13/ AK"
X _ d3 H_ & 6@PIPI6ogDB ? + :L8 ? 1 y 6.16 ? 6L: I9v/w/w #LR^Jog/A&/'1BI</46/E4 8: 6@4/ ? 47BI8@P!J6 ? %9B J4bgDBG45245/DA8L4 8:g(/ ?
4/G9<BI6@<5BQ1BI/=< 8><B7=8>PIP DF1/,=B k /'AK"% 4@/,DF<7/ 0 BI</'M / ? 4 I 8 ? A 8 ? ? J6 ? H6@<478@RPI/ <7/'134<7B7 4BI6 ? JH6@<4/@91/64/'BE< s :;8@P>9/&<76FJ/ AF<7/ BI1 45/2/6t&BI134/ ? J/2613478L47BI134B7=1g(B708@</8 ? JBIPIPI8@<DfH6@<:98><471 64/98>< 8>MO/J47/=<E" 6@< 45/ %eBhJ<7BG47/=<7BE6 ? >45/H6$PEPI6ogDB ? +<76/6$1BG47BE6 ? MK8/4@/'1/'8$1D4/ 13/,98>< 8L47//,4&16 49/BI/=< 8><B7=8@PPE/':@/'PI18 ? AKPIB ? 4&1 45/=M 476 49//E426/ 4/g(6@PI/e9<BI6@<E")91$4/ 8+@<7/=/'MO/ ? 464/PIPK<BI6@<BI1 ? 6@4821-=BE/ ? 4)=6 ? A&BE4BI6 ? 46 6@R&4 8>B ? 4/ 8+$</'/=MO/ ? 46 8 ? DBI/=< 8><B7=8>P^<7BE6$<,")/6/!6$1BG47/ BQ1)JPI/'8@<P D47<9//" )/:<76.6/ BI16$R.:BI6/918 ? A ? 6>4<7/66@<4/'AG/=<7//" ¡ / 198@PEP 13/'/ B ? /, 47BE6 ? " 4/5B ? 4/=<7/'134D6 4BQ1)H/ 8L49</:H6@<(9<BI6@<(=8@PEBIR< 8L47BE6 ? "
?,7A xEv "! "$#&%'#$ ()(+*-,/.0$#&213!45.67",&8h r nfq!;q π ; πJ m ; nlbp qqn π(θ) = π(θ1|θ2)π(θ2)
; πJ (θ) = πJ(θ1|θ2)πJ (θ2):9 n qn π1(θ) = π(θ1|θ2)πJ (θ2)
; π2(θ) = πJ (θ1|θ2)π(θ2)
<; !n 9e (π|yn) =
9e (π1|yn) +9e (π2|yn) − 1.
I " £ J
0 !!6$1/K46 8;:@/-/,4$/'Am
<7BI6@< 1π1, . . . , πm
" /6t4<6% 6.13BE4BI6 ? B ? A&B7=8>4/'15498>428 ? Du+$/=6@MO/=4<7B g/'B>+%$47/'AfJ6@M2RB ? 8L47BE6 ? 6 45/$<7BE6$<71
πiA&6/'1 ? 6@4 ? /=/'A 46OR/:/E4@/ AKB 8>PIPA<BI6@< 18><7/ B ? 8/+@<7/=/=MO/ ? 4
gDBG45 4/ A98L478?"=6$171BERP D 136$MO/-<BI6@< 1=8 ? R /G=6 ? %9B7 47B ? + g(BEPI/K45/J6$MRB ? 8>4BI6 ? 14 8D1 B ? 8+$</'/ sM / ? 4," ? 8J6$MRB ? 8>4BI6 ? BQ14'DB7=8@PEP D291/'A 476 6$R&478@B ? 8G+@PI6@R 8>P9<BI6@<:g(/ ? 13/':@/'<78@P B ? A&/6/ ? A/ ? 4/#tF /'<34 18><7/8;:L8>BIPI8@RPI//" /'/ § £ " £ " #F" x H6$< MO6$<//#tFPQ8 ? 8>4BI6 ? 1," )/~<76.6/BI1+$BE:$/ ? B ? %e!/ ? A&B>t § " " x "
?,7A x,£ "?>@!7BACEDF&#G#!7"AIHJ)KL DF7"KM7!NAGOP(QDFQ.R(ST.&7",&8e n6q π1, . . . , πm n m p p Θ ; n6q α1, . . . , αm n m <n q r m q ?;oq 0 < αi < 1 ∑m
i=1 αi = 1?9 n qn π(θ) ∝∏m
i=1 π(θ)αi
i
@; !n 9e J(π|yn) ≤
m∑
i=1
αi DACJ (πi|yn).
Aq q m n ?; E p %eBJ6/9PIAR /5/6tF /E 4/ A 476 4/ ? A 476 x g(9/ ? n 478/4@/ 1PQ8><5+@/ :L8>P /'1,"%( 4598>PIP>D 8@1* 8@<347B>+.8 ? 18D&1 Ix,/%£ J#jVU q!np pf/n r q[ n q$;FWen m qq!nl; q q m ( q9p 6 q θ q ?nlqn6p O(1) p O(n−1/2) X "ZY ? A/=<JPQ8@1713B7=8@P </,+/PQ8><7BE4'D[J6 ? A&BE4BI6 ? 126 ? 9<BI6@< A&/ ? 1BE4BI/'1 8 ? APEB 4@/'PEB 66&A 4/C8@1DM@46>47B ? 6@<7MK8>PIBG4'D 6
πJ (.|yn)196/PQA MK84$/ ? MO/=< 8L46$<8 ? A A&/ ? 6$MOB ? 8>46$<6
I " v J 478/4@/e=PE6.13/ :L8>P /'1," )BI18$1DM&46@4B7R/698;:BI6@<6 %eB gDBEPIP R / BEPIP 9147<78>4/ A-gDBG458 ? /#t8@M@9PE/ B ?4/$H6$PEPI6ogDB ? +?"
X9X [d)\^] a Fd _ \ V] \3] ! D _ !d \ $ ?? /6t4 1/, 47BE6 ? (g/F1 8>PIPJ6 ? 1BIA&/'<K4 8L4 πJ BI1<6% /'< I B9" //" ∫
Θ π(θ) dθ < ∞ Jr" 3DB '8>PIP>D DBG4BI1 4/'8@1/&g(/ ? Θ
BQ15A&BQ1=</=4/K6@< R6/ ? A&/'AK" 1 491 8>PIP>DπJ BQ15BIM<6% /'<," $ ? 49BI1'8@1/ 45/ A&/ ? 6@MOB ? 8L476@< 6
%eB BQ1 A&/ k9? / A2! 468 ? ? 4 ? 6og ? 8$AA&BE4BI:@/C=6 ? 13478 ? 4 8 ? A2g/ ? /=/ A 46 8$A844/&J<7BG47/=<7BE6 ? "eB8>PIPEB ? + 8>< s+%MO/ ? 4716$<B +@B ? 8>PIP D913/ AB ? 1 8D@/'1BQ8 ? MO6&A&/'P13/'PE/E 4BI6 ? >49BI1 8$A8&4 8L47BE6 ? gDBIPEP9R /45/19R/,J4 6 /, 47BE6 ? " #F"
− ! " #$% %&')(*' +-,/.103254687:9<;-=>568.<4 V .17687
S [d _ ] ] \(ca _¡0/R /,+@B ? gDBG45 4'g6 ABI15J<7/J47/A&BQ147<BIR!&47BE6 ? 1(H6$<g(B7
πJ BI1e<76//=<E" 6>47/5498>4 B ? 4/ 13/4'g6G'8@1/'1 %eBBI1 /#tFPIB =BG4E" )/ Hs A&BI:@/'<+$/ ? =/'1-=6 ? 1BIA/=<7/'Ah/=<7/F8@</ 8;:L8>BIPQ8>RPI/ B ? %!/ ? AB t § " " v " )9/ k < 14 /6t s8>MPI/ 8@R 6%&4O8 1 /=< ? 6/PIPIBA&BQ147<BIR!&47BE6 ? BQ1 4 84$/ ? H<76@M :L8 ? 1 y 6$16 ? 6L: I9v/w/w #>8%J 476 =6@M98@</ R6>458!9<6.8// 16" BI< 147P>Dg/-913/ 4/18@M / ? 9M /'<B7=8@P :L8>P /'128 ? A 6$R&478@B ? 45/178>MO/=6 ? =P> 13BI6 ? 1," /,=6 ? AP>D g/ 16og 498>454/
P− :L8@P>/-9<6FJ/ AF<7/G=8 ? 8%6=/6&4<7BI6@< 1$g(B 8@</O<7/ /E 4/ A RD %eB")BE< A&P D B ? 8 ?BEPIP 9147<78>4BI:@/-!9<6$1/g/<6% 6.13/ 8F16$<34134 AFD06/4/ [email protected]&476>47B OR/698;:BI6@< 6/ %eB" % 1g/G+// 11/'A 8 ? D[=6 ? % B J4A&BI178? / 8>< 1 8@1DM@46>47B '8>PIP>D%" )/13/EJ6 ? A0/6t8>MPI/ A&/'8@PI1gDBE4 8 MCPE4B ? 6@MOBQ8>PABI134<7BER?&4BI6 ? "$ 4 <7/J%9/E 4 145/ ? /'/'A 46f96.6.13/&=8><7/#9PEP D
πJ " $ ? 8@AA&BE4BI6 ? g/H6F# 1 6 ? 4/ /=:$6@P &4BI6 ? 6/ %eB¢g(/ ? 49/<7BE6$< MO/'8 ? BI1g/'PEP J/ ? 4/=<7/'A 6 ? 45/2A8>478 R!&4 45/@<BI6@< :L8><7BI8 ? J/5BQ1 MO6&A&B k /'AH" % k <7134 :BQ13BI6 ? 6 %eB 8$18e'8>PIBER< 8L47BE6 ? 466$P8/!/'8>< 1 8@1 8 A&B> /=<7/ ? =/3gDBE424/
P− :L8>P /<<76FJ/ AF<7/<<76/6$1/'A5RD :L8 ? 1 y 6$16 ? 6L:A"
() +*,.- xEv "@? n#pA oCBDB FE& /n BG /=4 yn
R /C8 ? HG HG G n− 18@M@9PE/H<6$M 8 1 /=< ? 6/PIPIBA&BQ134<7BER!4BI6 ? Br(θ)"¡0/C8@171MO/ 6 ? θ
8z<BI6@< 1 /J4 8A&BQ147<BIR!&47BE6 ? Be(α, β)6 ? [0, 1]
" 6>4/δn =
∑ni=1 yi
" )9/&¤$/# </,D1<BI6@<πJ BQ1D49/ Be(1/2, 1/2)
A/ ? 1BG4'D%")/ ? πJ (θ|yn) = Be(δn + 1/2, n− δn + 1/2)
" $ ? 45/ 178>MO/:J6 ? A&BG47BE6 ? 18$1B ? :;8 ? 1 y 6$16 ? 6L: I9v/w/w #>8 btK" v J 3g/266$1/
(α, β) = (5, 20)I 16
[θ] = 0.2J 49/ ? g/l+@/ ? /=< 8L47/8 178>MPI/6/ 1B '/
n = 10H<76@M
4/ Br(θ0 = 0.9)A&BI134<7BIR!&4BI6 ? "¡ / 6@R&4 8>B ? %eB J(α, β|yn) = 1.102
8 ? A g/GJ6 ? JP 9A&/O476 82<BI6@< s A8L4 8J6 ? %9B7 4,"b 6&A&B>DB ? +
θ0 = 0.25 g/:98;:@/ %eB J(α, β|yn) = 0.1026
g(B MO/'8 ? 1 I 8@1/#tF/,J4/'A^J8 ? 8+$</'/ sMO/ ? 4,"3B366.13B ? +
(α, β) = (1, 1)IπBI145914/$ ? B H6$<M <BI6@<rJPE/ 8@A1476245/58+$</'/=MO/ ? 4~H6@<8 ? D 1 /=< ? 6%PEPIB
A8L4 8@1/J4E")91)g/ 6@R&4 8>B ? 4/ 178>MO/J6 ? JP 91BE6 ? 18@1 :L8 ? 1 y 6.196 ? 6L: Ivw/w #>8 Jr"
¡0/@66$1/ ? 6ogθ0 = 0.7
8 ? A 4'g6 1B '/'1 n = 108 ? A n = 5
" B t&B ? +K45/<7BE6$< 13478 ? A8>< A A&/=:BQ8L47BE6 ? 476w " v g/8><7/PE66%4.B ? +H6@< 45/D< 8 ? +$/6/:L8@P>/)H6$<4/e<BI6@<MO/'8 ? µ0 = α/(α+β)
1?4598L4πBQ1 ? 6@4<J6 ? %9B J4 s
B ? +?" ¡0/ 6@R478>B ? µ0 ∈ [0.2, 0.96]H6@<
n = 108 ? A µ0 ∈ [0.23, 0.94]
H6$<n = 5
91B ? + %eB" 1 DO4/P− :L8>P /<76=/'AF9</ @45/ xEwI s /wI /=<J/ ? 47BEPI/ A&6$MK8>B ? BI1 µ0 ∈ [0.054, 0.928]
H6$<n = 10
8 ? A µ0 ∈ [0.056, 0.925]H6$<
n = 5" )/ # I s # I A&6@MK8@B ? </'MK8>B ? 1)JPI6$1/ 476
[0.048, 0.95]" )91 %eBBQ1=PE/ 8><7P>DKMO6@<7/ </ 147<B7 47BE:$//"
? 45/ 8@1DM@46>47B 1BIA&/ A/ ? 6@4B ? +0 < θ0 < 1
4/ 47</C:L8>P /C6 49/l 8>< 8>MO/J47/=< =g/C6$R&478@B ? 4/H6@PIPE6ogDB ? +[<6% 6.13BE4BI6 ? g(B7 J6 ?Fk <7M 498>4 %eB → 1
g(/ ? n4784$/'12PI8@<+$/ :L8>P / 1 I 4/ 9<66 BI1@+@BI:@/ ?
B ? %? / ? A&B>t § " " x Jr" )/@=6 ? :@/'<+$/ ? =/ BQ1 :@/=<5D 13PI6og R/,'891/26/PI6/+$8@<BE49M B74/=<7MK1 B ? 4/K8$1DM&46@4B7A&/=:$/=PI6/MO/ ? 46/ %eB J (α, β|yn)
" $ ? +8@RPE/ " x MO/'8 ? :;8@P>9/'1D6/ %eB J 8@</+$BE:$/ ? H6@< 13/':@/=< 8>P :L8@P>/ 1D6n8 ? A (α, β)
?g(/ ? θ0 = 0.9"
^P,7K x " ∀(α, β) > 0 pC;$A? 0 < q < 1 <nKJ?; n
(logn)q L [ 9e J (α, β|Yn) − 1]
n→∞−−−−→ 0.
/ ? 6>47B ? +Ψ45/ A&B +$8>MOMK8@ ? J4BI6 ? 8 ? A Aθ0(α, β) = Ψ(1/2)− Ψ(β) + θ0 Ψ(β) − Ψ(α) 94/ PIBIM BE4:L8>P / x BQ18/!<76$8/9/'AKRD?! /'<D6@<DPI6og/'<:;8@P>9/'1D8@1~H6$PEPI6og1,j
x "B>α = β
4/5PIBEMOBE4BI1D8>434 8>B ? /'A RD PI6og/'< I pn G ?! /'<J:L8@P>/ 1B α > 1/2
I pn G α < 1/2J
n (α, β) ! " #" ! $ $
10 %'& () %'&+*, ('& -. ('& -(102 / & %( / & (0 ('& ,1% ('& . /103 *& 0, *& )( ('& .2 ('& 21%105 *& . / *& ,1% ('& 21) ('& ) /1010 *& %1- *& / 0 ('& )) ('& 0(1020 *&+*) *&+*, ('& 01- ('& 0.1050 *& (, *& (% ('& 0) ('& 00
8>R9PE/ " x j=B6 ? :@/=<5+@/ ? J/ 16/ %eB J 476 x g(9/ ? θ0 = 0.9I 8;:@/'<78/+@/ AK6 ? x,w%w 178>MPI/'1Jr"
v "B α 6= β
49/ PEBIMOBG4 BQ1K8L4478>B ? / A RD PI6og/=<O:L8>P /'1 I pn G !!/=<rJ :L8>P /'1OB>Aθ0(α, β) < 0
I pn GAθ0(α, β) > 0
J £ "B
α 6= β8 ? A Aθ0(α, β) = 0
lim
n→∞Pθ0
DACJ (α, β|yn) − 1 ≤ 0
= 1/2
"B6 ? A&BE4BI6 ? Aθ0(α, β) ≥ 0
BQ15/ 0 9BE:L8>PI/ ? 45476 18D 4598L4πBQ1 8/,#M&PQ8L4B ? +CBE471 MK8$11 6 ? 45/KR6/ ? A91 649/
A&6$MK8>B ? [0, 1] A&/'1BE4/545/<7BE6$<DM / 8 ? '8 ? 136$M /=4BIMO/'1DR/ w " # I g(9/ ? α = β < 1/2
Jr" 6@MO/ /#t8@M@9PE/ 1D8><7/A&BQ1PI8D$/'AC6 ? B>+%</ " x " % MO6@<7/9198>P =8$13/BI1g(/ ? α I </ 1"
βJ478/4@/ 1B +/ :L8@P>/ 1eg(9BEPI/
βI <7/'1H"
αJ
<7/=MK8>B ? 113MK8>PIP")9/ ? 45/9<BI6@<M / 8 ? BQ1:@/'<DJPI6$1/D46 w 6$< x ")91 g(/ ? n BI1 k t&/'A .45/'1/ ? 8/,J/6478>R9PE/<7BI6@< 18@</ 8>P g8D11/=/ ? =6 ? %9B7 47B ? + RD 4/ %eBJ<7BG47/=<7BE6 ? "
Jeffreysa,b=0.1,0.1
a,b=0.1,10a,b=10,0.1
a,b=10,10
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
B>+%<7/ " x j Be(α, β)A&/ ? 1BG47BE/ 16"
( ) *9,$- xE£ " CB q A $E& ; BE& %n B G /J4
yn = (9, 5, 1)R/8 ? HG HG G 178>MPI/ H<6$MN49/ MCPE4B ? 6@MOBQ8>P.A&BI134<7BIR!&4BI6 ? M(p)
gDBE4p0 = (0.7, 0.25, 0.05)45/54<5/ :L8@P>/6
p" !!6$1/98;:B ? +6$1/ ? π(p)
8$1 8& BE<7B7PE/=4 Dir(α1, α2, α3)A&/ ? 13BE4'D/"(¡ /C=6 ? 1BIA/=<
16@MO/C4<7B PE/=4:L8>P / 1α = (α1, α2, α3)
1?N8$1 [p] = p0
egDBG45NB ? J<7/'8@1B ? + :L8><7BI8 ? J/%" 6 4 8L4<7BI6@<
− ! " #$% %&8 ? A0A8L4 8 B ? H6$<MK8L47BE6 ? 1 13/'/=M B ? 4BE4BI:@/'P>Dz=PE6.13/%"¡ /2459166$1/
α = ρ(7, 2.5, 0.5)" )9/&<7BI6@< :L8><7BI8 ? J/
6pA&/,=</ 8@1/'1@g(/ ? ρ
B ? J<7/'8$13/ 16" ¡ /uJ6$M@ 8><7/6 ? B +/<7/ " v 45/ /=:$6@P &4BI6 ? 1O6 %eB J g(/ ? πJ BI16$1/ ? 8@1 45/KMC9PG47BE:L8><7BQ8L4/ ? B H6$<M <BI6@< 8 ? AF4/KMO6$1348?<6%<7BI8>4/¤@/6<7/6D&1$<7BE6$< Dir(1/2, 1/2, 1/2)I ? 6@4/&6og/':@/'<D498>4 B ? 49BI1 MCPE4BQA&BIMO/ ? 1BE6 ? 8>P =8$13/ 4/@¤@/6<7/6D&19<BI6@<98@1 ? 6>4 4/ R / 14: 6.147/=<7BE6$<=6L:@/=< s8+$/=<76//=<4BI/'1 8>MO6 ? + 8;:L8>BIPI8@RPE/ ? 6 ? B ? H6$<MK8L47BE:$/=<7BE6$<71@1/=/8 ? + y 1 /'<+$/=< x,/ H6@<8>PE4/'< ? 8L4BI:@/36@B7J/ 15J#"
)/=<7/28@</1MK8>PIP AB /=<7/ ? =/'1 R /=4'g/'/ ? 4'g6f96@B7J/'1 6πJ j¤@/6<7/6D&1 8 ? Au ? B H6$<M <7BI6@< MO/ 8 ? 1 8><7/549/
18@MO/8 ? A 45/e¤$/# </,D1 :;8@<BQ8 ? J/BI1 0 BG47/ 1PEB +/.4P DPQ8><5+@/'<498 ? 4/e ? B>H6@<7M :L8><7BQ8 ? =/ I PI/'171498 ? 10−2 !!/=<B ? L2 ? 6@<7M-J#"e*6og/=:$/=< ?91B ? + 45/C ? B H6$<M A&BQ147<BIR!&47BE6 ? =8 ? PE/ 8@AC476 <7/#91/ :L8+%/C9<BI6@< 1 I g(9/ ? ρ → 0
Jg(B7 8><7/ PE/ 11B ? H6@<7MO8>4BI:@/K4598 ? πJ ? PIB>4$/ 45/f¤$/# </,D&1<7BE6$<," )9BI12BI128u4$/6D 8><5+/MO/ ? 45476u 8;:@6$<549/¤@/6<7/6D&1b9<BI6@<E" /,J6 ? A&P D ? 6>47/D4 8L4A98L47828 ? A9<BI6@<8/+@<7/=/D6 ? P>Dg(/ ? 4/<7BE6$<BQ1 ? 6@4466 B ? H6$<MK8L47BE:$//" $ ?4BQ1PQ8L4347/=<'8@1/ A8>478@98;:$/ 8 ? /,+@PIB J4BIRPE/ B ? %?/ ? J/ 6 ? 45/$ 6.147/=<7BE6$<ABI134<7BER?&4BI6 ? " )9BI1R/698;:BI6@<MK84$/'18147<6 ? +A&B> /=<7/ ? =/gDBE4 4/
P− :;8@P>9/e<76FJ/'A!</ R D :L8 ? 1 y 6$16 ? 6L: Ivw/w #>8 J /g(9B l=8 ? ? /=:$/=<A&/J47/, 48 ? DuJ6 ? %9B7 4 /':@/ ? B
π4/ ? A1 46 8 BI< 8/@<7BE6$<," )BQ1:H/'8L45<7/ 6/ %eB BQ158 ? 8@P>D&1/'A B ? 4/@H6$PEPI6ogDB ? + 8 ? A
478/4@/'18K1 /EJBQ8>P B ? 4/=<7/'134)H6@<45/<7BE6$<)'8>PIBER9<78>4BI6 ? "
rho
DA
C 1
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Jeffreys Uniform
B>+%</ " v j :@6$P>4BI6 ? 6 %eB J gDBE4 <7/'1 /E 45476ρ
I 45/G ? J/=<478@B ? 4'D[ D/=<598>< 8>MO/=4/=< 6/45/ BE<7B PI/J4<7BE6$<6/8 M&PG47B ? 6@MOBI8@P A&BQ147<BIR!&47BE6 ? J3H6@<45/6@B7J/ 1
πJ =M&PG47BE:L8@<BQ8L47/ ? B H6$<M 8 ? Al¤@/6<7/6D&1,"
S ] \ Fd \ _ \ cc ]`\(c a$ ? 6%<D6/9B ? BI6 ? !/,4B ? +CH6$<A8L4 8 s <7BE6$<~=6 ? %9B7 4DBQ16 98><4B7#9PI8@<B ? 4/'</ 14B ? B ? AF9134<7BQ8>P 1 8D@/ 13BQ8 ? 134 A&BE/ 16"$ ? 8K<7/=PIBI8@RBEPIBE4'Dl=6 ? 4/6t.4 45/C98@<78@MO/J4/'<1 8/J/
Θ'8 ? 6/ 4/ ? R/R6/ ? A&/'AK" Y 1 8>P+8@<+%MO/ ? 471 8><7/!D1B7=8>P
J6 ? 147<78@B ? 4 16"$ ? 4BQ1$=8$13/ πJ BQ1$<76//=<E" /#t45/#t8@M@9PE/ BEPIP 9147<78>4/ 1 45/OB ? 45BG47BE:$/2R/6 8;:.BI6@< 6 %eB 8 ? A
BG4 113478>R9BEPIBG4'DgDBG45 <7/'1/, 4476O4/5R6/ ? A16Θ
g(/ ? π BQ1D147<6 ? +@P DKB ? H6@<7MO8>4BI:@/%"
() +*,.- x " m ;oq A A p E ; B E /n BG /=4
yn
R /8 ? ? J/ ? 136$</ A2178>MPI/)H<76@M 8 N (θ, 1)A&BI134<7BIR!&4BI6 ? gDBE4
θ ∈ D = [Tl, Tm]" / ? 6>47/ θ0 4/D<7/'8@P:L8>P / 6/
θ"b¡ /$PQ8/J/ 6 ? θ 8 N (µ0, σ
20)
<7BE6$<,"πJ BI1)/'</ 49/: ? B H6$<MS<BI6@<
πJ (θ) = (Tm − Tl)−1
D(θ)"
)/ ? πJ (θ|yn)BI1 4/ N (yn, 1/n)
A&/ ? 1BG4'D </ 147<B7 47/'A 6 ? D " /'/ %? / ? A&B>t" xE£ #H6$<8 1BEMPI/gD<7BG47B ? +56/
e A > 9 1 2e
%eB J "=B396.6.13B ? +θ0 = 0
n = 5
Tm = −Tl = 15
8 ? A σ0 = 1g/ k <7134P DGJ6 ? 13BQA&/'<45/ /=:@6$P>4BI6 ? 6/ %eB
gDBE4 <7/'1/, 4 46µ0
B ? B>+%<7/ " £ ^H6@< 1/=:@/'<78@P :L8>P /'1 6σ0
" % 1DMOM /=4<7B /=:$6@P &4BI6 ? 8@<6% ? A θ0 8/!/'8>< 1? 8>4< 8>P g(9BEPI/K49/ PE/ ? +>45 64/8+@<7/=/'MO/ ? 4 A&6@MK8>B ? A&/EJ<7/'8@1/'1:g(/ ? σ0
B ? =</ 8@1/'1," ? B>+%<7/ " g/MO6&A&B>D
(Tl, Tm)1<B ? 4B ? + 8 ? A / ? PQ8><5+@B ? +21DM MO/=4<7B '8>PIP>D 49/:9<BI6@<A&6@MK8@B ? " )/58/+@<7/=/=MO/ ? 4A&6@MK8@B ? H6@<
µ08/! / 8>< 14/5MO6@<7/513/ ? 1BG47BE:$/ 476 82MO6&A&B k =8>4BI6 ? 6 4/5R6/ ? A1D8@1 σ0
478/4@/ 139B>+% :L8@P>/ 16"
')( R = &W = 7= V .1> . 2 W . W =9 9<; P RUT
$ ?C? MO/=<76/91e=8@1/'1πJ BQ1BIM<6% /'<,"~*/'</g/H6F#91/'1 /EJBQ8>PIP>D6 ? 49/+@/ ? /=< 8>PH'8@1/g(/=<7/
πJ BI1 A&/ k ? /'A! 468 ? ? 4 ? 6og ? M&PG47B>9PEB7=8>4BI:@/e=6 ? 13478 ? 4 cj " )BI1~J6 ? 13478 ? 4< 8@18 ? 8@A9A&BG47BE:$//# /,J4B ? 49/ A&/ ? 6@MOB ? 8L476@<6/ %eB
KL(πJ(.|yn) || πJ
)= − log cj +KL
(πJ(.|yn) || π′J)
g(/'</π′J(θ) ∝ πJ (θ)
" )91 %eB J BI1 ? 6@4~g/=PIP A&/ k9? /'AK"$ ? 45BQ11/,J4BI6 ? g/ k < 134P DKPEB ? 4 4BQ1A&B -6PE4'DO468 ? BQ171/Cg(B BQ1D4'DB7=8@PEP D / ? J6% ? 4/'</ AB ? 1 8D@/ 13BQ8 ? MO6&A&/=P 13/'PE/E 4BI6 ? " 6@MO/<76/6$178>PQ18><7/ A&6 ? / 46G+$/J48@<6% ? A4BQ1BQ171/ 9B ? JP 9A&B ? +K45/ A^q9p A r m J?n p ( q m#" )91eg/C<6% 6.13/ 8 ? B ? 47<B ? 1B :$/=< 13BI6 ? %eB AIJ 6 %eB J " )9BI1 8@A8/&478>4BI6 ? BQ1:J6@M98@</ AC46 4/ BQA&/'8@P'8@1/ 49<6%!+/ /6t8>MPI/'1 8 ? A 8f=8$13/ 1459AFD B ? :$6@PI:B ? +<7/'8@P¡ /=BIR!PIP A8>478C8 ? AFB ? AF9134<7BI8@P /#tF /'<3456%B ? BE6 ? 1 BI1$J6 ? 1BIA&/'</ AK" )9/ BQ171/ 1 <78@BI1/'AFR D 4/K8@A98&4 8L4BI6 ?8@</:B +/PIB>+%.4/'A 466^" % 6.11BER9PE/ BIM@9<6L:$/=MO/ ? 4)gDBEPIP R/5/=M!98$13B! =/ A B ? 1R 13/E 4BI6 ? " "
3 $ !\(c ]Kd FZ d3 d3 _ d ] Z d _ !d]P0 "!$#&% '( ' )* #+
BI<71343g/ 16og 6og 4/ 8@AABG47BE:$/J6 ? 13478 ? 4 cj '8 ? R/ <7/6+.8>< A&/'AO8$1 45/ M&PG47B>9PEB7=8>4BI:@/=6 ? 13478 ? 46 8 1 8D$/'1 8% 46$<," 6@<7MO8@PEP Dg/ MC913498;:@/
expKL
(πJ(.|yn) || π
)−KL
(πJ (.|yn) || πJ
)≤ 1.
I " J / ? 6@4/O45/ MK8><5+@B ? 8>PMO/'8$19</ 1 mJ(x) =
∫Θf(x|θ)πJ (θ) dθ
8 ? A m(x) =∫Θf(x|θ)π(θ) dθ
" )/KPI/# 4MO/=M2R /'<D6B ? / 0 98L47BE6 ? I " J478/4@/ 145/$H6@<7M
mJ (yn) exp
∫
Θ
πJ (θ|yn) logπJ (θ|yn)
π(θ)f(yn|θ)dθ
I " #/J
g(B7KA&6/'1 ? 6>4 ? /'/'A24/J6@M!478L47BE6 ? 64/6$134/'<BI6@<π(.|yn)
A/ ? 1BG4'D%")9/ ? g/=8 ? gD<7BG47/4/=<BE4/'<BI6 ? ? A&/=<45/ 8>PE4/'< ? 8>4BI:@/H6@<7M
DACJ2 (π|yn) = BJ,π(yn) exp
(KL
πJ (.|yn) || π(.|yn)
) I " Jg(/'</
BJ,π(yn)BQ149/ 1 8D$/'13 8/J46$<
BJ,π(yn) =m(yn)
mJ(yn).
)914/5<5PI/J6 ? 13BQ14 168/6=/6&47B ? +π8@1B ? 8+$</'/=MO/ ? 4)gDBG4545/ A8>478
yn
B DACJ
2 (π|yn) ≤ 1"
− ! " #$% %&
mu0
DA
C
−4 −2 0 2 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
sd = 1sd = 0.5sd = 0.25
B>+%</ " £ ji /'8 ? /=:$6@P &4BI6 ? 6/DACJ gDBG45 <7/'1 /E 446¥ 8>< 8>MO/J47/=< 1
µ08 ? A σ0
68¥<7BE6$<+$8/91713BQ8 ?A&BQ147<BIR!&47BE6 ? 6 ? 45/5MO/'8 ? 6/ 8OR6/ ? A/'Al+$8/911BQ8 ? MO6&A&/=P I 8;:@/'<78/+@/ AK6 ? £/w 178>MPI/'1Jr"
mu
0 v
alu
es
D=[ −5 , 5 ] D=[ −10 , 10 ] D=[ −20 , 20 ] D=[ −50 , 50 ]
sd = 1 sd = 0.5 sd = 0.25
−6
−4
−2
02
4
B>+%</ " j %+@<7/=/'MO/ ? 4A&6@MK8>B ? 1)H6$<4/C<BI6@<DMO/ 8 ? µ0gDBG45C</ 1/,J4D46O49/<BI6@<13478 ? 8>< A A&/=:BQ8L4BI6 ? σ08 ? A 16@MO/ M 6&A&B k '8L4BI6 ? 16 4/5R6/ ? A&/'ACA&6$MO8@B ? D = [Tl, Tm]
6 45/+$8/91713BQ8 ? 18@M@9PEB ? +OMO6A/=P"
e A > 9 1 2e
)/'</OBQ15136$M /OB ? 4/=<7/'134 46CB ? 47/=<5<7/J4 cj 8$1 4/KM&PE4B PEB7=8>4BI:@/ ? 4 ? 6og ? J6 ? 13478 ? 4 68 1 8D$/'1 8/J46@<E"$ ? A/=/'A BG45BQ1584'D 9B '8>PBI171/OB ? 6@R/E 47BE:$/2MO6&A&/=P 1/=PI/,J4BI6 ? 8>R6/&4$g(B7 8l!+$/2PIBG47/=< 8L45</ BQ15A&/'A&B7=8>4/ AK" /'/ 8@171 y 8o 47/=<5D I'xE/ #/J+8 ? A % ? A&<BI/6 n6q ; BG I9v/w/w?x JHH6@< 8 </':BE/,g" eMO/=<76/91 8/!<76$8/9/'1 8><7/3<76/6$1/'A4766@R478>B ? A&/6 8PE4 1 8D$/'1b 8% 476@< 1.8 ? AO45/ M 6.143 8>MO6/ 1 BQ1 45/ A/ k9? BE4BI6 ? 6 A^q9p A m? ;/n ; m q p I $ 1 1 /=<5+@/'< y /=<7<B7B x,%/ Jr"
)BQ15MO/J456&A&6@PI6/+%D BI15A/=/69P>D0PEB ? 4@/ A046C49/ ? 6>47BE6 ? 6/ E@ A DE ; B q9p; A A ; E B n I Jr" )/KMK8@B ?BQA&/'8CBQ1 476 913/- 8><47156/45/ A8>478 8@1 47<78@B ? B ? + 178>MPI/'1," /=MOB ? AF4598L4
πJ (θ)BI1 8 ? BEM<76//=<9<BI6@<E" /=4
x(l) = (x1, . . . , xq) ∈ χq A&/ ? 6>47/ 8298@<34 6/45/ A8>478yn
1? 498>4πJ(θ|x(l)) BQ182<76//=< A 8 ? A ? 6
1R91/J46/x(l)
PE/ 8@A1476 8<76//=<6$134/'<BI6@<E" % BQ1D459149/ MOB ? BIMK8>P 0 8 ? 4BE4'D 6/ A98L478@H6$<g(B7 8>PIP98@<78@MO/J4/'<71 B ? 45/OM 6&A&/'P8><7/ BIA&/ ? 4B k 8>RPI//" )91$J6 ? A&BE4BI6 ? B ? +
πJ 476 8l +$BE:$/'1 8l<6% /'<$ 6.147/=<7BE6$<'8>PIPE/ A8 qn6p p p p "
)/ ? 45/$ 1 BQ1A&/ k ? /'AC8@145/ 1 8D@/ 1= 8% 46$<J6 ? A&BE4BI6 ? 8@P 46K8- x(l)
R D
BIBFJ,π = BJ,π(yn)Bπ,J (x(l))
g(/'</
Bπ,J(x(l)) =mJ (x(l))
m(x(l)).
1 Dz=6 ? 134<5? 47BE6 ? BIBFJ,π
<7/=MO6L:$/'1 4/ 8@<RBE4< 8><7B ? /'171 B ? 4/-6$B =/O6cj
" +6C</ AF?=/245/ A&/, / ? A&/ ? J/O6 ? F913B ? +O45/ ;op q J E n#q mD8 ? A n ?n m q n5;op q J E n#q m $ 1
BAIJ,π(yn) = BJ,π(yn)
1
L
L∑
l=1
Bπ,J(x(l)),
BEAIJ,π (yn) = BJ,π(yn)
1
L
L∑
l=1
θn
[Bπ,J(x(l))]
MK84$/'1D1/ ? 1/ I gDBE4θn
4/C N8 ? A θ[.]
4/ /6tF /E 478>4BI6 ? ? A&/'< M(θ)Jr" 4/'<8;:@/'<78/+@/'1MK8D R/91/'A
I 8@1 49/ %n $E n6q9p m$ 1 J#" )/ zn ;$A $ 1 I 1 /'<+$/=< y /'<<7B 9B x,/% J 8? / 8><7/'A0</EJ/ ? 47P>DF8$1 6 ? /K6/45/MO6$13413478@RPI/ :$/=< 13BI6 ? 6/ 4/ $ 1 " B ? 8>PIP D&8 ? B7J/:<76//=<4'D26/ 49/ 8@<BE4MO/=4B7 $ 1 BQ1BE4718@1DM&46@4B7 / 0 9BE: s8@PE/ ? J/lgDBG45 8 <76//=< 1 8D$/'1C 8/J46@<K8@<BQ13B ? +[H<6$M ? /64< 8>P A?qp A r m p p I 8/! / 8><7B ? +0RD 8 g8@< 8/!<76tBIMK8L47BE6 ? Jr" )BQ1 BQ1 8134<76 ? + '9134B k =8>4BI6 ? 6 45/&9/6<7BI134B7" 6@< MO6@<7/C<7/,=BI1BI6 ? 1 13/'/ 8@171 I9vw%w!x Jr" 6$MO/ /6t8>MPI/'1)gDBE4CPEB>H/J47BEMO/5MO6&A&/=PQ18@</:+@BI:@/ ? B ? BEM y ? I9v/w/w%w Jr"
-9* " r A r m J Ap p m ;$A[p5n5 m nCq J!n m E q ;oq AA; B m q qDJ?n ? q J!n6f;opn Ap ?n6p G q nq J!;q A o p m ; n q J!n n A?qp A r m p p ;pn$; B <; p ?n#p G 9 nA q n (πIJ , πI) q J!n A^q9p A m Ap p ; E q q m ; BDB n 6F ; B nA?q$q (πJ , π) pfqDJ?n m $E q;q A Bπ,J(yn) G^ A m n πJ ;$A π ;pn2/n An AhqDJ?n
; E n !;p; E n#q n6p ?; m n q J!n A?qp A r m n 6 ;oq A # A ? n#p n6p n#prp m J "!##$5Gpn m nCq πI (θ) = π(θ),
πIJ (θ) = πJ (θ) L θ [Bπ,J(x(l))]
AA/n6pl; m AA q A n m J!;$A% n; 6DB q9 pfq J!n2 ; x(l) nn p n ; E B n 9 E m J !##$5 G p $E; J!n pn E ! A ? n6p n#p& (n6prp m J '!##%$ πIJ ( ; B <; p !n6pG r A m n π ( Ap ?n#p G
9 − ! " #$% %& " ) #+)' #
)/DB ? 47<B ? 1B /6<7BQ147B PI/'8 ? 1 6 ? 45/91/D6 1MK8>PIP 0 98 ? 4BE4'D6/ 4< 8>B ? B ? + A8L4 8 g(B7 8><7/6.13/ ? 8@M 6 ? + 49/6@R91/=<7:@/ A A8>478 I g(B7 /#tFPQ8>B ? 1 4/ 4/'<M A?qp A r mJ 476<7/'A&/ k9? /K8 14 8L4BQ134B7&g(9B BI1:H6$<MK8@P>D :L8>PIBQAFR!4</'MK8>B ? 1.B ? 8/J4A&B G#PE446O8@1713/ 11," $ ? 4/H6@PIPI6ogDB ? + %g/ 8><7/ MO6@<7/ B ? 4/=<7/'134/ AKB ? 8 ? B ? 47<B ? 1B 8@A8/&478>4BI6 ?6 %eB J 4598 ? B ? 8 ? 8@A8/&478>4BI6 ? 6 %eB J
2
" BE< 147P>D R/,=8/91/ 6/<@J6 ? 47/#t4BQ1 ? 6@4MO6&A&/=P13/'PE/E 47BE6 ? I 49/18@MPEB ? + MO6&A&/'P BQ1 45/ 18@M / B ? 4/ MO8@<+$B ? 8>P A/ ? 1BG47BE/ 16/
Bπ,JJ &1/,J6 ? A&P D2R/,'891/eg/ A/'1BE<7/D46&<7/'1/=<7:@/
4/5B ? 4BE4BI:@/51/ ? 1/ 6 4/=<BE4/'<BI6 ? 8 ? A BE471( 13/6PAH/'8>4<7/'1PIBI134/ A B ? § " £ " v "
c_! !d3 [
) ) #+
6$PEPI6ogDB ? + 4/ B ? 4<7B ? 13B7(/,<7BI134B7 .6/<BIA&/ 8 BQ1 4765478/4@/ 8 x(l)
8@M 6 ? + 49/ A8>478yn
46 6$R&478@B ? 89<6% /'<A&B>91/(<7BE6$<
πJ (θ|x(l)) 8 ? A24691/4/D<7/=MK8@B ? B ? +5A8>478yn(l) = yn/x(l)
46/E454/(9<6t&BIM BE4'D R/J4'g/=/ ?4/9<BI6@<B ? H6@<7MK8L4BI6 ? 8 ? A 45/ A8L4 8 B ? H6$<MK8>4BI6 ? " )/ A&BI:@/'<+$/ ? =/
KLπJ (.|yn(l)) || πJ (.|x(l))
BI1 ? 6og04/ MK8tBIMK8>P$:;8@P>9/bH6@< 45/ABE:$/=<5+@/ ? J/R /=4'g/'/ ? πJ (.|yn(l))8 ? A 45/8$11/'1713/ Ae9<BI6@<
π=6 ? ABG47BE6 ? /'A 46
4/DPI/'8@< ? B ? + B ? H6@<7MK8L4BI6 ? x(l) " )BQ1bJ6 ? A&BE4BI6 ? B ? + MC 14 R/A&6 ? /B ? 4/ ? MO/'<78>46@< 8$1B ? 49/A/ ? 6$M B ? 8>46@<6 %eB 476 8>4347/ ? 98L47/ 49/0BIM98/J464/ A&BI1349<RB ? + B ? H6$<MK8>4BI6 ? DBI/=PQA&/'ARD 49/ " 1 D8¥=<6.11 s:L8>PIBIA8>4BI6 ? 8@<+%MO/ ? 4 460</ AF?=/ 49/A&/, / ? A&/ ? J/ 6 ? x(l) <g/ A/ k9? / 6/9<28@A98&4 8L4BI6 ? 8$18 ? ;op q J E n#q m8;:@/'<78/+@/6L:$/=<
L8;:;8@BEPQ8>R9PE/ " / k ? /549/ A^q9p A r m ';p qDJ E n6q m 9e m p q n#p A RD
%eB AIJ(π|yn) =1
L
L∑
i=1
KLπJ(.|yn(l)) || π(.|x(l))
KL πJ (.|yn(l)) || πJ(.|x(l)) .I " J
)/ ? LB 4/13B! =/645/) <7/=MK8>B ? 1 13MK8@PEPgDBE4<7/'1/, 4 46n %eB AIJ <7/=MK8@B ? 1 1MO8@PEP g(/ ? π(θ)
BQ1 JPI6$1/46πJ (θ|yn)
" ¡./ ? π BQ1:$/=<5DCH/,gNB ? H6@<7MO8>4BI:@/ 8 ? AKBQ18><7RBG47<78@<D@6$1/ ? JPI6$1/46πJ %eB AIJ (π|yn) ' 1
g(B7 BQ11BEMOBIPI8@<47645/ R /,98;:BE6$<6/ %eB J " $ 4BI1/'8$1D 476 16og 4 8L44/H/'8>4<7/'1A&/'15J<7BIR / A B ? § " £ " v 8><7/<7/'1/=<7:@/'AH"bBPI/'8@<P D.49/=<7/5BI1 ? 6K<7/'134<7B7 4BI6 ? 4691/ 4BQ18$A8&4 8L47BE6 ? "
$ ? 8 ? B ? AF9134<7BQ8>Pe=6 ? 4/6t.4Gg(9/=<7/n
'8 ? R/F1MO8@PEP 4/ ? 9MR/=<L6/ 8;:;8@BEPQ8>R9PE/ '8 ? R/
<76@RPI/=MK8L47B 1B ? =/O4/KB ? 4<7B ? 13B7O8@A8/&478>4BI6 ? '8 ? </'MK8>B ? 134<76 ? +$P>D A/6/ ? A&/ ? 4H<6$M 16@MO/- =6 ? 478@B ?!sB ? + 6/&47PEBI/=< 1,"e*/ ? J/2BG4 BI1 A&/'1BI<78@RPE/ 476 B ? J<7/'8$13/ 4/ ? 9MR/=< 6/< " 1 /=<5+@/=< y /'<<7B 9B I9v/w/w v J)<76/6$1/13/':@/'<78@P BIA/'8@1 B ? 49BI1 13/ ? 1/ I / 1/,=BI8@PEP D2g(/ ? 1-=BE/ ? 4 13478L47BI134B7=1:=8 ? 1MOMO8@<B! =/ 4/OA8L4 8 B ? H6@<7MK8L47BE6 ? J8 ? AFB ? 47<6&AF?=/45/ ? 6>47BE6 ? 6/ n 6 nA?q ; B I J B ? JP 9A&B ? +l=/ ? 16@<7/'AFA8L4 8!" )91 49/ ? M2R /'< 6A8L4 8f=6@M2RB ? 8L4BI6 ? 1'8 ? B ? =</ 8@1/2A&< 8>MK8L47B '8>PIP>D%" 1 &4 4/K1B '/26/ 4/ '8 ? R/,=6@MO/ :@/=<5DCPI8@<+$/28 ? A4/OB ? H6@<7MK8L47BE6 ? DBI/=PQA&/'A0RD
yn(l)BQ1 PIB>4$/=P DC46CR / 8@<:H<6$M 4/ I BIA/'8>PJ B ? H6@<7MO8>4BI6 ? +@BI:@/ ? RD 4/-g(96@PI/
18@MPE/%" Y 1B ? + 45/1 /EJBQ8>P=/ ? 16@<7/'A%G¤$/# </,D1C<7BE6$<πJ
c
A&/ k9? / A RD) / 8 ? 4BQ1 n6q; BG Ivw/w?x J B ? 134/ 8@A645/C14 8 ? A8@<7A πJ BQ1O8u< 8/J4B7=8@P8@PG47/=< ? 8L47BE:$/GH6@<K1BEMPI/ M 6&A&/'PI1O8$145/ /#tF 6 ? / ? 47BI8@PABI134<7BER?&4BI6 ? j 8J/ ? 136$</ AK47BEMO/=8 ? R/,=6@MO/ 8 "
*6og/=:@/'<$49/ 1B '/q6/+8@ '8 ? <7/=MK8>B ? B +/fgDBG45 <7/'1 /E 446
n/'1 /EJBQ8>PIP>Dg(/ ? dim Θ
B ? =</ 8@1/'1,"$ 45BI1$JPI/'8@<P D 8PEBIMOBG4 8L4BI6 ? 64/-J<7BE4/=<7BI6 ? I " J 8 ? AFB ? JBQA&/ ? 478@PEP D BG45< 8>BQ13/ 145/<6$RPI/=M 64/OMOB ? BEMK8>P? MR/=<6A8L4 8H6$<g(B724/ ? 6>47BE6 ? 6/^J6 ? %9B7 4 gDBG45 8<7BE6$<+BQ1 MO8/4B ? + 1/ ? 1//" % ? 8>4< 8>PBQA&/'8:J6%PQA R/476J6 ? 13BQA&/'<D4/ MOB ? BEMK8>P ? MR/=< 6A8>478 1?C4598L4 4/& 6.147/=<7BE6$< A&BQ147<BIR!&47BE6 ? DBE/'PIA91 8KMOB ? BEMK8@P<7/,=BI1BI6 ?
e A > 9 1 2e
6 ? 8z ? 47BE6 ? 6 B ? 4/=<7/'134O46 /'134BIMK8L4/%" $ ? § " 8[6$1713BIRPI/ 8;:$/ ? /fH6@<OBEM<76L:B ? + 49/C8$A8478L47BE6 ? BQ1A&/ 1=<BIR/'AK"
P0(0( ! ' %&! '
( ) *9,$- x #F"@? n6pA oCBDB FE /n BGB6 ? 13BQA&/'<8+.8>B ? 49/:'8@1/ 6 1 /'< ? 6/PIPEB A8>478yn ∼ Br(θ)
g(9/=<7/eg/ 8$11MO/ 6 ? θ 8&<7BE6$< 1 /J4 8A&BQ134<7BER!4BI6 ?Be(α, β)
6 ? [0, 1]I r'" bt8@M@9PE/ xv Jr"¡ /66$1/
θ0 = 0.78@1 45/D4<5/:L8>P /D6
θ" )/¤$/# </,D1 9<BI6@< BQ149/
<76//=< Be(1/2, 1/2)A&BQ147<BIR!&47BE6 ? I MO6$</'6L:@/=< L45/'1/<7BE6$<718><7/eJ6 ? '!+$8>4/Jr") 1 g/=8 ? =6@M98@</ %eB J
8 ? A %eB AIJ 4<76/!+%45/e6$B =/6 4/e<BI6@< MO/ 8 ? µ0 = α/(α+β) k t&B ? + 49/<7BE6$< 14 8 ? A8@<7AOA&/':BI8>4BI6 ?
8>4 w " v " B +/<7/'1 " # 8 ? A " 16og"4/D/':@6@P &47BE6 ? 6/R6>45 %eB=<BE4/'<BQ8 >8;:$/=< 8+$/'A56 ? xEw 13BIMCPQ8L47/'AO178>MPI/'16/13B! =/ 1
n = 20I B +/9</ " #/J 8 ? A n = 10
I B +/9</ " J#" )/,D BEPIP> 147<78>4/2498>4 %eB AIJ 8 ? A %eB J '8 ?<76&AF?=/$=PE6.13/ 8+$</'/=MO/ ? 4A&6$MO8@B ? 1/':@/ ? B> n BQ1 ? 6>4B +/"
% 1/6tF /E 4/ A /=<BE4/'<BQ8 8@</MOB ? BEMOB! =/'A&g(/ ? 45/)<BI6@<MO/'8 ? µ0BQ1 JPI6$1/476 45/4<5/:L8>P /
θ0" 6@<7/=6L:@/'<
A&6$MK8>B ? 1D6 8/+@<7/=/=MO/ ? 4R/,=6@MO/1MK8>PIPE/'<eg(/ ? 4/&<BI6@< R /EJ6@MO/ 1MO6@<7/5B ? H6@<7MO8>4BI:@/gDBE4F</ 1/,J446K45/A8>478?")9BI1~H/ 8L49</98$1R /'/ ?C? 6@4B7J/'AB ? =t&8@MPE/ xE£ "
mu0
DA
C
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
DAC J DAC AIJ
B +/<7/ " #Fj /'8 ? /=:@6$P>4BI6 ? 6 %eB J 8 ? A %eB AIJ B ? ? 47BE6 ? 694/MO/'8 ? µ0
6/&4/~<7BE6$<1 /J4 8 A&BQ147<BIR!&47BE6 ? H6@<28 1 /'< ? 6/PIPEBM 6&A&/'P I gDBG45n = 20 1 /=< ? 6%PEPIB A8>478%J#"
mu0
DA
C
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
DAC J DAC AIJ
B +/9</ " j / 8 ? /=:$6@P &4BI6 ? 6/ %eB J 8 ? A %eB AIJ B ? ? J4BI6 ? 6/49/MO/'8 ? µ0
6&49/3<7BI6@<1 /=478 A&BQ134<7BER!4BI6 ? H6$<8 1 /=< ? 6/9PEPIB MO6&A&/=P I gDBE4n = 10 1 /'< ? 6/PIPEB A98L478 Jr"
− ! " #$% %&() +*,.- x, " L AnA?q ; BE /n B "¡0/D178>MPI/'A 8 x,w s 1B '/'A HG HG G A98L478 s 1/J4 yn
H<6$M 8 ? /#tF6 ? / ? 4BQ8>P&A&BQ147<BIR!&47BE6 ? gDBE4@98@<78@M /=4/'<λ0 = 150−1 j
yn = (142.76, 142.99, 470.3, 419.09, 185.20, 84.41, 8.13, 27.15, 573.17, 17.12)" )/@ y = λ−1
n = 207BI1 8 M B ? BIMK8>P 1GJBI/ ? 4 13478>4BQ147B 8 ? A 6L:@/'</ 147BEMK8>4/'14/ <7/'8>P :L8>P //"* /=<7/πJ (λ) ∝ λ−1 BI1 49/ 13478 ? A8>< A
¤@/6<7/6D&1~<7BE6$<8 ? A 8 BI1 8 1B ? +@PI/ :;8@P>9//" ¡0/96.6.13/ 45/J6 ? '!+$8>4/:<7BE6$<
λ ∼ G (a, axe)
g(/=<7/a/=M2R 6&A&BI/'14/ 13B! =/ 6 8 :BI<34598>P18@MPE/ 6 MO/'8 ? xe
" /=/$B3 8&47/=< £ H6@<M 6$</ /#tFPQ8 ? 8>4BI6 ? 1," 6>47/498>4
πBQ1(/=<H/,J4P D B ? 8/+@<7/=/=MO/ ? 4(gDBE4
yn
B a = n = 10
8 ? A xe = y = λ−1n = 207
" ¡ /+@BI:@/ B ? 8>RH" " 4/O:L8>P / 1 6 %eB AIJ H6@<51/=:$/=< 8>P<BI6@<515J/ ? 8><7BEB" )9/O8$A8478L47BE6 ? <7/ /E 471 6 ? P>DuH/6g B ? H6$<MK8L47BE:$/&9<BI6@< 1H6@<(g(B7
xeBQ1D:@/'<D- 8@<)H<6$M 45/ A8L4 8 I 1MOMO8@<B! =/ A R D 4/5MOB ? BIMO8@P 1GJBI/ ? 4 14 8L47BI134B7 y J#" :$6@P &4BI6 ? 1
6 %eB AIJ 8><7/KA&BQ1PQ8D@/ A 6 ? B>+%</ 1 " 8 ? A " x,w "f¡0/ ? 6>47B =/K4/18@M / 8% 428$15B ? 4/ BQA&/ 8>P<'8@1/K6bt8>MPI/ x,£ j /':@/ ? 8@+@66&A/#tF /'<34:L8@P>/'8 ? R / k9? 8>PIP>D</ /, 47/'AB
πBI14766KB ? H6$<MK8>4BI:@//" $ ? 6>45/=<(g6@< A1
%eB AIJ 8@1 %eB J ABI15=8@<7A1 ? RBI8$13/ Ai<7BE6$<71gDBG45 1MO8@PEP:L8><7BI8 ? J/ 18 ? A :@/'<D0RBI8$13/ Ai<7BE6$<71gDBG45"PQ8><5+@/:L8><7BI8 ? J/ 16"
6@<7/=6L:$/=< <g/u<76L:BIA&/ B ? 8>RH" " 4/F8/+@<7/=/'M / ? 4 A&6@MK8>B ? 1GH6@<xe
J6$M@?&4/ ARD %eB AIJ 8 ? A 49/P− :;8@P>9/(<6F=/'AF<7/(<76/6$1/'A2RD :L8 ? 1 y 6$16 ? 6L: I9v/w/w #>8%J#" )BQ1=9<6FJ/ AF<7/DBI1A&/'15J<7BIR / A2B ? 8+$/ ? /'<78@PJ/ ? 136$</ A =8@1/ B ? bt8>MPI/ x%x " )/ 8+@<7/=/'MO/ ? 4 A&6@MK8@B ? BQ1 45/
(5%, 95%) /'<5=/ ? 4BIPI/ B ? 47/=<7:;8@P 6/54/
MK8><5+@B ? 8>P ABI134<7BER?&4BI6 ? 6/ 4/&9198>P MOB ? BIMK8>P1-=BE/ ? 4 13478L47BI134B7"¡0/26@R91/=<7:@/ 1BEMOBIPI8@<P D46=t&8@MPE/ xEv 498>4BG4 8%6=/6&4 1)<BI6@< 1147<6 ? +@P DK<7/ /,J4/ A RD 45/ %eB.<6F=/'AF<7//"
e A > 9 1 2e "
a xe
, - & 0 / ('& %% (&+*. (& -% / &+*.* *& // ('& ,2 (& , / (& ,* ('& 21%('& , *& ( / ('& 21( (& .. (& .1% ('& .)('& / , ('& 0) ('& )( (& 21) (& 2- ('& 2 /
8>R " j 8>P /'1O6/ %eB AIJ H6@< 1/=:$/=< 8>P~<7BE6$<15J/ ? 8><7BEB"
a AIJ P− & (5% − 95%)
/ ,1, ,1(1( %2! . / ,, 01(" %)1% - * ,*(2 * ( / %,1, ,1% -)(* ( *1*)" % / ( .,! -.(*, * %2 / 01( 22! -1-(
8>R " j %e+@<7/=/'M / ? 4A&6$MK8>B ? 1~H6@<xe
"
expert central value xe
DA
C A
IJ
0 100 200 300 400 500
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a=0.5a=1
a=2a=5
a=10a=20
B +/<7/ " j :@6@P &47BE6 ? 1O6 %eB AIJ gDBE4 </ 1/,J4O46D /=<598@<78@M /=4/'<71(xe, a)
6 4/ J6 ? '!+.8L4/l<7BE6$<A&BQ134<7BER!4BI6 ? 6 4/5/6t6 ? / ? 4BQ8>P MO6&A&/=P"
expert central value xe
DA
C A
IJ
0 100 200 300 400 500
0.0
0.5
1.0
1.5
2.0
2.5
3.0
a=30a=40a=50
a=70a=80a=100
B +/<7/ " x,w j :@6@P &47BE6 ? 16 %eB AIJ gDBE4C<7/'1 /E 446(xe, a)
"
− ! " #$% %& X H_ba (_ ] !:a d!Z FZ d aaCc d _ d !d \ ¡0/J6 ? 13BQA&/=< 4/<7B +/.4 s J/ ? 136$</ A<7/'8@PPIB H/=4BIM / A8L4 8
yn
In = 18
JH<76@M +8@RPI/B" xCI %!/ ? A&B t § B" x J#")/,DJ6$<<7/'1 6 ? AC476f 8@BEP <7/ 4BIM / 156@<51346%!B ? + 4BIM / 1J6$PEPI/,J4/'A06 ? 16@MO/ 1BIM BIPQ8>< A&/':.B7J/ 1 ∑ R /'PE6 ? +@B ? + 46 49/13/EJ6 ? A8><5D g8L4/'<JBI<56BE4 6 ? ?=PE/ 8><PQ8 ? 4 16" B H/=4BIMO/'1 8@</+@BI:@/ ? B ? MO6 ? 491," 6@<!D&13B7=8@P </ 8@16 ? 1 8 ? A8/,J6@< A&B ? + 46 8KPI8@<+$/J6 ? 13/ ? 191946.13/2A8L4 8 8><7/8@171MO/'A 468><7BI1/$H<6$M 8¡0/'BER?PEPA&BQ147<BIR!&47BE6 ? W(η, β)gDBG45 A&/ ? 1BE4'D
f(x|η, β) =β
η
(x
η
)β−1
exp
−(x
η
)β
x≥0.
)/ BQ1(ηn, βn) = (140.8, 4.51)
gDBE4"/'134BIMO8>4/ A 14 8 ? A8@<7A A&/':BI8>4BI6 ? 1 σn = (7.3, 1.8)" )/GB +/
:L8>P / 6βn
BQ1H ? 9198@P@8 ? A B ? :BG4 1 I PIB>4$/ 49/ PI8@<+$//'134BIMO8>4/ A514 8 ? A8@<7A A&/':BI8>4BI6 ? J^H6$<+8 1 8D@/ 13BQ8 ? /'134BIMK8L4BI6 ? "
g6<7BE6$< 6/9B ? BI6 ? 1 6 ? 45/ PIB>H/J4BIMO/ 8@</8;:L8>BIPQ8>RPI/D6 ? A&/':B =/ ∑ +@BI:@/ ? RD2B ? A&/, / ? A&/ ? 4/#tF /'<34 1 E18 ? A
E2" )/,D 8@</ 1MOMK8><7B '/'ACB ? +8@RPE/B" v I %!/ ? A&B t § B" x Jr" )/ E1
6/B ? BI6 ? BQ1MC^ M 6$</5B ? H6$<MK8L47BE:$/498 ? E2
8 ? AFR 6@4 8><7/<7B>+%$4 s 1B> 4/ AugDBE4F<7/'1/, 4 4649/OA98L478?" 6@<7/=6L:$/=<D45/ /#tF /'<34 1 8><7/ ? 6@4 8$14$/'A 8>44/ 18@MO/ PE/':@/'P 6b<7/,=BI1BI6 ? " E1
BQ1 8 ? ^JPI/'8>< 6//=< 8L46$< 8 ? A 1/'8/41H6@< 8-98><4B7#9PI8@<eJ6$M@6 ? / ? 4eg(BIPE/ E2=8 ? R/1/=/ ? 8$1 8$=6@M 6 ? / ? 4 <76A!?J/'<g(6.13/6/9B ? BI6 ? 478/4@/'1 B ? 46 8/6=6/ ? 4 8 :L8@<BI/J4'D56/ <5 ?? B ? +$J6 ? A&BG47BE6 ? 1,"
6>47/ 498>41B ? J/ 4/:¡ /=BIR!PIP A&BI134<7BIR!&4BI6 ? A&6/'1 ? 6@48$A&MOBG4=6 ? '!+.8L47/<7BI6@< I 6$PI8 ? A xE// J $4/$6$134/ s<BI6@<b=6@M!&4 8L4BI6 ? BQ16 4/ ? 9/'8;:D 8 ? A ? /'/'A1 ? MO/'<B7B ? 4/,+@< 8L4BI6 ? I B ? +%!<5g8@PEPQ8 y ? x,%/ J 6$<178>MPIB ? +8>P +@6$<BE4MK1 8@1: 6 ? 4/GB8><7PI6 8><54@6L: B398>B ? 1 I B~[B)Jr" $ ? 6%< 8!PIB7=8L47BE6 ? 1g/C 13/ A BIM@6@<478 ? J/ 18@M sPIB ? + I r<B398H"#%Jr"
¡0/<76/6$1/ 46-J6 ? 1BIA&/'<4/$<7BI6@< 1η ∼ G(a, b),β ∼ G(c, d).
B ? J/4/OA&/=:B7J/ ∑ BQ1 1R9M BE4347/'A 46C8+@B ? + 8 ? 9198>PA&6@MK8@B ? 6/MK8>B ? :L8><7BI8>4BI6 ? 1H6@< 4/ :L8>P /'1 6βBI1
Dβ = [1, 5]I #'"=B3 8&47/=< £ Jr" B ? =/
ηBQ145/ %£ 45 6@< A&/=<~/=<J/ ? 47BEPI/ 6/ 4/:¡0/'BER?PEP ABI134<7BER?&4BI6 ? BE4BQ1M 6$</
4< 8/J478>R9PE/)H<76@M 49/ /#tF/=<46/9B ? BI6 ? 4598 ? β " ¡0/ 4< 8 ? 1PI8>4/8/!<76tBIMK8L47BE:$/=P D 49/ /'<5=/ ? 4BIPI/'1 6 ? X B ? 476549/ /'<5=/ ? 4BIPI/'1 6 ? η 91B ? + 45/(¡ /=BIR!PIP? A k tB ? +
β = 3")9BI1 4< 8 ? 1PI8>4/ AC4 ? 6ogDPI/'AF+$/D8 ? A249/J6@<7<7/'1 6 ? A&B ? +
:L8>P /'16/a8 ? A b I 8@1713/ 11/'A R D PE/ 8@134D1 0 98><7/ </,+@<7/'1713BI6 ? J 8><7/$+@BI:@/ ? B ? 8>R9PE/ " v "
X η a b
!#"$ %&('*),+.-0/21#'*34$ -0%5+.-0/ 6' !#"$ %&('*)(+0-./ 1#'*37$ -.%5+0-./ 6'8 E1 / (1(9 %(1(: / ,( // -9 %%. / )1( ..& % ('& / %8 E2 *(1(9 ,1(1(: / ,( ** / 9 ,1.( / )1( - & . ('& ('*,
; 8>R9PE/ " v j6 <7BI6@<A&6$MO8@B ? 16 ? X 8 ? A η 8 ? AD /'< 8>< 8>MO/J4/'<71:L8>P /'1~H6$<π1(η)
"
? Ix,% J5</EJ6@MOMO/ ? A&/'A 476[ 13/ 4/ </6H/=<7/ ? =/G<7BE6$<πJ g(/J45/=< 6 ? /6$<R6>4 98>< 8>MO/=4/=< 18@</ 6
B ? 47/=<7/'134 I / 1/,=BI8@PEP D-g(/ ? 1MO8@PEP 178>MPI/'18><7/91/'A?J#" $ ? 98@<347B 6PI8@<< g(/ ? R6>4f98>< 8>MO/=4/=< 18@</ 6 B ? 4/'</ 14<4BQ1)<7BE6$<BQ149/: ? B 0 /51/,=6 ? A s 6@< A&/'<MK8L4B ? +&<7BI6@<E"* /=<7/:g/:PI8%J/ ? 6BI/=< 8>< D R/J4'g/=/ ? 49/: 8>< 8>M s/J47/=< 1< ? 8>MO/=P D2g/2A&6 ? 6@4J6 ? 1BIA&/'<45/<7/#H/'</ ? J/9<BI6@<
πJ B ? </ 13/ ? J/56/8 ? BQ18 ? J/ ∗ 98@<78@M /=4/'<," B ? =/πJ BI1BIM<6% /'<<Fg/$98;:@/ 46-=6@M!&47/ %eB AIJ " % ? ? =/ ? 16@<7/'Af=;
x(l)BI1D8=6/!PI/ 6 :;8@P>9/'1
(xi, xj)
e A > 9 1 2e
1? 498>4xi 6= xj
8 ? A xi, xj > 1I ? x,% J#" 6$<345 ? 8>4/=P D< πJ (θ|xi, xj)
BI1 /#tFPIB =BG4<g(B7 13BIMPIB k / 1=6 ? 1BIA/=< 8>RP D 45/@=6@M!&4 8L47BE6 ? " /=/ %!/ ? AB t § " " £ " v H6@< A&/J4 8>BIPI1,":¡./ ? 4/=; =6 ? 478@B ? =/ ? 16@<7/'AA8>478 <4/ 1 /EJBQ8>PH¤$/# </,D&139<BI6@<
πJc
/=PIB7JBE4/'A B ? B398&47/=< £2I § £ " v " v " x J BQ1)91/'AK"
/ ? 6@4/
π(η, β) = π(η)πJ (β) ∝ 1
βπ(η),
˜π(η, β) = πJ(η)π(β) ∝ 1
ηπ(β).
<6$M I " £ J<g/$+@/=4 %eB AIJ(π|yn) =
%eB AIJ(a, b|yn) + %eB AIJ(c, d|yn) − 1.
I " Jg(/'</
%eB AIJ(a, b|yn) =1
L
L∑
i=1
KLπJ(.|yn(l)) || π(.|x(l))
KL πJ (.|yn(l)) || πJ (.|x(l)) ,
%eB AIJ (c, d|yn) =1
L
L∑
i=1
KLπJ(.|yn(l)) || ˜π(.|x(l))
KL πJ (.|yn(l)) || πJ (.|x(l)) .
;)91<Kg/=8 ? /E4π(η)
13/,98>< 8L47/=P DfH<76@M 8 ? Dl6$171BERPI/π(β)
"@¡ /96$1/'AL = 30
I 8@MO6 ? +n(n − 1)/26$1713BIRPI/- ? =/ ? 16@<7/'A 8 ? A =/ ? 16@<7/'A=; Jr" 6@< /6tF /'<34 E1
<g/ 6@R478>B ? / A %eB AIJ(a, b|yn) = 3.41" 6$<
/6t/=<4 E2<g/6$R&478@B ? /'A %eB AIJ (a, b|yn) = 1.76
" ;)91<Ag/ A&/J47/,J4 45/@=6 ? B J4 R/J4'g/=/ ? 4/ A8L4 8 8 ? A45/5/#tF/=<4716 ? 49/5PEB>H/J47BEMO/ 15=8>PI//"
6@4/498>4245/u8 8>MOMK8u<7BI6@< 6 ? η<bH6$< 4/ k < 14O/#tF/=<4 6%B ? BE6 ? < BI1 :$/=<5D /'8/4@/ A"8 ? A=8 ? R/fg/=PIP
8/!<76tBIMK8L47/'AugDBE4 8 ? 6$<MK8@PA&BQ147<BIR!&47BE6 ? I 13B ? =/a > 30
Jr" <6$M / 0 98>4BI6 ? I " J < BE4BQ15:BI1BIRPE/O4598L4 ? 66$B =/D6
π(β)</=:$/ ? 8 8>43<7BI6@<<@8>PIPE6og1 46545/=6@MPI/J47/<7BE6$<
π462R / ? 6>4~=6 ? B J4B ? +?"¡0/ 4B ? 4 4 8L4B ?
8 ? B ? AF9134<7BI8@P!J6 ? 47/#t4<$1? 8 13BE498>4BI6 ? MC9134 R / ? 6@4B7J/ AR/#H6$</4/ 1 8D$/'1BI8 ? B ? H/'</ ? J/ @4BQ1 A&BQ1=</,98 ? 6D<7/ /E 4 1D8 A/=/6CA&BQ1 8><7BG4'D R/J4'g/=/ ? 45/ 4'g6O16/9<5=/'16B ? H6@<7MO8>4BI6 ? I A8>478O8 ? A /6t/=<4J#"
;)/ 13/EJ6 ? A2/6tF /'<346%B ? BE6 ? BI1 ? 6@4B ? 4BQ1<'8@1//" ;)/ 15=8@PE/(98@<78@MO/J4/'< BQ18o /,J4/ AKRD 8 1BIM BIPQ8><<=6 ? B7 4<R!4BG4<7/=MK8>B ? 1)6$1713BIRPI/ 46-/,4 498>4D4/=6@MPI/J4/9<BI6@<BQ1 k9? 8@PEP D ? 6@4eJ6 ? B J4B ? +?j g/5MC9134(98;:@/
π(β)1?498>4 %eB AIJ(c, d|yn) ≤ 0.24.
<6$M 45/K8 ? 8@P>D&134 :BI/6g(6@B ? 4< 49/ /6tF /'<34 1 8@</ 6/&47BEMOBQ147B &gDBE4 </ 1/,J4 47645/OA8L4 8 I 4/'MO1/=PI:@/ 1 A&6 ? 6>44 ? 6og 49/ A8>478%J#" 6549/6D 1/=/=M 46C 8;:@6$<816 48+@B ? +56/ 4/ A&/=:B7J/ ∑ I 8213BIMPI/ <7/'8@16 ? BQ1 4598L44/,DOB ? 4/ s+$<78>4/ 136$M /4 ? 6ogDPI/'A!+@/58>R6/&4D45/ 4/E ? B7=8@P /=:$6@P &4BI6 ? B ? 45/=BI<6/9B ? BI6 ? 15J#" 6$<D4BQ1D<7/'8$136 ? <4/ 1 8D$/'1BI8 ?8 ? 8>P D&1416%PQA&66.13/4//#tF/,J478L47BE6 ? c/d 6
π(β)B ? [1, 2]
" 6@<B ? 13478 ? J/<;45/D8 ? 8@P>D&1341/=PI/,J471 [β] = 1.5
";)/ ? 49/MO6&A&/=PIPEB ? +6/&4/13/EJ6 ? A /#tF/=<4+6/9B ? BI6 ? BQ1 ? 6>4 J6 ? B J4B ? +(H6$<+:L8>P / 1
c1? 4 8L4 8@<
[β] ≥ 0.683<
16CH6$<c ≤ 0.65
"
;)BQ1/6t8>MPI/ BQ1<91/#9PB ? 4'g628$1/,J471," $ 4D196og1 4 8L4 %eB =8 ? R /:91/'AH6$<~'8>PIBER< 8L47BE6 ? </g(9/ ? 8 k <713498@<34649/(<BI6@<BQ1 8@1713/ 11/'A28 ? A49/13/EJ6 ? AC98@<34 BQ1 98><4BQ8>PIP D 8@1713/ 11/'AK"Y 1B ? + %eB 8@1 8:<7BI6@<b=8@PEBIR< 8L4BI6 ?476.6$P BI145/ 1R/,J46+4/ ? /6t4 13/E 4BI6 ? "
− ! " #$% %&')( =9 V .<4= . W =9 = WN PSRUT
$ ? </':BE6%91 /#t8>MPI/'1$g/-+@/ ? /=< 8>PIP DuJ6 ? 13BQA&/=<7/'AF4 8L4π
g8$1 / ? 4BI<7/=P DF8@1713/ 11/'AH" ;)/-<7BE6$<5:;8@<BQ8 ? J/@g8$1k t&/'A 8 ? A 49/<BI6@<)6@B ? 4'gDBI1/ /'134BIMO8>4BI6 ? θe
6/θ
I 8-=/ ? 4< 8>P :L8>P /58@145/<7BE6$<MO/ 8 ? J3g8@1(/E4@/'AgDBG45</ 1/,J4 46 4/ A8L4 8CPE6F=8>4BI6 ? " ;)91< g/ 6$R&478@B ? /'A 8 ? 8/+@<7/=/'M / ? 4A&6@MK8>B ? H6@<
θe"u*6og/=:@/'<<+/'1 /EJBQ8>PIP>D
g(/ ? 49/@9<BI6@< BQ1 1R/E 47BE:$/O8 ? A BQ1 /=PIB7JBE4/'AzH<76@M 8 ? /#tF/=<456/B ? BI6 ? < θe</ /,J471 8l /'<716 ? 8@P+:BI/6g(6@B ? 4
8 ? A BQ1b9198>PIP D/'8$1D46 8$11/'1716" 1 &4<J6$M MO6 ? 9<BI6@<b ? J/=<478@B ? 4'D2M / 8@1<7/'1<@8@1 45/:L8><7BI8 ? J/<@8><7/D6 47/ ? MO6@<7/A&B -6PE4D46 k t I B ? 98@<347B 6PQ8><)g(/ ? 45/5/#tF/=<46/B ? BI6 ? BI1D8@R 6%&4
X8 ? A ? 6>4 θ J#"
$ ? 8><4B7#PQ8><< 82<7BE6$< MO6&A&/'PEPIB ? + /'PEB7JBE4/ A[H<6$M 4/K/6tF /'<34 6%B ? BE6 ? <HgDBE46%&4CJ<7BG47B '8>Pbg6@<542H<6$M 49/8 ? 8@P>D&134<=8 ? R/ 134<76 ? +$P>D B ? H6$<MK8L47BE:$//" ;)BI1 BI1 45/'8@1/-gDBG45 4/ /6tF /'<34 6%B ? BE6 ? E1
B ? § " F" £ "f 6@<7/+@/ ? /=< 8>PIP>D< BG4 8!/ ? 6 47/ ? 498>4 A m pn B n A p E ;q A BI158;:L8@BEPQ8>RPI/ 6 ? 4/K/6t/=<4," $ ? 46.13/-'8@1/'1< 8A&/# 8/PE4<PI/:H6@< k t&B ? + 4/9<BI6@<D:L8><7BQ8 ? =/ B ? 8&9<6% /'<(g8DKBQ1DA&/'1BI<78@RPE/%"
Y 1B ? + %eB =8 ? /'P>O4768 ? 1g/=<45BI1 0 /'134BI6 ? " / ? 6@4B ? +ω4/e<BI6@<< D/=<598>< 8>MO/=4/=<<>A&/6 8PE4:L8>P / 1
H6@<ω
=8 ? R/$H6/ ? AC1?498>4 %eB
(ω|yn) = 1
? A&/'< 4/G=6 ? 134< 8>B ? 4 g(ω) = θeg(/=<7/< 4'D 9B '8>PIP>D<
g(ω) =[θ]
"%(J4B ? +CB ? 1? 82g8D<Kg/G66$1/24/MO6$134 B ? H6@<7MK8L47BE:$/&<7BI6@< B ? 8/,J6@< A8 ? J/gDBE4049/ /6tF /'<3456/9B ? BI6 ? 8 ? A ? 6>4=6 ? B7 47B ? +fgDBE404/KA8>478?" $ ?8>R91/ ? J/6/+816@P &4BI6 ? < θe
16/9PIAKR/ M 6&A&B k / AGgDBE48 ? /#t47/=< ? 8>P^4 ? 6ogDPE/ AF+@/%" ;)<76/?+/6%&4/#t8@M@9PE/ 1<91B ? +4/5/6tF 6 ? / ? 4BQ8>P 8 ? A 4/¡ /=BIR!PIP A&BQ147<BIR!&47BE6 ? 1<!g/ BIPEP 9134< 8L47/ 16@MO/ 49BI1134< 8L47/6+%DK6b=8>PIBIR<78>4BI6 ? "
-* xEw " qnqDJ?;oq 9e AIJ Ann q m E q nq J!n qn6p p%nA r q9 π(θ|x(l)) J!n6pn x(l) ( ;$ ; G; J (( ; B DE@ q ;oq A¥q q J!n m E q ;oq AA; B J?;AA An & qDJ?nG;/%; q ;oq A J?nA¥; Ap p JF !n6p !;p; E n6qn6p( A A A G ; J (( p5n5; BDB m A nA nA?q A B Azq J!n m ; n Cm A %;qn E /n B ;$A Ap p G
a d ] _ d \F\ Z!"! \ ! # d _bac d _$&% d ' d \$ ? 4/ /#tF6 ? / ? 4BQ8>PH=8$13/<Fg/$98;:$/ 4691/ %eB AIJ "3B6 ? 1BQA&/=<D8+.8>B ? 45/<7BE6$<)<6% 6.13/ A B ? bt8>MPI/ x,
λ ∼ G(a, axe)
g(/=<7/aBI1 45/K13B! =/O6/8 :BI<34598>P B" B9" AK"O18@M@9PE/gDBE4 M / 8 ? xe
8 ? AF:L8><7BQ8 ? =/ax2
e
" ;)91a/=MR6&A&BI/'1 4/
/#tF/=<4e ? =/=<478>B ? 4'DAj3=8@PEBIR< 8L47B ? +πBQ1e96.6.13B ? +
a" $ ? 49/H6@PIPI6ogDB ? + <9A&/ ? 6>47/ a 45/2134<7B7 4P Dl 6.13BE4BI:@/5:L8>P /
1^ 498>4 %eB AIJ(a|yn) = 1" ;)9/2/6tBQ134/ ? J/28 ? A ? B =BG4'D 6/
a'8 ? R / R98@1B7=8>PIP Df<76L:@/ A291B ? +49/CH6$P s
PE6ogDB ? +58><5+/9M / ? 471," B ? J/ %eB AIJ(a|yn)BQ149/ 19M 6/6$1BG47BE:$/J6 ? :$/#t ? 47BE6 ? 1 6/
a<@BE4BQ1=J6 ? :@/#t2B ? BE471
8><5+/MO/ ? 4 a " ;)/'<//#t&BI134718 ? B 0 /MOB ? BEMOB! =/=< a∗ = argmin %eB AIJ (a|yn)
" B ? J/ %eB AIJ(a|yn) → ∞g(/ ? a → ∞ <H6@< 8 k t&/'An > 0
<Hg/@ 8;:@/a∗ = 0
6@<0 < a∗ < ∞ " ;)/G'8@1/
a∗ = 0J6$<<7/'16 ? A1 46
π(λ) ∝ πJ(λ)8 ? A %eB AIJ(a∗|yn) = 1
" ;)BQ1+BQ1J6 ? 47<78$A&B J46$<D:gDBG45 45/< 8% 4 498>416@MO/3<BI6@< 1=8 ? R / ? 6>4J6 ? B7 4B ? +^" ;)91
a∗ > 0" 1 DJ6 ? :@/#t&BE4'D<=4/'<//#t&BI13471+8( ? B 0 /:L8@P>/
a > 01? 4598L4 %eB AIJ(a|yn) = 1
"
) # % &! ' $ ? B +/9</ " x%x <94/2/=:@6$P>4BI6 ? 6aBQ1 A&BQ19PI8D$/'ACB ? ? J4BI6 ? 6/
xe<A91B ? +
4/ A98L478 6/ bt8@M@9PE/ xE "&$ ? 4/ 178>MO/ k +%</<g/PI8%J/ 8 R6/ ? A98><5D PEB ? / 8>4 a = n = 10<+1B ? J/
nBQ1 8
? 8L45< 8>P PIBEMOBE4:;8@P>9/H6$<a
"
: o R +
xe
a
0 100 200 300 400 500
010
2030
40
size n
;+8@R " x%x j :$6@P &4BI6 ? 16/aB ? ? J4BI6 ? 6
xeB ? 8 ? B>+%R 6$<66&A 6/
y = 207"
) # % & ! ' %M C? 6og 4 8L4yn
=6 ? 478@B ? n − r<7B>+%.4 s =? 36$< A k t A I 4'D 6s $J :L8@P>
cn−r = (c1, . . . , cn−r)" ;) (k <34A&B> < '? gDBE4O45 < :BI6/<=8 B 4 8L4 n B ? 6 MO6$< 8 < < '? PIBEMOBE4
:L8>P H6$<a
" % =6 ? <7:L8L4BI: 6$B BeJP 8><7P D 4 ? MR < r 6/ ? =? 36$< A A98L478?" ;) J6 ? A AB < =? B 4 96@B7 6
πJ " %(6=6@< A&B ? +476& 8 ? 4B n#q); BG I9v/w/w?x J <.45 '? 36$< Af¤ < D~<BI6@<
πJc (λ) ∝ λ−1
√√√√n−n−r∑
i=1
exp(−λci)
I § £ " v " v J<DB PQAD8 R 434 <(H< 0 '? 47B34J6L: < 8+ 498 ? 45 3478 ? A8>< Af¤ < D~<BI6@<πJ H6$<4 634 <BI6@<
1 8D B ? 4 <:L8>P ,"
B6 ? 3BQA <45 PIB> 4BIM '? 36$< A A8>478 4=H<76@M ; 8>R9P " £ " ;) ηn = 1/λn = 13, 115" ? ;+8@RP
" g < =? 4 34BIMK8L47BE6 ? 6a
J6$M@?&4 Af3B ? + 4 9<BI6@<πJ 8 ? A πJ
c
"b¡ 5? 6>4 498>43B ? +πJ H6@<45
=6 ? : '? B '? 6/ BE4J6 ? '!+$8>4BI6 ? <76/ <4B '8 ? A&< 8>MK8L47B '8>PIP>D MO6&A&B>Da8 ? AB> 44 8+@< M =? 4 A&6@MK8@B ?
H6$<xe
476 45 PI8@<+ 34 :L8>P 6/X
" ^0< PE4:B +/PIB>+%.4 45 O? A 68 k9? 96@B7 6 45 R '? MK8><54? 6 ? B ? H6@<7MK8L4BI: <7BE6$<~g( =? 4 A8>478 8>< 66@<7P>DKB ? H6@<7MK8L47BE: <&8 ? A 4 ? A6B ? + 45 PEP A98L478 B ? H6@< sMK8L47BE6 ? : '? BE4BB ? =6@MP 4 "
a d] _ d \ F\ Z! !d 'aa&c d _$ % d ' d \¡ =6 ? BIA <8+$8@B ? 45 ¡ BER!9PEPF '? 6@< A A8>478H<6$M § " F" £ "H¡ 98 ? + 6%<<7BI6@<<96.63B ? + 8eB < 8><B '8>P<7BI6@<DA #k ? A B ? B3 8&4 < £
µ|β ∼ G(a, b(a, β)),
β ∼ Be(p, q)
g( < µ = η−β <?g( < 4 1 4 8 BeA&B 34<7BER!4BI6 ? BA #k9? A 6 ? [βl, βr]
8 ? A b(a, β)B A #k ? A RD
b(a, β) = (21/a − 1)−1xβe ,
xeR B ? + 8@P6 89<BI6@< 34BIMK8L47BE6 ? 6/+45 M A&BQ8 ? PIB 4BIM " 47BEPIP 45 < < a + 4D8 '? 6/8 :BI<34598>P78>MP
B " ;) 4 J6@MP 4 <7BE6$<5BCD <598@<78@M 4<7B AFR D(a, xe, p, q, βl, βr)
"f¡ A&6 ? 6>4CJ6 ? BIA < <
− ! " #$% %&
.2*9 )( / 9 *,*2 -%1)(9 -%1)(9 -)*)9 , / ,1.9, / ,.9 .'*% / 9.*% /;! #" " £ j tF%$& &(')!"*"+)> ')+, &$- ./. 0'
a1bis"
xe πJ πJc
%9 ((1( (& / / & ,,9 ((1( *1& 21, ,& ,*(9 (1(( 2& ) .'& /*, 9 (1(( * / & , *& ,
;! #" " j ')+,10')+$2&3$a
g( &4' '5!& . 2- . !& . '5 &$- . ¤ 6- D)#-)+$2-3!- . "
' $2,#" ' tF -'7$%#)8&9)8$&:+2)+: &;)+& ;! 9" " v <<#,)+&!+ )+,#" D='9!'xe = 250
"@>$2- $0: -<Ag @k t(βl, βr) = (1, 5)
2& . $2, :02"> ?$/ [β] = βl + (βr − βl)p/(p + q)
")$&@' !, ) . >'9!'A)+&B'5 tFC$2& &')+2"3 <bg !- !" $D"+$($%4E)8&!+uF$2- . 9"G':0!" CF$2-
(a, p, q)")$!' '590' πJ(β)
) &#-$% -H$&[βl, βr]
" $C)8&?+&=; )+& %eB AIJ )I- 0 #)8- . $&#" DH'$H . '' πJ (θ′|β)g( - θ′ )J!&D@9!-5!, '-)! !')+$2&
!,$2&?+ µ, η, λ " ;)K'5 )! $/LC=; x(l)
) x " )+,)8"!-" D?'$ § " !" £ </g 2& '5 . Dπ(β)
!& . π(θ′|β) 9!-50' " D<)8&?+ 0 90')8$&
%eB AIJ(π|yn) = %eB AIJ (a|yn) +
%eB AIJ(p, q|yn) − 1.I " J
)& M % ) ' FN Mπ(θ′|β)
;) tF -'($2,,$2&2 &'-2"*:0!" xe = 250) O: -5D1$/P')8,) ')
gO)G'51- Q'R'$S'5 T. !'?"¡ (k & . a = 0.25'<(S&9$2& s Q$& )7')+&!+#-)+$2-R$&2&D 5!" 92-2, '-) 0')8$&
2&#&#$!'3D() " . ,$2- )8&!F$2-,!')+$2&'59!&1/4
$*'5 )+&FF$2-,10')+$2&-DE) " .. D )8&!+" U. 0'5!" ;)#)<#-)8$-R)V: -D 0'E"
)& M % )' N Mπ(β)
¡ &#$2' σ'5 #-)8$- '5!& . 2- .S. :E)+!')+$2&W$2& β ^U'590' %eB AIJ(p, q|yn) =
1!& . (p, q)
' $FQ)0' D -%!-5!, ' -S:0!" ," $& ;<2 #" " X'5 :0!" 2- $2,!P' . F$2- $2, Q")72"#$) T$
[β])+&D- "+)+2 #)+"8)8''D/" ;) "+) )G' .D. &)G') U2- A. ) #"D . $2& ) +/9- " xEv " $2' '9!'
g( & [β] → βn
'5 & σ . Q- <g(#)7;, !&7'9!'?$2&#" Dz9-)+$2-,$ . "+"8)+&!+D '-$2&?+>+2)+&!+=!- &#$2'Q$& )7')+&!+gO)8'1'5 T. !'?" T$!') '9!'I&9!'#-5!"^#$2)7 F$-
(p, q)<()+&X! &? $L!&DQ-)G') !"#)+&FF$-,10')8$&
! C$/P'J' tF -'3$/9)8&#)+$2& <P)(1, 1)
" $& .. <!g $ P'2)8&4@#&#)>F$2-, . )'-)+ !P')+$2&4$2&[βl, βr] = [1, 5]
" ;) &' A. /#"8'U:0!" $/
σ)
0.53" ;)<%#,)8&?+
[β] = 2<%' - )U&#$X$2' -#$) '92& "+) )G')8&?+
(p, q): -DG"8$ '$ I9w " < x " /JV'$$2 #'!)+&Y&#$2& s Q$& )7')8&!+@#-)8$-,"
8[β] σ (p, q)
/ ('& , (& , 9+*& ,% (& -- / & (9 / & (:- (& / 2 2& / 9 / & -
;! #" " Fj #"8':0!" ~F$-3' 9-)+$2- '2& . !- .Z. :E)+!')+$2&Y!& . D -92-2, ' -J$/π(β)
"
# A
beta
dens
ities
1 2 3 4 5
E=2E=3E=4
;<2 " xv j 1 '5 . &)G') π(β|p, q) gO)G'54- '3'$& =[β]
"
' FN M )#! ' # !!C$ '9!'R' 1 D )!&H2&9!" D'R- $/+&#) '590'R C$!'5 tF -'9: - )8: . !'J" '3$2& !)+" #- ')+, )8&/' 9'E" '')8&?+
a = 1F$2-J' $2,,$2& tF -'3, . )!& <g
$ P'2)8& %eB AIJ(a|yn) = 1.16" !!C$ '9!'U'5 1 D )+2&>!&92">D':#$E$ [β] = 4
I g(#)7D, 2&U'-$2&!+">D=%6 " -!' . +)8&!+Jr" % #,)+&!+/'9!'W'5 Q$,@9" ' #-)8$-W- ,12)8&U&#$2'$2& ) Q')+&!+ <Hg $2 P'5!)+& "+$og -3 %$%#& . σmin
$2&σ)>g gJ2&('J'$-9:
%eB AIJ (π|yn) ≤ 1.
-$, I " J <E'9)O"+$og -3 %$%#& . 2&/ Q$,!P' . ?/'%0' %eB AIJ(p, q|yn) = 0.84.
¡ $2 P'5!)+& k &9!"+">D σmin = 0.286)+&
(p, q) = (6.1, 2.03)"
')( R 2 9<7 7= 46 =032 9<6 0 6 CW
$& § v <!' ) R-5!) . DH' W.#k &#)G')8$&X$ %eB AIJ %: &9- &' . ; DA2- )+2"8" D . '$1' ) q $ K' =; <^g#$ H. )'9- #)+&/)8&F$-,1!')+$2& 2&> '$E$X)+& &(')!"$2&=' . #'0')8$& $&$#-V)+& . '-)!" t#!,#" <2'#)I) F$2-'9&90' " DA&9$!'R' F$2-I' tC$2& &')+2"%2& . ¡ )+ #"+"%,$ . " <)8& q) R- ')8: " D x !& . v !& . n - ,1!)+& "+$ '$ x,w )+&1' gJ$2-' ;<)8'I) . )8-5! 9" '$ . -
' 2,A$#&('U$ . )'9- #)+&X)+&F$-,10')8$&uDE) " .. D4' =; )+&('$ %eB AIJ $&D'#) U2)8, <Ag 'T'$<n !JFq ' )+& - & Cg#) @" . '$ $2,#' πJ (θ|x(l)) !& . π(θ|x(l)) )+&('$=' )+&('-)8&) . P'!')+$2& ; ''gJ$%$' -)8$- 2&4 ,$ . ) k. )8&('$ n qn6p p F$2"+"+$og
!#"%$&'x(l) = x1, . . . , xq ( &*)+ HG HG G-,./ "
01"32&5476+&8'9& <n Jq n B n B J # (: Lp(x(l); θ) =
(q∏
i=1
f(xi|θ))p/q "
;1"32&5476+&8'9& <n Jq n5 qn6p p (:
πJp (θ|x(l)) ∝ Lp(x(l); θ) πJ (θ),
πp(θ|x(l)) ∝ Lp(x(l); θ) π(θ).<>=@?BADCEGF>HJILK@MNCOADPQEGHJRTSBUBV>F>SXWTYZ[MN\D]F>H#^`_abRHcE $! MNPSXde?BRCb?fMN\ $Ng ADPEG?BFhTUBA EbFiSBA jF>HbF>PQE#VRPQEbF>kTElRNmEG?FiCbEbUBSY
RNmn RCbEbFHJADRH#VRPCGADCbEbF>PBVoY
− ! " #$% %& " 6
p)L'59&l6+O6 ) ) & 6+
IN9*'9)'
πJp (θ|x(l)) )+
πp(θ|x(l)) )& & "
2-)+"< p ≤ q ;<' ,1<$9' "+) "+) #$E$ . $ 9' =;
x(l)) &9"+2- . 2& . ' . $%$ s
' -)+$2- #$#" . E) " . ?"+$og -I)+&F$-,10')8$&1'%!&/' - !" C$' -)+$2- $!' '9!'I' :2" $ p ) & -5!"+""8)+,)G' . ' #$) $ πJ )+& 9-$ - #-)+$2- π 2&/2- #)8'-5!-)+"A #$ &Z- #"+2- &#$X'$A#, '9!'
πp(θ|x(l))) #-$ -F$2-2&
p ≥ 0 ;#) , ) )+"4 . P'2 #" '$ &$- . ; <#,$ . ) E)+&
' g ) (' . "+) "+) #$E$ . !& . C$)+ #" πJ $2' '%0'g & πJ ) X#-$ -<g 9: " !-" p = 0@ $2-
,$ . " <g $2 P'5!)+& πJ1 (θ|x(l)) = πJ (θ|t) g - t )' 9!"E,)+&#)8,12" ) &(' '!') ') "! ' , 2&
)8&@2& HG HG G $# $ x(l) % ,$2- )8&(' - ')8& '7) ) p ≤ qF$-7, 9"G') . )+, &)+$2&92"R$2-&#$& $& >(0'
,$ . " R $-O)8&'5!& < p = 1F$2-J' ¡ )8 #"8" ,A$ . "0< 9$($)+& πJ I' ¤ 6- #-)8$-<P!"8'9$ q = 2
& tE' #-$%$)8')+$2& <Fg#) 4) #-$0: . )8& % & . )>t § x
^P,7K x n6q (xi 6= xj) > 1 n%G& p@qDJ?n('@n #CBDB E& %n BG A r /n6p@qDJ?n )%p*% E n#qp + %oq A(µ, β) J!n6pn µ = η−β G J n πJ (µ, β) ∝ (µβ)−1 G & J!nA πJ
p (µ, β|xi, xj) ( Ap ?n6p p,% BDB p > 0 e A)%prq m CB %p p p = 1 %AA
µ|β, xi, xj ∼ G(p, (p/2)
xβ
i + xβj
),
πJp (β|xi, xj) ∝ βp−1(xixj)
pβ/2(xβ
i + xβj
)−p
.
$2, $2' -?-#" 3 $#" . #-$C$ . '$ 9$($ p 7=- 2"R:0!" )+&' . $ O!&@)+&(' -7:02" s )!"+"<<)8'?) 7$3)+&(' - ''$ k t
0 < p < 1'$=$2 #'!)+&BY:0 & #,12-*`#-)+$2- πJ
p (θ|x(l)) g &@)G'H)%$)+ #" ;eg #$#" . $&) . -O'#) #-)8$-U-E) " . )+&Z' 2, )8&F$-,1!')+$2&='90'SF- ')8$&>$ R$& )+&" T. 0' =tP2,#" x F$-I2&X)8"+" '-!')+$2& /. P' X?-#" - ,1!)+& . )2 #"G'I'$A K$ ')+: "
021 43)576 x '@n 6CBDB E /n BG¡ F$2"+"+$og ' ¡ )+ #"+" t#!,#" 9- &' . !' § v Y)+&/' !, 9-)+$2- . ) '-)+ P')8$& <Kg #$og : -'-#& 0' π(µ|β)
$2&[0, 500]
;πJ ) #-$ -T2& . %-8 J !&Y [ $2,#' . F$-O' $, %!-)$2&lgO)G'
%-8 BIJ I )>tP)8& a = 2!& . R!- [β] = 0.025
<g !- "+$($E)8& F$2-H' Y. $,2)8&H$&@' 9-)+$2-H, . )!&xe
2& . ' #-)+$2-T, !& β0 Y'9!' π(µ|β)
2& . π(β)!- &#$2' $& ) ')+&GgO)8'
yn
<)+& . : "8$#, &('! # ;<2 #" )+& . ) !' J' 7. $,2)8& $2,P' . gO)8' %-8 J $2- %-8 AIJ ; )+&('-)8&) . P'!')+$2&) $,P' . )8& . $C$' -)+$2- πJ
p (.|x(l)) .6k & . )+&IR-$C$)G')8$& x #<lF$- : -5!"V:02" 7$ p $&9!-') 9"+2-eg [ #$E$ p = q = 2
< )8& )8'W) U' 9!" g - πJp (.|x(l)) = πJ (.|x(l)) ¡ $ -:
'9!'H' Y. - )8& #$2) $ p ,19 7' - , &(' . $,1!)+& ')8,1!' . %-8 AIJ "+$ -H'$ ' - - & W. $2,1!)+& ')8,1!' . %-8 J ;< '2 #"+)#)+&1H-#" '$g ) (' πJ (θ|x(l)) !-I'$ 9 '3$- 2"*)8&(' - '
J AIJ
p = 2 p = 1 p = 1/2 p = 1/4
η %2! / - / -, / %. -% / %) -(" / -( %0" / -'*β *1& - / -'& 0 / *1& -0 - & 21) *1& -," - & ) / *& -% - & ), *& - / - & )1)
;<2 #" j% 2- , &(' . $2,1!)+& $2,P' . %-8 J !& . %-8 AIJ ! F$- : -2"C:02" 3$ p #
'
')( T 9 lV 4 7 =9
Y)+& t -'W$#)+&#)+$2&U)F- &'W)8& )8& . '-)+2"<) eg - A. 0'5Z2- $2"+" ' . gO)8' . ) #"8' ; # ')+: - P')8$& !&$, ')+, U !-F-$, ' $2 ')+: )8&F$2-,!')+$2& E) " .. 74' A. 0' <7F$2-)+&'!& F$-?' #&9) !"J- $2& ; E)+& . ) - %!& $-H!& 2- , &(' 2)+: H2&B)+&F$2-,10')+$2&g#) 2&> . /' . )!&=!&92"' ; %-8 -)8' -)+$2& 2&> . F$-'#) . ' ')+$2& $ 'U) 2&D!"8' -&9!')+: #-$ Y'$/' P− :2" #-$ . #- #-$C$ . R:0!& y =$9$2&#$0: ! v/w/w 2#< vw%w 0 # $ '9!& . )8& ! - )!"+"%#) #"+) ' . )8&4' $&' tE'O$ i#) -5!- #) !"7#-)8$- # ) <#)8&4$9- & <# k -'2-#, &('J)+& :2$-J$ %-8 . #'O' - !- '#- ,1!)+& . ) - & ''g &/' -#"
% k -' . )> - & ) ?'90' %-8 )8& . ) 0' H'#- #$" . :2" [F$2-A' -9!-5!, ' -< '9!'π
2& .yn
!- &#$2' $& ) ')+& << #' @:0!" ?- ,2)8& #& . ) . 2 #" F$2-7' P− :02" #-$ . #- ;< %-8 !&D- ': - / #)!) . 9-)+$2-'90'W' P− :0!" #-$ . #- T P'< k tP)8& 9!"+"/' '#- #$" .P− :0!" I'$ (5%, 95%)
- &(')+"
% $& .4. ) - & )J'90'< $&('-5!- H'$A' P− :0!" #-$ . 9- < %-8 P'3$&#"%#-)+$2- . ) '-)8 s')8$&)gO)G' $, 9-$,A) )8&A:0!-)+2& 2& . 9)+ $og : -<g &/' #-)8$-#& -'!)+&(' X) O- $2&92 #" <E %$2'#-$ I" . '$A' !, $& " )8$&
% k &9!" !'#- $ %-8 ) I'9!'I)8' !&X . J'$($"%$ 2"8)+ #-50')8$& <2)+& . ) !')+&H . #"G'3:0!" F$2-' :0!-)+2& $ e#-)+$2-O)+&F$-,10')8: ,$ . "8"+)8&
;@g '#)+&) '90'& %-8 ) !&B)+&(' - ')+&;!"8' -&9!')+: '$>' P− :02" #-$ . 9- 2& . " c#", '#$ . '$ #" )8&Z' '$E$"E)8'I$ *' . )+2&1!&92"' $&3c#& ')+$2&/$ *' :0!)+"! #" )+&F$2-,10')+$2&X2 %$P'' % $& . )8')+$2&7$ 3' t -)+, &('H2& . ' t -' - . )+ #)8"+)8' <<' . )+2&;2&9!"' $#" . $2-- '' # ')+: "8) J$2-J* F$-I$2' - t -)+, &('R'$ 9& . -'!& . ' W. ) - 9!& %&/$ &X) $9" . . ' ')+&$2, $P'"+) -3$2-J' '$E$H)+& &(')!" . !'H)8&4' 2, 9" &(')+2" $2,P'50')8$&3$ %-8 <)+& - )+&A$2-3-5!& . $,) )8&H' 7. 0'5 '
$og : -<E' - - ,2)8& $2, )+, C$2-'2&(' . )2 #"G') O'$3 %-8 ; [ #$2) $<' &#$&#)8&F$2-,!')+: & 9,2-* #-)8$- πJ , 'J U. $2& !- c9"8"<E )+&7' 9-)+$2-3gO)G'Z '1F- &(')' $0: -5 $ *' C$ s' -)8$- . )+&(' -:0!" #- )+&('-)8&) . P'50')8$& 2& F - $ %' ,1!"+")! n 2& . ' #) . )8, &)+$2&A$ ' ,$ . " % C$)8 #" )8,#-$0: , &('#9 & #-$C$ . F$-'#) "+'i%$)8&('<!!&H: & F$2-lcP'#- gI$-*
')( 2 2 6 >5=,
%!\ \ _ d d \ __ c %!\)\ _$&D'#) 9 ')8$& <*' ,12)8&`#-$%$)8')+$2&U)+&('-$ . . $- . )+&D' ' tE'7!- - !"+" . 2& . 9-$E$ S2- )8: &
P?67K #+! ) )#! N #+ # ! ' ' # & n#q π1, . . . , πm n m p p AΘ %$A B n6q α1, . . . , αm n m <n Jq r m J q J)%q 0 < αi < 1 ∑m
i=1 αi = 1 G 9 nA q n π(θ) ∝ ∏mi=1 π(θ)αi
i G
− ! " #$% %&& J!nA
9e J(π|yn) ≤m∑
i=1
αi DACJ(πi|yn).
#&# &#$!')8&& [.]' , !&O)8'Y- '3'$ πJ (θ|yn)
<9)G') 1'$ '90'
KLπJ (.|yn) || π
)=
[logπJ(.|yn) −
m∑
i=1
αi logπ(θ)i
]+ log
∫
Θ
m∏
i=1
π(θ)αi
i dθ,
=
m∑
i=1
αi KLπJ (.|yn) || πi
)+ log ∆
- ∆ =
∫
Θ
m∏
i=1
πi(θ)αi dθ
Y)+&A' & -2"8)! . " . -O)+& 92"8)8' < $2 P'5!)+&
∆ ≤m∏
i=1
(∫
Θ
πi(θ) dθ
)αi
.
;∆ ≤ 1
)+& ' πi2- . &)G') O!& .
KLπJ (.|yn) || π
)≤
m∑
i=1
αi KLπJ(.|yn ||πi
)
#) /" . '-2) ('F$2-3!- . "A'$H' '0' , &('O$ ' #-$%$)8')+$2&
^P,7K # ' ' &! #&% 'F Xn %$A HG HG G % E B E % ? A oCBDB Br(θ)
( # A G' % rCE A θ % ? %- ( # A Be(α, β) A [0, 1] G & J W πJ ( J Be(1/2, 1/2) A r G δn =
∑ni=1 Xi G 9 A A 0 < θ0 < 1
J % B J )% % E G & J A ∀(α, β) > 0 %$A? 0 < q < 1 (logn)q L [DACJ (α, β|Xn) − 1
]→ 0.
#&# ; =$)8:E- 9"+ ! $2- 8 ; # ' $2- , )8& . ) 0' 3'%0' #& . - Br(θ0)<
δn − nθ0√nθ0(1 − θ0)
L−→ N (0, 1).
; . &9$!' δn = nθ0 + Un
√n
- UnL−→ N (0, θ0(1 − θ0))
T$!' Ψ' . ) 2,A,1 c9& ')8$& ! ' "+$ . -)8:00')8: $ ' (!,,1
c#& ')+$2& # %' - $2, : !" #"0')8$& )+&1*E,P'$2') F$"8"+$O)+& . : "+$#, &('
Ψ(n+ 1) = log n+1
2n− 1
12n2+ o(n−3),
log Γ(n+ 1) =1
2log 2π +
(n+
1
2
)logn− n+
α
12n,
- 0 < α < 1
#) !&Z S. -)8: . F-$2, %T #-5!,$O)G' T!& . ' 9& ! <5 / # !& . ' ' ')8-"+)8&[F$2-, 9"+)8: &Z)+& %-')+& ! % <5 # <P- ')8: "< & . F$2- n > max( 1
2θ0, 1
2(1−θ0))'90'
KLπJ (.|Xn) | πJ
= (n+ 1/2) logn− nΨ(1/2)
+ Un
√n 4 − 2Ψ(1/2)+Kθ0(1/2, 1/2) + o(1),
KLπJ(.|Xn) | π
= (n+ 3/2− α− β) log n− n (Ψ(β) + θ0 Ψ(α) − Ψ(β))
+ Un
√n 4 − Ψ(α) − Ψ(β) +Kθ0(α, β) + o(1).
-
Kθ0(α, β) = logΓ(α)Γ(β)
Γ(α+ β)− log
√2π +
(α− 1
2
)log
Ψ(α)
θ0+
(β − 1
2
)log
Ψ(β)
1 − θ0.
; &Z' *E,P'$2') W. : "8$#, &('O$ DACJ )8:
DACJ (α, β|Xn) = 1 +Aθ0(α, β)
logn
1 − Ψ(1/2)
logn
+ Un
B(α, β)√n logn
1 − Ψ(1/2)
logn
−UnCθ0(α, β)√n (logn)
2 +D(α, β)
n+ o(n−1) ! #
-
Aθ0(α, β) = Ψ(1/2)− Ψ(β) + θ0 Ψ(β) − Ψ(α) ,B(α, β) = 2Ψ(1/2)− Ψ(α) − Ψ(β),
Cθ0(α, β) = Aθ0(α, β) 4 − 2Ψ(1/2) ,D(α, β) = 1 − α− β.
T$!' '9!'W0'W" 'T$2& ' -, )+&=' A. : "+$#, &' ! # ) &#$& -$ < P' & π ≡ πJ ! ⇔ α = β =
1/2 ⇔ DACJ = 1 # $& .. <
Aθ0(α, β) = 0,B(α, β) = 0,D(α, β) = 0
⇔
Ψ(α) = Ψ(β),α+ β = 1 ⇔ α = β = 1/2.
;<$#-$0: '90'
(logn)q [ %-8 J(α, β|Xn) − 1
]→ 0
F$-32&0 < q < 1
<E)8') &9$X'$ $2&('-$2" [Vn] - Vn = Un [n(logn)q ]
−1 % ) &(' $2& . )G')8$&Z) '$ #$ '9!'
[Vn] → 0 & n )+& - ;#) 2&/ 7. $2& F$2"+"8$ &#$!' Zn = δn/n
¡ )8'
Vn =(Zn − θ0)
(logn)q ,
$2 P'5!)+&/ >2-*$0: 3)+& %!"+)G' <F$2-!&M > 0
< [|Vn| |Vn|≥M
]≤ M−1 [|Vn|2
],
≤ M−1
[Z2
n] + 2θ0[Zn] + θ20
(logn)2q ,
≤ M−1 θ0 (1 − θ0 + 2nθ0)
n (logn)2q
− ! " #$% %&#) $2 E:E)8$" ' & . X'$ & n → ∞ F$2"+"8$ .
M → ∞ ;9)/- 9"G' &#- 1'90' Vn)
% E % BDB A E B A % B ; & <F-$, :0!& . -!R!-' ! #< ; $2- , # < 9:
limn→∞
[Vn] = lim
n→∞
[U ]
n(logn)q
- U ∼ N (0, θ0(1 − θ0)) ; &
[Vn] → 0!& . ' '50' , &('$ <' #-$C$)8')+$2& F$2"+"8$
; -)G' -)8$& %-8 J $2, J'$ )+&4, !&Y % -T$2-O"+$ -O:2" ¡ ' ' . 4*E, #'$!') !"+"' 9-$ 9! #)+"+)G' Pθ0(
%-8 J1 (α, β|Xn) − 1 ≤ 0)
'$ $2, '$ X"+$ -O:2" &#$!' Φ' #,[#"+!')+:
c#& ')+$2&Y$ <' - . . &#$-,12" . )'-)8 P')+$2&
% # C$ -'"1'90' Aθ0(α, β) 6= 0 ¡ & B(α, β) > 0
<F-$, ! # <
Pθ0
%-8 J(α, β|Xn) − 1 ≤ 0
n→∞' Φ
(− Aθ0(α, β)
√n
B(α, β)√θ0(1 − θ0)
).
$
limn→∞
Pθ0
DACJ (α, β|yn) − 1 ≤ 0
= 0 ⇔ Aθ0(α, β) > 0,
limn→∞
Pθ0
DACJ (α, β|yn) − 1 ≤ 0
= 1 ⇔ Aθ0(α, β) < 0.
8I$&(: - "< & B(α, β) < 0<
Pθ0
%-8 J(α, β|Xn) − 1 ≤ 0
n→∞' 1 − Φ
(Aθ0(α, β)
√n
|B(α, β)|√θ0(1 − θ0)
).
#) 2)+: Z' !, $2&E: - & Z$2& ' !, $& . )8')+$2&Y$& Aθ0(α, β) )+&92"8"< C$
B(a, β) = 0 ; & <#) Aθ0(α, β) > 0
<E' &
Pθ0
%-8 J (α, β|Xn) − 1 ≤ 0
n→∞' 1 − Φ
( √n logn
4 − 2Ψ(1/2)
)n→∞−−−−→ 0.
)+,)8"!-"< & Aθ0(α, β) < 0<E' &
Pθ0
%-8 J(α, β|Xn) − 1 ≤ 0
n→∞' Φ
( √n logn
4 − 2Ψ(1/2)
)n→∞−−−−→ 1.
. # $2& . "<%$ '%0' Aθ0(α, β) = 0 ! $ Cθ0(α, β) = 0 # ; & <#) B(α, β) > 0<
Pθ0
%-8 J (α, β|Xn) − 1 ≤ 0
n→∞' Φ
((1 − α− β) log n
B(α, β)√n
)n→∞−−−−→ 1/2.
¡ $2 P'5!)+&>' 2, - #"8' & B(α, β) < 0 ; "' ) & B(α, β) = 0
$ '? !- & α = β
; & <*) α < 1/2<C' #- )8$& ! # T'9!'U' "8)+,)G') W!''5!)+& . 7 -
:0!" 8I$&(: - "<) α > 1/2<E' "8)+,)G') O0''2)8& . 7Z"8$ -3:0!"
FN ' ' ! # % 'F (xi 6= xj) > 1 % HG HGG E@ A DE % B % A A% % E B & J ' CBDB E& B )% % E + (µ, β) G J πJ(µ, β) ∝ (µβ)−1 G 9 A
πJp (µ, β|xi, xj) ∝ L(xi, xj ;µ, β)p/2
πJ (µ, β)
J L(x;µ, β) ( J ' CBDB B B J # G & J A πJp (µ, β|xi, xj) ( % BDB p > 0 ( A% CB % p = 1 %$A
µ|β, xi, xj ∼ G
(p, (p/2)
xβ
i + xβj
),
πJp (β|xi, xj) ∝ βp−1(xixj)
pβ/2(xβ
i + xβj
)−p
.
#&# ¡ 9:
πJp (µ, β|xi, xj) ∝ µp−1βp−1 (xixj)
pβ/2exp
−pµ
2
(xβ
i + xβj
) (µ,β)≥0.
; &
µ|β, xi, xj ∼ G(p, (p/2)
xβ
i + xβj
),
πJp (β|xi, xj) ∝ βp−1 (xixj)
pβ/2
(xβ
i + xβj
)p .
$ (xi, xj) > 1
2& . xi 6= xj' & α = max(xi, xj)(
√xixj)
−1 > 1 ;5
βp−1 (xixj)pβ/2
(xβ
i + xβj
)p ≤ βp−1 (xixj)pβ/2
max(xi, xj)pβ,
≤ βp−1 exp
(−pβ log
max(xi, xj)√xixj
),
≤ βp−1 exp (−pβ logα) .
)+& α > 1<logα > 0
*' & - $&#) '%0'W' -) (' 9!& . ) . $ R' "+'S)8& 9!"+)8' 4) #-$C$2- ')8$&9!"%'$S2,, . &)8' O)G' 9 9!-5!, ' - p 2& . 2" 9!-5!, ' - p logα
; $ P'2)8&Z'9!'p > 0 ⇒ πJ
p (β|xi, xj)) #-$ -
'"!#! "$ ]&%('*) !,+ -! ! %/. +10! %&2 ! ] !,3 $, $ ' #- )8$& - f' - !&Z . -)+: . F-$, &#& ! # Ψ
)I' W. ) (!,,1 c#& ')+$2& ! ' "+$ . -)+:00')8: $ <' (!,,1[c#& ')8$& #
? % ( A G ' B(a, b)2& . B(c, d)
' J$ . ' . ) '-)+ P')8$& ; &
KL B(a, b) || B(c, d) = logΓ(a+ b)Γ(c)Γ(d)
Γ(c+ d)Γ(a)Γ(b)+ (a− c) Ψ(a) − Ψ(c) + (b− d) Ψ(b) − Ψ(d) .
− ! " #$% %&9 J B o( A G ' D1 = Dir(a1, . . . , ad)
!& . D2 = Dir(b1, . . . , bd) ' J$U)+-) #" ' . )'-)
P')8$& T$!' a =∑d
i=1 ai!& . b =
∑di=1 bi
; &
KL D1 || D2 = logΓ(a)
Γ(b)+
d∑
i=1
logΓ(bi)
Γ(ai)+
d∑
i=1
(ai − bi)(Γ(ai) − Γ(bi)).
A % % A E % B A ( A G ' N1 = N (µ1, σ21)2& . N2 = N (µ2, σ
22) ' J$ #&#)+:2-)0'
&#$2-,1!" . ) '-)+ P')8$& . & . $&Z' C$#& ..Y. $2,12)8& [Tl, Tm] T$!'
αi = αi(Tl, Tm, µi, σi) =
(∫ Tm
Tl
exp
− 1
2σ2i
(x− µi)2
dx
)−1
,
Ni(x) = αi exp
(− (x− µi)
2
2σ2i
).
; & ! . &#$2')+& [.]' , 2&O)8'Y- '3'$ N1
# <
KL N1 || N2 = logα1
α2+
(1
2σ22
− 1
2σ21
) [(X − µ1)
2] +µ2
2 − µ21
2σ22
+µ1 − µ2
σ22
[X ]
-
[(X − µ1)
2] = σ21 1 + (Tl − µ1)N1(Tl) + (µ1 − Tm)N1(Tm) ,
[X ] = µ1 + σ2
1 N1(Tl) −N1(Tm) .
A % % A E % B " A E o( A G ' N1 = N (µ1, σ21) #&#)+:0!-)+!' W. )'-)+ P')+$2&Z$2&
[Tl, Tm] ' U ' #&#) F$2-, . )'-)8 P')+$2&4$2& [Tl, Tm]
¡ )G'Y' &#$2'!')+$2& #- :E)+$" . & . <
KL N1 || U = logα1 + log(Tm − Tl) −1
2σ21
[(X − µ1)
2].
% E E % A % % A E % B"( A G ' G = G(γ, β) S!,,1 . ) '-)+ P')8$& O)8'
, !& γ/β2& . :0!-)+2& γ/β2 ' N = N (µ, σ2)
[9&#)8:02-)0' S. )'-)8 P')+$2& ; &
KL(G || N) = logβ√
2πσ2
Γ(γ)+ (γ − 1)Ψ(γ) − 1
σ2
[γ
β
(1
β− µ
)+µ2
2
].
% E E % o( A G $-i = 1, 2
<E" ' Gi = G(αi, βi) ' J$ S2,A,1 . ) '-)+ P')8$& O)G'Y, !& αi/βi!& . :0!-)+2& αi/β
2i
; &
KL G1 || G2 = (α1 − α2)Ψ(α1) + α2 logβ1
β2+ log
Γ(α2)
Γ(α1)+ α1
(β2
β1− 1
).
3 ! % + % % + % 3$&7'#) & . ) < 2)+: ' F$2-,1!" #- )+$2& $ #' 3. ) #-)+$2-<#-$C$ . F$2- . &#)8&' )+&('-)8&) . P'!')+$2&/$ ' &#$&#)+&F$2-,10')+: #-)8$- πJ )8&4' %$& &(')+2"L!& . ¡ )8 #"8"
P #+ ' % FN #+
)+-'<C$ '90'3' =; x(l) = xi
- xi) 3 )8&2" HG HGG . 0' F-$2, !& #& &$2- . !,#"
yn
; & πJ ) J' ¤ C- #-)8$-O!& .
λ|x(l) ∼ E(xi),
λ|yn(l) ∼ G
n− 1,
n∑
j=1
yj − xi
.
$& . "<%$ '9!' x(l) = ci - ci ) J?-) (' &$2- . .Z. !' &#$2' c1 . . . , cn−r
' n− r &$2- ./. 0'5H)+&('$
yn
R -$, !&(') % B G ! $# < ' )!"H¤ 6- #-)+$2-
πJc (λ) ∝ 1
λ
√√√√n−n−r∑
j=1
exp(−λcj).
; &
πJc (λ|ci) ∝
√√√√n−n−r∑
j=1
exp(−λcj) exp (−λci) .
;<$#-$0: '90' πJc (λ|ci)
) #-$ -< $&) . -J' ,1!-2)+&92" . &)G'
mJc(ci) =
∫
IR+
πJc (λ|ci)S(ci|λ) dλ
- S )J' #-:E)+:0!" c#& ')+$2& $ ') &#$Z'$#$ '9!'mJc(ci) <∞ )+&
1
λ
√√√√n−n−r∑
j=1
exp(−λcj) < λ−1/2
√√√√n
n−r∑
j=1
cj ∀λ > 0,
' - )' α > 0 /'90'
mJc(ci) < α
∫
IR+
λ−1/2
√√√√n
n−r∑
j=1
cj exp (−λci) dλ = α√nπ
√√√√n−r∑
j=1
cjci
<∞.
P ( N ' '% N #
¡ C$ '$ 9: !,#" yn = (xr, cn−r)
- xr = (x1, . . . , xr)
!- #& &$- . ) ) . . !'O)8' . &)G' c#& ')+$2& f(x|η, β) = βη−βxβ−1 exp(−η−βxβ)
F$-x ≥ 0
2& .cn−r
)U 'W$ n − r .-) (' o &$- .@. 0'5 $2"+"8$O)+& #& ! # < $2&) . -A' $2& . $- . -?,1!' #)+&;- - & #-)+$2-πJ(η, β) ∝ (ηβ)−1 % 9& &$- . ;
x(l)F$2-3' ¡ )8 9"8"<,A$ . "L) $#" (xi, xj) > 1
< ! xi 6= xi#
T$!' '%0' O. & &#$ #) -5!- 7S ' &7' ' J$ 9!-5!, ' -<&92, " 3. $S&#$2' $2&) . -' - - & #-)+$2-3)8&4' $ KA&59)!& 9!-5!, ' -
$2' '90'F$-V!& =; (xi, xj)
' C$' -)+$2-K$ 6' - - & #-)8$-R) #"+) )G'<2'9!&) '$7S P') c9"- #"8'3$ . - - % BG ! #
πij(η, β) = (2(xixj)| log xi/xj |)−1 (xixj)β−1
βη−2β−1 exp(−η−β
(xβ
i + xβj
))
− ! " #$% %&; & $2&) . -3' & 92-2, '-) 0')8$& η → µ = η−β < β → β
O)8' ¤( $2 #)!& J(µ, β) = βµ1+1/β ; $-- %$& . )+&?&9$2&#)+&F$2-,10')8: 9-)+$2-3) πJ (µ, β) ∝ (µβ2)−1 ; πij(µ, β) = πij(µ|β) πij(β)
O)8'
µ|β ∼ G(2, xβ
i + xβj
),
πij(β) =(xixj)
β−2
2| logxi/xj |(xβ
i + xβj
)2 .
; $2,#'0')8$&B$ %-8 AIJ(π|yn)& . H'$ $, P' ' C$' -)+$2- . &)G' @$
πJ $2& . )G')8$&9!"+";'$
yn(ij) = (y(ij)1, . . . , y(ij)n)' !,#"
yn
#$ $, C$2& &(' xi2& . xj
9: &;- ,$0: . &#$!' )+,)8"!-"
xn(ij) ; C$' -)+$2- . &)8') O2-
πij(µ, β|yn(ij)) = πij(µ|β,yn(ij)) πij(β|yn(ij))
O)G'
µ|β,yn(ij) ∼ G(r,
n∑
k=1
yβ(ij)k
),
π(β|yn(ij)) ∝ βr
(r∏
k=1
x(ij)k
)β
(n∑
k=1
yβ(ij)k
)r .
Chapitre 5
W!&O" % #)8'- #-! . &'<P&9$3&#$ $2,, F$ 2"8) " 9-O" Q' .7. " $2&9&9!) !& -' '3" 2"8)+ #-50')8$& . #& , #- . #-$ 9! 9)8"+)G'! % $# $# < & 92&('J:$2"+$2&('2)8- , &(' $ )+" & " ,% Q'#$ . . !" 9"* 9& ) & <E:$' )( ')+, - % $# $# #& F$2& ')+$2& . )+&'! -!*Q' 8 "+" o ))+&(' -:E) &#& &'92-!)+"8" 9- . 2&3" !" #" . -)8',+- %-8 <9-$%$" 9#)G'- $2 ' . 9-, &' 9#)8'- ' . $& S. #-! &(' - 'T!&%!" -3" 3,% Q'#$ . &59,- -) J&#$ -, ''5!&(' . , & -O" !" #" J&' 2)8- .( #& 2&9!" 9& ) &9& $2,#"/+Q'
0 $ "! ! ! ] $ ! !13254 687:98;=<3>@?BAC9$DE<'7 2F2G2F2G2F2F2G2F2G2F2F2G2F2G2F2F2G2G2G2F2G2F2F2G2F2G2F2F2G2F2G2F2F2G2H4JI:K132EL MONEPQ<Q;RD/9=SUTWVYXZ>[]\YA^S`_Q7'9$D/NaN/<'7U73_bPQVc2F2G2F2G2F2F2G2G2G2F2G2F2F2G2F2G2F2F2G2F2G2F2F2G2H4JI'L132Ed ef[]_bN/PQ<Q;RD/9=SBTgVihGjlknm?U73Vc\&9$?B>BVoAC;RD/9$D/pB?`Vq2G2G2G2F2G2F2F2G2F2G2F2F2G2F2G2F2F2G2H4JI3r132]I ef[]_bN/PQ<Q;RD/9=SBTgVisf<:t@?BNE_b9$D/<:7uk<'7:98VJv,wx_b;RN/<y2F2G2G2G2F2G2F2F2G2F2G2F2F2G2F2G2F2F2G2H4&1QL132E1 wx<'7BACN5?BX$D/<'7`X2F2F2G2F2G2F2F2G2F2G2F2F2G2F2G2F2F2G2G2G2F2G2F2F2G2F2G2F2F2G2F2G2F2F2G2H4&zYI
^
− "$% % $ @( 9 > =9
8I$2&) . -$2&#& 9!&(')+"+"8$& .H. $2&#&' U$2 -:& yn
< 9) $2&(' C$, x*'- H. T- 9-, &'50')8$&. #& :0!-)+2 #" !"/ !'$2)+- U.. &)8', f(x|θ) < . 92-2,-+'- θ ∈ Θ
!&I" . - 9& ) & < θ 'J" #)L)%$" =2"E 0'$)8- 'X, 9&#) . #& >. &)G'! π(θ) . $,A)+&' 9!-#& , 9- =. -! Q - & ν #-1" 9 , #-2 #" Θ ')+,10')+$2&4 9Y ) &#& 'F$2& . #-O""+$2) % $# $# .7. &)8',
π(θ|yn) ∝ π(θ)L(yn; θ)
$ L(yn; θ) . ) & "+A:E-2) ,H #"!& W. " 9!&(')8"+"8$& yn
8 9 #)8'- ' $2& -! W ,% Q'#$ . O&#,% -) -, ''5!&(' . ')+, - %!" $# $#O#& F$& ')8$&. )8&(', -,*' Φ(θ)
$# & & &' -5!".< . !&X9& . - . ' . ! #)+"+)' < Φ(θ) = θ<Φ(θ) =
#[X |θ] ! . #-! X. :E)
,$ &9& $# $ Φ(θ) = S(t|θ) $ t ' #& :0!" #- . ' , ! 9-:E) ;' , t # $ ')+,10')8$& φ"+`#" H'-5 . )8')+$2&9& "+" '1-, !"+), $H" % '& % % #(& ! $2 -' $# $" 2)8' . " B -5!& %" $# $#
φ =
∫
Θ
Φ(θ)π(θ|yn) dθ,
=
∫Θ
Φ(θ)L(yn; θ)π(θ) dθ∫Θ L(yn; θ)π(θ) dθ
.
9" $#: &('< V)8&('! T2-5!" R& $2&('19 9"8) )8' 'I)+" 'V)+& . ) &2 #" U. - $9-)+- ( . R,% Q'#$ . . #-$ )8,1!')+$2& ;<-$)J' . ,% Q'#$ . $2&('O'' $2-) , &(' P')8 #" . F$#-&9)8- 9& C$2&#& ') ,10')+$2& . 3)+&'! T-2" ) I,% '#$ . . #-$ )+,10')8$& . )8& . " ! # ' . #" ! ;3) -& % BG * # <F$2& . #-J" . : "+$ , &' E,P'$2') . 92&')G'! )()+&(', T- - + ) V,% Q'#$ . . )8&('! T2-50')8$&1&#,% -) W. ' -, 3'$& 8I$!' < #& /. P''#< ' #)6$106- &' #" P'22'. C$2&?-, #"8'!'S"+$2- d = dim Θ ' ')8' < ,1!) #-! &(' &(' . )+&'2 #)8"+)8', ?&#,% -) ! . 9 * #
0 O!" 2$-)8'9, . 9!&(')8"+"8$&#&9 3 "+$2& . 9 ! * # <9" . #- ,)E+- ,% Q'#$ . T- ' &(' D . #', x(1&9$!'- . - ?. Q' . <C$ D" . $2&9&' yn
$2&(' &32)8 #" &#$2,H #- ' &#-! <E $2&('-2)8- U. J,- '#$ . . 92&')8"+"+$2&#&9 4 !& 9 9)G'- <P&#$ $2&) . -$2& " )+: , &(' . -&#)/+-
"8)+:E- . $ -'65 8J "8" ! 17 # $106- #& 92&#$2-5!,1 $2,#" ' . S,% '#$ . . 9!&(')8"+"8$&#&9 ' $2&')G' ">-! Q - & ,18 #- /. &#$2'- 9#)G'- 3 ) 2&%!"+$2&H" !-') " & F-5!&9!) . #)8"+"+)8& % BG! # #)R$ 0C- #& )8&('-$ . ')+$2& '-!+ "!)+- (4" $2&'- ')+$2& . W,% Q'#$ . < &u :E)8'2&' ;,1 ) , #, " J&#$2'!')+$2&J,10'' ,10') <E!)+&) #&Y -:9Q%#)'$-) U. " 9- . : "8$ , &('J )+& . "
$,A,[#&9P', S 9Y ) &#&
<; F>EJEbF1IF>CbUBHbFeCGF>HGMADPSBUBA EbFF>E#PBRPFoknBHJADI<>FSfMNPBC#\DF>ClADPQE=<?>HbMN\DF>CeSBFV>FV ?fMNnA EGHJF nRUH#m MNV>AD\DA EGF>HO\ M \DF>VoEGUHbF
3,% Q'9$ . . %!&(')+"8"+$2&9&9 #: &(' "+" ,%*, . )+:() - & . 0'! T2$-)
• " E J $A % B !% J)%DA % ! 8 8 # <• " 3,- '#$ . . J%A # BDB $A A % A # B ! $ . /# E %A % E B # A%$#
!&R" . < "+" R:E) &('f( )8,[#" - #&c 9!&(')8"+"8$& θM = (θ1, . . . , θM ) ." . )'-)+ P')+$2& .U. &)8',
π(θ|yn) . !" , &'<P$2 P' &#)+-#&4' " %!&(')+"8"+$2& -, 'O" ')+,10')+$2&
φM =1
M
M∑
i=1
Φ(X |θi)
#) $&(: - : - φ #- 9- , &(' 9!& . M → ∞ < . 9!-W"+X"8$) . -2& . T&#$2,H #- ')8,1!')+$2&. $&('3" #-! ) )8$& P' *'- ')+,- 9!-
σ2φM
=1
M − 1
M∑
i=1
Φ(θi) − φM2.
8 & . !&('< )<" $ 8 ') 'U" ,%*, <#-! )$2& [ . ,% Q'#$ . $&('92- $&'- ')8$&Y'-!+. ) 0U - &(' 4 I,% Q'#$ . 8 8 $2&('I)8', -!')+: ' 9-$ . #) &(' (" )G'! -50')+$2& t 9&o 92&(')+"8"+$2&#&% θM(t)
#) $&' . ?-, !"+)0')8$& . #& 9& Z. >!- 2$0:;!&(' C$#- . ) '-)+ P')8$& '50')+$2&9&9!)+- π(θ|yn)
2" $2-)G'#, . 92&(')+"8"+$2&#&% #-! Q - &(') "9-$ . #) &(' #&o 9!&(')+"+"8$& )+, #"/ "+$2&#& T. )'-)8 P')+$2& # A CE A % B < .O. &)8', J) . 2" , &('i9-$ 3. π(θ|yn)
-'2)8& e9!-') . " 9!&(')+"+"8$& #: &(' &#)8' *'- $2& -:Y <0 #-!+ -! R 9!&(')8"+"8$&#&9 < . ' "+" 9Q$& " 92&(')+"8"+$2& &92"9)8: #-$ )8,1!')+: , &'π(θ|yn)
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π ' . yn
"8) & 'T!)+&) !)8'U: " 9#)8'- 7 <!$ 'V)8&('-$ . #)8' 'f Q' . )/ T" -)G'!+- %-8 #)F$#-&9)G' . ' "8" V)8& . ) 0')+$2& K2-V!)+"8" 9-<2&#$2)$2& #& -'5!)+&=&#$2,H #- ?. 9-$%$)8')+$2& %$#-U#"+) -T"X,% Q'#$ . ,$ . "8) !')+$2&T 9& ) &9& . )+ #"+" . : "8$` 3 9#)8'- F
T'-$2) )E+, 9!-') 3. % #)8'- ' $& -! (T" #')+"8) !')+$2& . " !" 2$-)8'#, CB % # A $A % B ! 8 # <()+&'-$ . 9)G' 9!- 8J B % BG ! 7 # #)I!,% "+)+$2-! 9!- $ % BG ! # ' 8 " % B G ! * # %#-,+ #& U. -) P')+$2& . "7,% Q'#$ . ! !)+&) #& $2,92-2)$2& . $2&'- ')+$2&X: . # &#$f Q' . )+$2& $2&Z#"+) 0')8$& . !&I" X . - T. J,$ . "+)0')8$&V %& ) &#& . )+ #"+" . : "+$B V 9 9)G'- Y& $2,9!-5!) $2& '2)G' <P:E)+A" "8$2) ) " )+& . Y" 3 <%: " O,% Q'#$ . (8 8
"8$& A. 2&U&#$!'- . - A. ' . <6" %!&(')+"8"+$2&yn
' $#: &' $2&')G'' .H. $&#&' W,2& 92&' < $2,#- &%!&(' : &' "+" , &(' . . $2&#&3 J)+& $,#"E+' ! :$2)+- % #)8'- # , $!'$2& z
" . $&#&' ,1!& %!&(' . #-$2 #"/+, ' k(z|yn, θ)
" #- . &)G'! $2& . )8')+$2&9& "+" , &(' (T" 9!&(')8"+"8$&?$ -:Y 2&i#&
+ − "$% % $ $&' ' .U. $&#&' .T. #-, T. :E) &#-, ( . -$)G' <" . $&#&' K,1!& 92&(' z
$2&('V" n− r ' ,.T. T!)+"8"!& #-: &%!&('V #-!+R" K' , . &9- $ -:Y c1, . . . , cn−r
<C$, 1& &' -5!" , &(' )8& . T & . !,, &(' . " B -) & %"+$2-
k(z|yn, θ) =n−r∏
i=1
f(zi|θ)S(ci|θ)
]ci,+∞[(zi)
$ S 'J" F$2& ')+$2& . #-:E) U. 4,$ . +" #& 9Q$2& & &' -5!" <"+ . &)8', . . $&#&' J$2 -:& P'
-)+- $3"[F$2-, ,12-)8&92" fy(yn|θ) =
∫g(yn, z|θ) dz
$ g
' . $2& "+ . &)8', . . $&#&' $2,#"/+Q' ;<$P'I1"8$& . 9#)8'- <!&#$V T $2- . $2&i9& #" )8,C$2-'2&' (H" #-) & $, #' . . $2&9&' 3,2& 9!&(' . 2&3" 3,% Q'9$ . . ')+,10')8$&
@( 4<9 = 0 687 > . ' =449 .<6
. !,3
P'- &#$2'- -, T - & #-)+& ) 92" "8)+:E- U. $ -' 5 8J "+" ! 17 # <!$&'-$#: 3. !& . 9 ! * ##& -! #,% "+2)8- . #-)+& ) 9!" H,% Q'#$ . 8 8 !)+&) /. A- ,1!- . P')8"+) !')+$2& , $H& &!) $&J) ) #&Y #- -5 "
#-)8& ) Y &' -2" . #& ,% Q'#$ . 8 8 '7" 9)8:0!&(' K!-'!&(' . #& Z. )'-)8 P')+$2& .X. &)G'! π0(θ|yn)
<C"/,- '#$ . #-$ . #)8' #& 9& H. >!- 2$0: . -, !"+)0')8$& θ(1), . . . , θ(t), . . . < #)K3C$#- . ) '-)8 P')+$2&'50')8$&#&9!)+- π(θ|yn)
)+&).<#" $2&E: - & U. " ')+,10')+$2&
φT =1
T
T∑
t=1
Φ(θ(t))
: - φ 'R#-, %!- " -2$ . ) )8', . '' 9& 3. >!- 2$0: R" eF$2-, "8" , &'< &A&#$!'5!&(' t0 " ' ,. 9 0 . "+ %& $2)8'" &#$,? #- 7. )G'! -50')8$& t (9!-')+- . " " 9& 7. >2-*$0: P'G*'- $&) . -, &4-! T)8, '50')8$&#&9!)+- $&4
1
(T − t0 + 1)
T∑
t=t0
Φ(θ(t))
p.s.−−→ φ.
Y& ,% Q'#$ . 8 8 ' &(')E+- , &(' . ' -,)+&' 9!-W" 9$2) =. #& A)% E % # A < " $& P' $, 9- & . - $,, " & &' -5!"+) !')+$2&B . - $&')8& . " ,!'-) 4. '-2&)8')+$2& . 9& 9& /.>!- 2$0:c(- Q'50' . ) - ' & P'" 7. &#)+-<#/' , t . "H,% '#$ . <9!-
K (θ, A) = P(θ(t) ∈ A|θ(t−1) = θ
),
=
∫
A
κ(θ, θ) dθ
$ θ 7→ κ(θ, θ)
' #& 7. &)8', #- Θ !. )8' A # %A # # A# 2-,)*" O,% '#$ . 8 8 <9" !" 2$ -)8'9, . ')8& '-$C$2"+) ! ')+& # 'I" !" 2$-)8'#, O. W)+ # ! ,12& 5 ,2& 7 # $2&('" 9" P')8"+) " '$2&(' . $2&#&3 Z"+) . &#$2,H #- !" 2$-)8'9, . -)+:& "+$2) ) " . Y" 3 )8& . Y- 3! 3 9) " 92"G' - % BG 7# -, ' . )8,[#" - . 9& 8 (8 ' . ')+, -H" 92-2,%+Q'- . #&,$ . +" <92-) 92&')8"+"+$2&#&9 S. W)+ #
+ 'M+# !$ % M # #+'
8 'R2" $2-)G'#, #-$ . #)8'#& 9& O. >2-*$0: . &9$ KH
&[!) 2&('K ".< (T" )8', -!')+$2& t→ t+1<
( . Q'5
$2& )8,[#" /. ?-! 2"8) !')+$2& . θ & P')8"+)!&(' #& &#$( ,1!- 2$0:E) & % # B # % # < ./. &)8', κ(θt, .)!- #)G'-2)8- , &'
$2& #-$ + . ( 9& ' . T P'!')+$2& - 8 ' #- X-! 2"8) !')+$2& "8$& #& #-$ 9! 9)8"+)G'! p(θt, .). T & . !&(' 7. κ(θt, .)
"+ & . 9 )G'! -50')+$2& <e9& %!-') 4. A:2" #- )8,[#"/ %( " )8', -50')8$& #-! . &' ' $& -:& ' #& P'- 9!-') '?- ,) ( 8$#- # %3 '' $& . Q'5 &92"0<<" Z 92&(')+"8"+$2& #-$ . 9)G'& $2&('%O# G # G G #-$2 92 #)8"+)8', p(θ, θ) ' $2&'-#)8' H. 9Q$& ( "+/"+$2) )+ #" π(θ|yn)
$)G'X#& "8$))+&E:2-)!&(' %$#-U" %& H. &#$
KH< " $)G'W" #$2) Y. κ(θ, θ) " ' & . 2&('U&' !)+-
" -%$-'
π(θ|yn)/κ(θ|θ)$2)8' $2&#&(`#& $&'!&(' , 9"G') #"+) 0')8: 9-,+ H)+& . & . 2&' 4. θ K$)8-" % #)8'- D. $2 -' 58J "8" ! 17 # C$9-3#" . #-, ) )+$2& )8', -!')+$2& t → t + 1 . " 2" $2-)G'#, -)8'F$2-, "+" , &(' $2,, #)8' &4-5 "+" "A&#$!'50')+$2&
π(θ|yn) = L(yn; θ) π(θ).
'M+# !$ M # #+' ')' 69+
t→ t+ 1!#" 6 5&
θ ∼ κ(θ(t), .) 01" 6 5&
u ∼ U[0,1] ;1" ) )& 5) ) ( ) ( 66')&')' 69 +
p(θ(t), θ) =
min
1,
π(θ|yn)/κ(θ(t), θ)
π(θ(t)|yn)/κ(θ, θ(t))
6π(θ(t)|yn)κ(θ(t), θ) 6= 0,
16++"
")&θ(t+1) =
θ
6u ≤ p(θ(t), θ),
θ(t)6+ +
.
"
;<-,+ "8" , &'< $2& #$))8' κ(θ, θ) = κ(θ)< #) -, ' . )+, 9"8) -A" -)G'#- 4. "+`#-$2 92 #)8"+)8',
. P'!')+$2& . &)8', κ(θ) . $2)8' 2"8$-<,)8, - $2-- ' , &(' " $2,C$2-' , &(' . π(θ|yn)C$9- #- -
#& C$2&9& $&(: - & W. " !" 2$-)8'9,
+ + 'M+# !$ % FNN
9 #)8'- <&#$V:2$& - ,1!- ' " '- '9- T. "+ . )'-)+ P')+$2& % $# $# . 92-2,-+'- (µ, β) .
Y,$ . +" 7. )8 #"8"
µ|yn, β ∼ G(a, b(β)),
β|yn ∼ π(β|yn)
# − "$% % $ -, ''2)G' . )8,#"+) -A"+)+, 9"+!')+$2& . $#" (µ, β)
92&(' . A-! 2"8) !')+$2& . β <K)+" '1!) , . )+, #" - . -, !"+)0')8$& . µ . 9!-A" $2&#&%!) !& #"+) )8' =. π(µ|yn, β)
8 ' #- . ' "8" #-$#-)E ', .7. T & . !& X $2& . )G')8$&#& "+" &('- 9!-5!,%+Q'- " $2& $2&'-#)8'O" !" 2$-)8'9, . W)+ #
R" & &3 -5!" , &('< )*" $2& P') -)+- θ1, . . . , θp#& 9!-')8')+$2& . /: ' 9- . 9!-5!,%+'- θ C$#-3"
,$ . +" ' . )/ < ' )6" $&% P' ` ) - $,#"E+' , &('3" J"+$2) . $2,C$2-' , &' %- $# $# $2& . )8')+$2& & "+"
π(θ1|θ2, . . . , θp,yn),
π(θ2|θ1, θ3 . . . , θp,yn),
. . .
π(θp−1|θ1, . . . , θp−2, θp,yn),
π(θp|θ1, . . . , θp−1,yn)
!"+$2-K$2&% P' )+" , &(' )+, #" -<91%!-1%< . V-! 2"8) 0')+$2& $& . )8')+$2&#& "8" . θ1, . . . , θp4# &/)G'! -5!&('
" #-$ . #&%-2& . &#$2,H #- W. F$)<2" %& U. >!- 2$0: #-$ . 9)G' 92- )+, #"0')8$&R-! TB Q'! J C$9-. ) '-)+ P')8$& '!')+$2&#&92)8- π(θ|yn)
$-, "8" , &('<P" !" 2$-)8'#, -)8' $,A, #)8' ' M# ! N<N ' )' 6+
t→ t+ 1
!@" 6 5&θ(t+1)1 ∼ π(θ1|θ(t)2 , . . . , θ
(t)p ,yn)
0 " j = 2, . . . , p− 1
65&θ(t+1)j ∼ π(θj |θ(t+1)
1 , . . . , θ(t+1)j−1 , θ
(t+1)j+1 , . . . , θ(t)p ,yn)
; " 6 5&θ(t+1)p ∼ π(θp|θ(t+1)
1 , . . . , θ(t+1)p−1 ,yn)
"
8 & . 2&('<!)+"9!--)8: -5!- , &(' " $2&#) ` ) - $2,#"/+Q' , &('K'$P' " K"8$) $2& . )G')8$&#& "+" 3 )3" $& . )8')+$2&#& , &(' P'-*'- $P')+,), . 9Q$& (>$ P' &#)8-" 9" -2& . &#$2,H #- C$)+ #" 4. "+$2) #"+) )G' <)+" ' $9: &('V&3 !)+- T. $9" - 'O!" 2$2-)8'#, ( . Q'5 . ')+& '-$%$"8) C$9-$2 P' &#)+-"+ )+, 9"+!')+$2& . #& "+$2) $& . )8')+$2&#& "8" $2&9&5 (#& $2&'5!&(' , #"8') #"+) 0')+: #-!+ '$P'eF$2-, . $2& 9&Y2" $2-)G'#, E #-) . #).<92-F$)< P'x*'- "+$9- .4. P')+$2& + ) ' % !# %
# & $2&('- %!-') ?. #& - "+!')+: )+"8)8', . )8,#"/ , &('50')+$2& <C" U,% Q'#$ . 8 8 #: &'X#-! &(' -. . )2 9"G'! O)+,%$-'5!&('
$2&E: - & ' ( . )+- " !'' )8&(' S. " #)8"+)+ #- . " %& W. D!- 2$0: #' *Q'- '-!+J" &(' <9-'$P' "8$- π(θ|yn)
' . )2 )+" (W #-$ )+, -#%!-e#& O. &)8', . '-5!&)8')+$2& κ )+" , &(' )8,[#"! #" 9!-J!)+"8" 9-<9& 2)8 9" ,2&#)! #)+"8)8', . " #- )+$2& . π(θ|yn)- & .3 '' X $&E: - & %!-F$2) . ) )+" (
#-$#: -
2-1!)+"8" #-<<,%*, ) $2&E: - & )8" B#< . ' ' -" ' , . 9 "0 'A"+$2)+& . *Q'- :E) . &(' Y& )8,C$2-'2&' "+)G'', -50'#- 'I!)+&) $2& -, ("+?- - O. -)G'!+- . !--!*Q' . R,% Q'#$ . 8 8 K$2)+-J2)8&)6" J"8)+:E- . $ -' ! # ' $2 -' 5 8J "8" ! 7 < 9 # 'J" !-') " W. 8 9#: 5 ) C$2"8' ! #
. !3 . ] $ # + ! ! $ 2 !" % % ! # + ! ! U2" $2-)G'#, . 9!&(')8"+"8$&#&9 #-! Q - &') "0<C$ # E %A % E B # A% < $&(' . T,% Q'#$ . . )
,[#"+!')+$2& #).<B(92-')8- . #& 92&(')+"8"+$2&u# G # G G θM = θ1, . . . , θM9-$0: &92&' . #& H. &)8', W# A CE A % B
$ . /# E " %$A ρ(θ) <%:2$&'#-$C$ -#& ')8,1!')+$2& . φ "# & 0 '<#)
φ =
∫
Θ
Φ(θ)π(θ|yn)
ρ(θ)ρ(θ) dθ,
#& ')8,1!')+$2&Y&90'#- "+" . φ '
φM =M∑
i=1
ωiΦ(θi)! $#
$
ωi =ωi
M∑j=1
ωj
,
ωi = ω(θi|yn) =π(θi|yn)
ρ(θi)=
L(yn; θi) π(θi)
ρ(θi). ! #
# &1#"+) 0')+$2& . )+- ' . "+S"+$2) . 2-5!& . &#$2,H #- < $ " 5C$!''+ 3 (π(θ|yn)) ⊂ 3 (ρ(θ))
' ∫Θ
Φ(θ)π(θ|yn) dθ <∞ <9$2&Y
φMp.s.−−→ φ.
3 ) . #" ∫Θ
Φ2(θ)π(θ|yn)
ρ(θ)π(θ) dθ <∞ <92"8$-
√M φM − φ L−→ N
(0, σ2
φ
)
$ σ2
φ =
∫
Θ
π(θ|yn) dθ
−2∫
Θ
(Φ(θ) − φ)2ω(θ|yn)π(θ|yn) dθ
8 ' . ,% Q'9$ . #'92-+'- )+&Y &#) ,12)F-5 2)+" H. 2& . #-!') <6" ')+,10')+$2& φM '!&('V- & . F$2-' , &(' . & . !&(' T. 1%$) . ωi & #'I-! #, -R)+&F$2-, "+" , &('1#&1 C$2& #$) 1. ρ
• ρ(θ) .$2)8'R- ' -1#-$ O. π(θ|yn)
'$P' &c '2&' )+" ( )+, #" - . !& <!" C$2) . ωi- ' &('
- "+!')+: , &') 9)8"+)8 9-, 'O-, . #) &('3":0!-)+2 #)+"8)8', .! #
• "+ . &)8', ρ(θ) . $2)8'C$, . - . O4,$)8&O)*"+$#- . "+" . π(θ|yn)
;<$P'O" 2-' . " 9!&(')8"+"8$&#&9 9-, T - &(') " $2&)' . $& ( 9$2) )+- & , &' 9& . &)8', ρ #)0< & #" . :& -) -T" $& . )8')+$2& ) . < . $2)8' 9-, &' - . 2- '! -)') . "+) 0' F(Z:Y -) - ! )8"iP' %!- ,#" Q'2 #"+)8- . -)G'!+- . #-$ )8,)8', T: π(θ|yn)
< . &)8', " $2&1& $&#&9+'1% #"8) )8' , &(' # 8 #) #"+) " 9!&(')8"+"8$&#&9 #-! Q - &(') " !)8' Q'! " . "!) " <9!-" 9, <# #-$ '. K,% Q'#$ . 8 8 8 & . 2&'< . ) 0B - &(' V,% Q'9$ . . $2&'- ')+$2& . ρ(θ) $2&(' Q'! #-$C$, & P'" , T%!- - & . !', 2$2-)
− "$% % $ H,- '#$ . % #(& <<$ ρ(θ) ' . Q' -,)+&' 4. 9Q$2& #&#) & 9-, !"! #" (>" )8,[#"+!')+$2&
K!-,) "+" <6"Z,% Q'#$ . . !' R# !J # B # J" ? %: # <@#-$C$, %!- , 3'$2& ! # ', 3'$& 5 ' -* ! 7 # $2&'-#)8' ρ(θ) . 9$2&4&#$2& 92-2,- '-) <9!-#- &(') ( 92-')8-. " )+&F$2-,10')8$&Z C$2-', 92-
yn
7. %'5 8 " ! * # $2&(' C$2)+&(', U" . )2 9"G'! V'' $2-) ' $2& -,+' - & $&'-, %!-" #')+"8) !')+$2& . '' ,- '#$ . " $&('#-$C$, l(1"+3#"+ " !" 2$2-)8'#, . ! ? % # %$A % # $A % # E # + % # A# < #)72)G'O" $ 8 ' . 9& Q' . -)8') S. !&I" ')+$2&#)+:2&(' 8 " 9) o ) ' )+,#" . P')8"+) !')+$2& <*-! -:& A ,$ . +" Z( . $&#&' W,1!& 92&' '& ., ' 9 . % $# $#R)+, 9-$#- " ,H #" X. $2& . P', X 9-$ #"E+, 9Y ) &S)+& . '-) "?,1!) - & $2&('- S. 9-$ #"E+, I'' $2-) 'T#"+) '
,% Q'9$ . % % % # <$ ρ(θ) ' $&'-#)G' %!- )G'! -50')8$& )8: < & F$& ')8$& . " . '50')+$2& . )+, 9"+!')+$2& $#-5!&(' θ1, . . . , θM ∼ ρ(θ)
(;"+ . &)8', )8 9" π(θ|yn) Y& ' .>.
" 2" $2-)G'#, 8 ! : CB % # $A A % B # '!)+&)O#-$C$, § 7
; -5 . )8')+$2&9& "+" , &('<P" O2" $2-)G'#, . 92&')8"+"+$2&#&9 #-! Q - &(') " $&'$ )/ ([9& Q'5 7. -, 92&(')+"8"+$2&#&% %!- . &)G'! . #-! Q - & <%&#$!'! 3 ! % E B # A E %A % E B # A% < # #)+& 5 # < #)i -, ' . $ P' &#)+-F( 9!-')+- . θM ∼ ρ(θ)
#& 92&')8"+"+$2& θP <6&#$& # G # GG < . '!)+"+" P < M #) 9)G'T#-$ )+,10')8: , &(' π(θ|yn)
& 9-$ + .. "H,12&#)E+- #)+:2&('
' M# ! 7
!@")97) 67& 5&6ωi
&+ωi = ωi/
M∑
j=1
ωj"
0 " / 6 5& 5& 6+ 767)'& (i1, . . . , iM ) ∈ INM &7+ 5) 76('76+l6f) 5& MM (.|ω1, . . . , ωM )
'& )& M∑k=1
ik = P"
; " / 5&)f' 69+5+&(θ1, . . . , θP ) = (θ1, . . . , θ1︸ ︷︷ ︸
i1 fois
, θ2, . . . , θ2︸ ︷︷ ︸i2 fois
, . . . , θM , . . . , θM︸ ︷︷ ︸iM fois
)"
8 2)!&('<592-I" #"+) 0')8$& . "7"+$2) . 1-2& . R&9$2,? 9- < θP #)8'O #-$ )8,1!')+: , &' π(θ|yn)"8$-
M/P → ∞ # #)+& ! #1 $2&) . +- ! 2& 8>') !')+$2&4'3 $2-) $# 9&=-%$-' M/P = 20 ' Y &' -2" , &(' !&(' " !&3" O #"+) 0')8$&<P$2& #$2) )+-5 M/P = 30
6)3 6 # # 1B E J J)%A # BDB A A% A # B % " A B 7% # A # %$A A J)%$A # BDB A # E %A # B # % B B & A # E # π(θ)
A π∗(θ) )%
E B " A A" # # B # G # &
π∗(θ|yn) ∝ π∗(θ)
π(θ)π(θ|yn),
# B % # A # A A B # E % % $# $# A E # %A B # ωi A
ω∗i =
π∗(θi)
π(θi)ωi.
# A" # A % E A B # B # % # A % $# $# A % #(& " # E % $# $# A A # A # A G
@( *.<4<9 = 0 6 6 5>56# = = 56
+ + . ! !#$ . ! !" 2$-)8'9, . #-$%$ " $2&'- ')+$2& . " . &)8', S)+&'-9, &('!" ρ(θ) ( 9!-')+- . " )8&F$2-,
')8$&> C$2-', 9!-" . $2&#&3 yn
,1!) & #')+"8) 2&('T )<" )+&F$2-,10')+$2& % $# $#L,$ . "+), %!- π(θ) "
& ' 9"8) 2 #" . 2&O" . - . O,$ . +" )( . $2&9&' O,1!& 92&' 8 . $&#&' 3,1!& 92&(' $2&('- $2&')8'' .( 9!-')+- . #& )8,[#"+!')+$2&/,2-)8&%!" %! $# $# < #)*)+, 9"8) π . $)G')*'- 9-$#- #-)+& ) A. " !" 2$2-)8'#, 'W" 9)8:0!&('
&>,1 )+,A) !&('U"X:E-5!) ,? #"!& 7. M 9!&(')8"+"8$&- $2&') '' <#$2&=$ P') &('T2"8$-#& 9!&(')8"+"8$& θM . ')+,10')+$2& . θ
# & #$))!&(' ρ(θ) $,A, " . &)8', .θM
! $ $,A, 9& ')+,10' #-T&9$2& 92-2,% Q'-) 7.[ '' H. &)G'! # < ρ(θ) 'W)8&('#)G')8: , &(' P')+ #" . *Q'- #-$ . π(θ|yn)
3 "+$2&4" #'x(10'' )+& . - <P$& #' $#" - '' ,- '#$ . : W. ,- '#$ . . -! R 9!&(')+"+"8$&9 $#" . 3 'T!)+&) $2& )8"+"E XC$#-O$ P' &#)+- . O-! 2"8) 0')+$2& . "A"8$) % $# $# K$2)+- . 9 ! *#< 9 9)G'- # C$#-#" . #-! ) )8$& #-O" #$) ( 0 ' -
3 $)G'z" . $2&9&' V,1!& %!&(' 4 !" 2$2-)8'#, -)G'3!"+$2-<: " V&#$2'!')+$2&1#-, ) " . !&I" )+&'-$
. ')+$2& . X % #)8'- <
!#" i = 1, . . . ,M
!#"$ 65&θi ∼ π(θ)
!%&$ 65&zi ∼ k(z|yn, θi)
! , $ ))5&θi = arg max
Θg(yn, zj|θ)
"01"('f+ B9)676B'
ρ$ & 5) &+6B' & ' 6$)*)&)
θM = (θ1, . . . , θM )"
# & & &' -5!".<C"+ . &)8', ρ & ' 9 #"8) )8' 'X9& ')+,!')+$2&;&#$2&`9!-5!,% Q'-) ρ 'S&3 !)+- 8 2)!&('<#$2& "8) S"1-! T92-')G')8$& . ) -!+Q' ?. T:0!" #- θ1, . . . , θM
9!-U" 9)+2) . F$2& ')+$2& $&(')+&5 &('-! #-O" θi
H' #&#) P')+"8) , '"H,% '#$ .7. 3&#$ ) & #) ')8, ρ(θ) 92-
ρ(θ) =1
M |F |
M∑
i=1
N(F−1(θi − θ)
) ! #
$ F '#& ,10'-) X !--! &#$&)8&#"+)E+- W. $2- . - d = dim Θ ' N(x)
'J" . &)G'! . "H"8$) N (0, Id): Id " ) . &(')8', & . )+, &)8$& d 1 U,!'-) F ' #$)) "8$&?" 9-)+& ) J. A"+) , )8,[#, ! ; -- " #+ #) - (H,)+&#)+,A) -
#[(∫
Θ
ρ(θ) dθ − ρ(θ)
)2].
C$2& &(') " ' #& #')+$2& < ρ '!&(' #"+) )8' < ' !)8'7" $2 "8 ' . " $ 9!-') #)+:0!&(' " -, ' . ,$2&('- - " !-5 '!+- )8&('9)G') " . " !" 2$-)8'#, P'c*Q'- '-$2, #- ' ρ & ',1!" #- , &(' %I'$8$#- #&Z C$2& #$) X.. &)8', . )8,C$2-'2& !)8'<P" -$ . Q #' . . ' . - & . - " C$2) . ωi
. "+F$-,[#" ! # 9!-F$2) 3'-!+ . T & . !&(' . T:0!" #- θM 3 %$$& %!- ,#" )+" 2)G' ./. $2&#&3 S,12& 9!&(' $ 2& 3 <L" #
θi$&''-,+ #-$ S" 9&
. HP'- < '1" . ρ(θ) -) &(' . *Q'- #" A"E &+- "+" . π(θ|yn) 8 3 &#$,-+&
'U2,#"8) %!-T" #$) =. &9$ ) & "# & $2&" & <9"1:0!-)!& ,#)+-) ?. " ')+,!')+$2&-) S. *'- ! )+: , &(' 2)8 #"
− $ + '"! ,3 ! % $ ' + 3
& . $& . , #- -3" . ) #"8', O'' $-) . . <#&#$. Q' . )8$& $2&Y 9"8) !')+$2&= 4,$ . +" C$2& &') " " 2)8' . #& *'-,+ )+,#" <%$#-L" "(" . &)8', ρ(θ) . (θj)
' #"+) )G' '<& &' )G' 9 . $2&'-#)+- #& ')+,10')8$&H&#$& 92-2,% Q'-) "+$2- π ' $2&8>' . ) & &(' & . <0&#$ & :2$&&#"V $2)+& . #& !" 2$-)8'9, X. 9!&(')+"+"8$&#&9 #) π(θ|yn) 'H!"+$2- &(')E+- , &(' $&#& 'A!) " , &(' )+, #"! #" , $,$2&('-$2& #- )8,#" ρ :Y+- #& !& . ) . !' . !& - 2-< C$2-&' 9- Θ
3 $2)8'f(x|λ) " . &)8', . 4,$ . +" %$& &(') " E(λ)
&%9"+ 9- λ 9& %! $# $# $2&8>' G(a, b) .. &)8',
π(λ) =ba
Γ(a)λa−1 exp (−bλ)
λ>0.
& C$ ( " 92&(')+"8"+$2&B$ -:& yn
$2&(') &' 0 ≤ r < n:0!" #- (x1, . . . , xr)
&#$2& &#-! 'cn−r = (c1, . . . , cn−r)
:0!" 9- &#-! ! ' # "8$-< &Y&#$2'2&' t(yn) =∑n
i=1 yi<#$2&Y
λi =n
t(yn) + ti
$ ti =
∑n−rk=1 z
ik '< . 92-O"+ #-$9-)/ Q'! . 2 & . ,% ,$)8- <
zk|λ ∼ E(λ),
ti|λ ∼ G(n− r, λ).
. &)8', )+& $2& . )8')+$2&9& "+" . ti 'T2"8$-
fT (t) =
∫
IR+
fT (t|λ)π(λ) dλ,
=Γ(n− r + a)ba
Γ(n− r)Γ(a) (b+ t)a+n−r tn−r−1.
K!-#& )8,#" 9!& , &(' . :0!-)+2 #" <P$2&/'-$9:
ρ(λ) =Γ(n− r + a)
Γ(n− r)Γ(a)
((n− t(yn)λ)
−1
(n− (t(yn) − b)λ)a
)n−r
nbaλa−1 0≤λ<n/t(yn).
. &)8', . # - $2&')8'' ' . $2& C$2-&' #-[0, n/t(yn)[
< #)K)+& . 9)G' " . &)G'! o )+ #" -, "+" S. " 9!&(')8"+"8$&#&9 'O"A- '-) ')8$& . π(λ|yn)
#-[0, n/t(yn)[
)8&).<ρ- #-, &(' '' $-)
, &(' #&4,#:0!) #$2) /.7. &)G'! . )8,C$2-'!& 8 & . 2&('< 7. T P'W-, #"8' H. !)8' " $& #-)8:E)+"E 2) "+ $2&#&%!) !& . " . &)8', ρ(θ) ('$P'
- $&'- ')+$2&D&#$2&`9!-5!,% Q'-) 8 '' " '#- , $ . " !" 2$-)8'9, -, ' -' . , ''- &:0!" #- "F-5 )8"+)8', . . <!,1!) $2& P'R)8,12)+& -i#&A %$& 9$2) ?. &#$ &#$2&A C$2-&' <0: J. #'! 9" < #) -, ''-5!)8' . :0 - '' ;. ) #"8', >'' $-) P')+"8) 0')+$2& . &#$ ( ) &. !& ! # 2&'U' & . !& ( Y - -W"+ $2& &('-50')8$& . # - $&')8'' < . S&#$ D. 8J 7.. &)8',
ρC(θ|α, θ0) =(απ
) 1
(θ − θ0)2 + α2
9!-5!) &' P')8 9" . !,% "+)8$- -S" ')+,10')+$2& &#$& 9!-5!,% Q'-) . ρ $# & 0 '<<" "+$2) . 8J $&(' $2&9&5 C$#-V& :$2)+-K&#) B -5!& &#)%:2-)!& < 'V" V:2-)! 9" . 8J 7H& $2 B ) &(' 9R '' $ -,+, &('-5!"K"+)+,A)8' L"+ #-$2 %! #)+"8)8', . $2 #' &9)8- . S:0!" #- "+$2) 2&3 . θ0 ' &`2)G''-$ " :& C$9-
-, ''- (X"X,$ &#& , 9)8-) H.[ $2&E: - - )+&).< '' "8$) !-5 '! -) 92-#& F$2-'x Q'5!" , &(' ' . J'-,+ 3"8$#- . 63 6 % % # BDB % # A & ,% J)%: # - A% $# $#x# A $# B π %A B % # J $A A G # A # E E B A B !##$ * A # % # !& B ρ A B %! ? %A B % A A # B E # E A A % B E A B % E % % # B % E %: π(.|yn) G
#& 9$2&Y &' -2" <0" P')+"+)0')8$& . . ' )+, 9" ! ' %$)8 9" , &('I!,% "+)+$2-5! #" $# <0,2) $&2-$. T P' ' . F$#-&#)+-%$2' &(') "8" , &(' . iC$2) . ωi
'-$ . , #)+"8)+ #-! &[F$& ')8$& . " "+$2) 2& , &(' &('- π ' yn
& 9-$%$ 7. $& 7. #-, . -" ,#"+$2) . . . !" #" . -)G'!+- 8 . &#)< 9 9)G'- 7 < #) ,? 9" #) &/ . #', S 3#-$2 9"E+, 3 )48!,1!) 8
> 1$ )6" C$9- &(' U.W. $&#&' &#-!
'2)8 #" <2$2&#-$ -)G'K" P')8"+) !')+$2& . . ' 9&X P'- #$) .. &)8', . )+, C$2-'2& . $2)8' *Q'- !)8' & 2, & :E) " . "#-$ )+,)G'! &('- )8&F$2-,!')+$2& %! $# $# 'T$2 8 ')8: #' - "+$2- " . )8, & )8$& . θ '- '- )+&(' ( $ !& . .7. )+, &)8$&B -) #- < "+" S. :E) &(',A$)8&. :E) . &(' (#-! T & . -< . $ 4" P')+"+)G'! . 8
! . ! 3 + + 3 % ' ! $ ! ! . ! % % ! ! !& " . - R. 9#)G'- <$&S . &#) π(β)
#-l#& $,9 ' [βl, βr]C$9-L" ,A$ . +" I. )+ #"+" 2&
,%*, 9#)G'- <!&#$2)$2& #& #$2) % $# $##&#$ -, ''!&(' . $ P' &#)8- #"+) )G' , &(' π(µ|β,yn)
µ|β ∼ G(a, b(a, β)),
β ∼ Be(p, q). &#) #- [βl, βr].
)8&).<#$2& #- "+ . &)G'! )8&'-#, &('5!"
ρR(µ, β) = π(θ′|β,yn)ρR(β),
β ∼ U[βl,βr]
- ' " $2& . )8')+$2& $92)G'! 92- ! 7# ' $2& -: . K'$8$#-1#" V"+$9- . π(θ|yn)
&Y!"+$2-ω(θ′, β|yn) = ω(β|yn)
'
ω(β|yn) = βr
(r∏
i=1
xi
)β
(β − βl)p−1(βr − β)q−1 ba(a, β)
(b(a, β) +
n∑i=1
yβi
)a+rβl≤β≤βr.
8 9$2) . -, T - & &#$ -, ' . $2& <('-5: - . #"8) !')+$2&V&#,% -) < . #-! ) -V" $, C$2- ' , &(' . " . &)G'! )8&'-#, &'5!" #-$%$" %!- .
$ %&$ +#3 3 & )+, 9" o 9!&(')8"+"8$& . '5!)+"8" n = 20
"+$2& " ,A$ . +" X. )8 9"8" . 9!-5!,%+'- (η0, β0) =
(112, 3) ! $2)8' . ` -2& ' . :2-)!& 362 # &- ,#"+ n−r :02" #- . ' 92&')8"+"+$2& 9!- 9& &#- )+, #"/ "+$2&A"+S"+$2)9&#) F$-, #- [110, 130]
9!'- %$#- &(' . &#- -$2&(' $2&) . -!
I < I < I < I & [βl, βr] = [1, 5] 'p = q
)+&)0<π(β) '?'$8$#- &'-, #-
β0 &
− $
#$2) )G' . :0!" 9- p = 1.5 ' p = 15< #) &('-5& σβ = 1 ' σβ = 0.36
" P'- 9!-'<9$2& %$ $&#&9'- "+,% . )+2& ,2-)8&%!" %! $# $# xe
& P')+"+) . $2& < . #-!+J" 9 9)G'- Z <P"[F$2-, #" b(a, β) = (21/a − 1)−1xβ
e .
& #$2) )8' xe = 100< #) ' %!-!)8' , &' &> $2- . : " . $2&#&3 < ' xe = 50
< #)L'! ,$2) 2& . #& -') '-,+ )8,) ' "# & & < a $-- %$& . 2&'.( #& '2)8"+" 7. 9!&(')8"+"8$& 7 ') 4<#$&, " ')+$2&#& a = 3 ' a = 10
!&J" W. )E+, < )+"X F$-' &#- <(" %- $# $#CC$2-' 9& )+&F$-,10')8$& #-θ′#" 3)8,C$2-'2&' " . $2&#&3
2&S" U'! 9" ' <C$2& #-! &(' %$#- 9 #$2) >.1. &)8', A)8&'-#, &'5!" < "+$2& 9" ) 9- !-5 ', -) ') . π <5#&1'-) #" ' . :0!" #- $-- C$2& . 2&('V ')+,10')+$2& . >$2&(' 8J2-"+$ . ". #-! S. :E) ,$ &9& %!" $# $# ! " B -5!& 9-, . ) ')+: 9- X # <E&#$!'! D < . $2&W 2-' ' σD
'O" %$#- &('5 . #-:E) #-! . ) ') ( t0 = 130 ! $2)8'I"7:0!" #-I,1 )8,12" T. " &9- # <2&9$!', S130
8 ') ,10')+$2& $2&(' 2" #"/ I #-!+ #& ' 3 #-V" )+, #"0')8$& . . ) 0U - &(' . &)G'! I)8&'-#, &('5!" <(92-')8- . M = 104 92-') #" ')8'- H. $2,92-2)$2& <#$2&>)+& . ) 2" , &('T" ')+,10')8$&>,$ &9& . 9!&(')8', 9!-O,1 )+,)0')8$& . "A:E-2) ,H #"+2& ! # #
9Q$& 9) *'! ,10') <6"+Y,% Q'#$ . . $ ')8, σD 8I$,, 9-, :5 < '' ')8,1!')+$2&
P'.*Q'- F$-' , &(' --$&' "+$2- "+ &9- '3)+, C$2-'2&(' ' " %- $# $# ' "+$2) 2&' . . $2&9&' !J '! 9" < +, "8) 2& $# ')8,10')8$& . D $ 06- "+" ) . $2& )G'C$!' &(') " &('- S. $2&9&' ' %' $# $# $2- " %' $# $# '!)+ #" , &('S)8&F$-,1!') V:E) ( :E) . . $2&#&' ! a = 3
<σβ = 1 # <C" ')
,10')+$2& . D%!- . - ' '-!+ . T & . 2&(' 7. xe = 50
K!-U!)+"8" #-< . QP' . ')8,1!')+$2& ' .#-$ )+,)G'! : π(θ|yn)
)8& . #) &(' . 2& ' ,#" . J'5 Y. 9-:E) #-! :0!" '
a σβ & 8 ρR % ('& %. / , *1* / %0 %'* **( -(%( ** ( -( %*
,( * / 0%0 -) * / ) -( -) * / . -. -, *( * , * ((%, / * *('* %1% / * *(1(%2 / * / , *1*, %0 %, ** / %, %'* **1* - / %*
,( * %% -% , / * // %1) - / * / % -0,* ;<-) #" ' (D, σD, S130)
%$#- #& %! $# $# &> $- . : " . $2&#&' ! xe = 100 # ). $ 0C- #& ')+,10')+$2&=-5!) $2&#&92 #" S. %!&(')8', %6" $# $# <6:E) ( :E) . " ')+,!')+$2& . -, T - & !. -&9)E+- $2"+$2&#& #
a σβ & 8 ρR % * / , *1*%%0 %% ** / %2 % / *(1) -) %1(
,( * / 2 -( -) * / - %--- * / ).'* -% 21, * . / ,1% 21( *%)%, .( *%1% ** - . /
('& %. / , *1** %1- %( ** / %2 % / *(, -( / 2 ,( * %( - / -0 * / 2 %1) -2 * / % -) - / 21, *,1) -2 2* *,1( -2 .1- *,- .'* .*
*( * , * ( / %, // *(1- %10 / , )1. -, * . / , *1* - %2 %% **, %10%1- 02,, / - ,( * / 0 - * -) *%'* ,( -) ** 0)'* %1.
;<-) #" ' (D, σD , S130)C$9- 9& `#-)8$-) . !"/ Y: " . $2&9&' ! xe = 50 # /. $
')8, F$-' , &(' σD. !& . &#$2,H #- < 'S" ')+,10')8$& . "+ . #-! H. :E) D '92-F$) --$&'
! A. -&#)E+- J"8) 2& # <#:E) ( :() . " ')+,10')8$& . -! Q - & ! . -&#)/+- X $2"+$2&#& $#
+ − $ .<4 <9 = 0 6 9 2 54. =9 9 6
+ + - - . #& %$&#& 7. &)8', ?)8&'-#, &('5!" ρ(θ) 'T" &8 . '' ')+$2& , $T" :2$&J:#-! . ,, &('<" #$) X. ρ(θ) 'I"+$2)+& . *Q'- :E) . &('<(,1!" 2-! T" )8,#"+) )G'! . $2& . )G')8$& . 8J B % ! 17 # <$" &#$, . 2" $2-)G'#, 8 ! % # % # <*$2&('#-$C$, . )8&('-$
. #)+- #& Y. )+, &)8$&B)G'! -50')8: /. 2&?" 9-$ . ')+$2& . 9!&(')+"+"8$& 9-, T - &(') " . 2&H" P' . . P' -ρ(θ)
(Z"+ . &)G'! )8 #" π(θ|yn) 8 '' . P'50')+$2& 'W)8,#"/ , &('! H. 9$2& , &') "8"
%t .
" !" 2$-)8'9, < M -, !"+)0')8$& . θ $&' )+, #"/ (92-')8- . #& H. &)G'! 7)+&'-9, &('!" $#-5!&(' 7. T & . !&(' . M
-! !"+) !')+$2& . %9 t− 1 " Q'5 &%!" t = T
<(" M . -&#)/+- V-! 2"8) 0')+$2& $&(' P')8"+), %$#- ')+, - φM
"+$2&4" 9- )8$& ! $#
c#- ' (?, #- S. J)8', -50')8$&<" %$&Z$/" ,1 #:02) $,%$-' , &(' . " . &)8', S)8&'-#, &'5!" $#-2&(' ' 8>& O(A'-5: -J"+ . )'-)+ P')+$2& . C$2) . . )+, C$2-'2& < 'U) . !" , &('<#"+ F$-, ?. ρ(θ) -#-$ . "+" W. π(θ|yn)
:2&(' . '' ,% Q'#$ . ' . -, ''- #& 2-5!& . Y &' -2"8)8', . #$2) )8&'-#, &'5 ( 9 9 . " !" 2$2-)8'#, < ' . ! C$P')8- C$!' &(') "+" , &(' (%#& , )8"+" #- )8', " I2" $2-)G'#, ") . 9!&(')8"+"8$&#&9 #-! Q - &(') " K!-J!)+"8" 9-<9&%2)8&1)+,%$-'5!&(' ."+H,% Q'#$ . 'J" $ P'#" P' 2!'!)+ #" W. )8', -50')8$& & t :0!&(' . %!-: &9)8- (?" '2 #)+"8)8', . "+ . ) '-)+ P')8$&)8&'-#, &'5!"
&1'-$#: U. 2&I" !-') " T. #)+"8"+)8& % ! # #&Z-! #,% O'-!+ "!)+- . " $2&'- ')8$& . 8 '#& $2,92-2)$2& . " 2" $2-)G'#, : " O,% '#$ . 8 8#" C$2& $2,%$-' , &(' . 8 :E) ( :E) . -, #"8'!'%!-F$2) O)8&!') >2)!&(' . ,% '#$ . 8 8 'U2)8&)L,) &=" #,)/+- $#- . '' ')8$& <2&#$ #-$ . 9)8-$2& " $2,92-2)$2& &59,- -) < . !&V&#$!'- $2&(' ' W. ' . )+& . '-) "8" < #)*)8"+" '- -$&('3"a( )*" , )+"+" #- $, C$2-' , &(' . 8
!#"%$!&'#)( * "8$- .,+ 'X#& !" 2$-)8'#, A. 9!&(')8"+"8$&#&9 '!') !- 8 '
%. #(& )+"T2)8' . ,1!&9)E+- . P'!')+: , $: --$& #& ')8-5 (% $# $# '1&' 2)8- C$9-
8 & . !&(' " #) o )%& -' (" )+&#)8')!"+)0')8$& . " 2" $2-)G'#, & #' . $& P')8"+) - 8LC$9- ')+, - #& 2-5!& . #- % $# $# - %$#-J"+ "+" " %- $# $# '3&#$2&Z)8&F$-,1!') - (?"+ . ) 0B - & S. .,+ /)8" ' . )+, #" -V)8&#)8')!" , &' #& "+$2)2)8 #" , &'I)8&F$-,1!')+: K2-V!)+"8" 9- - .,+ & ' P')+"8) ! #" &#-! & W.7. $2&#&' 3,1!& %!&(' 0P" P')8"+) !')+$2& . 8 & '93- '- )+&('
1 32 32 465789YM 3:; )8,#"/ ,<&('50')+$2& . <S" 2" $2-)G'#,<>= !-') #"?< P'$#- . <" )G'! -50')+$2& . <@=. Q'7<@=A=#)8:02&'B<C=
$&=)+, #"?< M :02-)! #"?<C=K2"E 0'$2)+-D<@= ! $ % # # =#)+:2&(' #&E< . <&F=)8', #$2)G=)?< % $# $#P$ $#-5!&('B<H $& 2" #"?<I"?< C$2) . = . < 9 F<192-') #"<,=#)8:02&' !3I # <'K$2&?-! R 9!&(')8"+"8$&#&E< %!-,A)E"?<C= M 92-') #"?<C= -#-$C$2-')+$2&9&E<"+"<,<&(' ( <C=C$2) . = - %$#-O$2 #'D<&9)8-#&Y&#$#:!<"@ 92&(')+"8"+$2&
&B&#$!'B< FM,t" <&7=J<,? #"?< . <@= 92-') #"?<C= θ1,t, . . . , θM,t
<' . <@= C$2) . = ω1,t, . . . , ωM,t !" #"/ C=O(=" )8',
-!')+$2&t .
<X" !" 2$-)8'9,K< 8<C=M
9!-') #"?<C=L=$&'72)8&F=)M=)8,[#"/ N<C=L=J<"8$& M . )G='-)+ P')8$&F=)+&F='-9,K<&('!"?<C=ρi,t(θ)
! i = 1, . . . ,M # -(. <&F=)G'! C= 9)'=$2&(' . $& $2&F='-#)8'D<@== 92&(' FM,t−14# &O<0'<' - '$#'D<C=I"?<C=I)8&F$-,1
')+$2&F= . %!&(')+"8"+$2&9&9 P<A<Q' . <W-, ^ 92&')8"+"+$2&#&9!<x(H" )8', -!')+$2& t− 1 .<S" !" 2$-)8'#,<Q=$2&('%$2'D<&')<"8"?<,<&'
2E1
G=<@=X B<ρ(θ)
( !! "#$t $F=% $ B<&<'$(D< <C=( <'$F=!! C= ρi,t(θ)
< G=$) <* $$G=D=$ <!=B=" C< =*+&,?=, N<X- FM,t
<". "8</ § I 7 7
0 132 5F 5476 8 I >*98 2 3:; !#";: '&$=<
t = 0 4)6&+ B9)6 %! $# $# &
θ1,0, . . . , θM,0"
01";: '&$=<t = 1, . . . , T
) i = 1, . . . ,M
! ) $ <6 5&θi,t ∼ ρi,t(θ)
&' ) )$5&ωi,t =
L(yn; θi,t) π(θi,t)
ρi,t(θi,t)
! ( $ ) )$5& &-<6< +9 ) 6<-<ωi
&' B9)+' 677++&95&-< )' 65&-< )+'&-<&+ ' 66<)+' 5&-< 6<
ωi ! $ +=<B'7O6&
ρi,t+1< )B9)+' FM,t
"
$%<@=%&>D<T-/!=*?& E<'&K<'$? =$7=@?=E< φ <C=AB =@$$3
φM,T =1
T
T∑
t=1
M∑
i=1
ωi,t φ(θi,t).
C#$ <D < < <@=%&>D<T-@' ,<$ T <M E@?<T-B= - F t = 1, . . . , T - =7=G <@= $-
$F= '<THB<C=A C!<C= - ∑Mi=1 ωi,tΦ(θi,t)
$+C!<'P< <'$ --!! >C!<'B= φ < -$I' '!+&< <'$E&J!B<= E< -$- M → ∞ K L %? ! I#M
1 2 + 5 #"89$ :; $ 4 #" N-&K<<@=%(&J <$H8%L$?OL-$E<c " '<> I<@=($$' N<@=O&>$ L-$B<C= z
=$?*B< $F=AL' C<C=C+ =&PL$ L$?D<> <C= -"&%+D<@= M 8<'<'L % ! * # =J<'$?OL-$E<;C!<'B=$QE< 8 L&J&K<BL$c N-$?$$ !</E<"R-F=H<'$L-?=$?S?<C=
z<S?<C=
θ=J<'$J<'LB=GG=%L$F= $-$-$E<<@=
B<C=7< CP<C= M $ T3A<$ <0'</<(U"N L$F<*$LCP<<* <'$F=!! *$F=AL&<$?"?<
ρJi,t(θ, z) = k(z|yn, θi,t−1) ρi,t(θ|z) !3IM 7 #
ρi,t(θ|z)
.*B<DV-U"NE<R <π(θ|yn, z)
M <OVE=F$$$&>G=" = 'U = ! C@ $LUW$XLU"N-VB<DV,L?<C=/$"#$7= #
ωi,t =g(yn, zt,i|θi,t)π(θi,t)
k(zt,i|yn, θi,t−1)ρi,t(θi,t|zi,t). ! IM I#
<@ YN&<OVB<$- PT5&<>=AL-C#$?B< M
0 132 5F 5476 8 * M >*98[ZI\ 5+ ##" 89$ :; $ 4 #" :?]_^`7a-b=<
t = 1, . . . , T
ced>bf]gi = 1, . . . ,M
!h $ <ia-]-j?`glk`nm h-o fpqi7pk b`pk h p+^`zi,t ∼ k(z|yn, θi,t−1)
`^θi,t ∼ ρi,t (θ|zi,t)
!%r$tsh j s ]j?`g_j`lb-f-i#k< !"u d u $
2E1 I −
d>g s-h p^=ij+j?fp+p`g j?`-< b h g+^,i s ]j?`< s f]g h p^`-< <`j?fp j`-< b-fi#k< p-fga h j-i<-< d s fp=< ^+g?] i g`
ρi,t+1< hs-h p^ FM,t
b-f]gi = 1, . . . ,M
!s mGd § u dFdc d $ d
LUN<$F=OV-E+'&<C= - )B<'U$F=AL$H <C=P$-$' N<@=*&>$?L-$B<C=O$ <C=AV-P=*! 'NB<>VL(L$F<$$E< <@=%&>#$J-& @=A?<$$F< - V-H< <&JV?< # / <@= YN&<@=E D!=A$YP= <VG=3UWT M <?<L M ?=DY& '$' "?<&<$? - U <AD<K<C=A&>#$ <C=AR;U< - CD<*&JV=D=< - < =D<VC!<R <C=DV,P=B=A, @= <*U&.V-E $ L$)&.:+'<Z(J$$' N<@=&>$?L-$?D<C=@LX+V'<
z|yn, θ ∼ k(z|yn, θ) ∝ g(yn, z|θ),?L=T3U!B<*<(U'UL E<C=@V,E= - <C=A&J&>Y!<7?< M
1 32 \ ; 4C 5 \ &E$ $ ! $ $E5 1 $EL 131 "@$ 4C 5 JY"$ <*C#, ( <@=VF= IM I - UUN<$?L' C<*V-BJV! C=D<$U <* <$$' C<C=&>$L$?D<@=A=A&L-E C<C= -
V7<'L *D<*&J$+L' C<><$ U$F=: '$?FL$F<(VV &>$ <*J <$7=A, *$F=AL&<$?<O&>Y$-?<O < θi,t
ρJi,t(θ) =
∫ρJ
i,t(θ, z) dz. IM *
# $ < 0 < - OC $U < <*?=AL$ ρJi,t(θ)
=LE <$7=J<'&?</ <C=θi,t
<C=A $T <'LD< (OUN<<BE<*G=AL $8%$B<
ρJi,t(θ, z)
=L <$7=J<'&?<G <@= UL-V<@=(θi,t, zi,t)
- <FV- ?<SN' ,+'&<G < + ! U#" < <'!T3$-5 3 &J!N - + 7<'A 5 !=D< - 17 - V M ^ M$ $-$?LF</VL@UUD<B =%", R <C= =U"N' &>!=G LY &<$?$I <>$$3 N<C= - U <AD< =A"#! Y<.VAB<J<J$& <J U#" <'G=$ $7=O D<'&J$Y<. <C= DM 3 U <AD< =%, 'Y?< $E<S&J <FV-!= <@=eV-V/ ! C= <GU$+C!<'YP<$-UN< < - <'?<SB<%?L?< UN<V'<$-$?BV=D=, GE< =, V-D<' =&PL#$(E<C= $-$' N<@=e&>$L$?D<@=0<E <@=eC#?<L-B= UL-$B<C= LV-&%+B<
θ M
$E< <C=A&>#$ <& IM * V- # ." ! .' # (' V7<'&<) < ! L?=D<X <'$F=J<'&?<QE<C=)$-$' N<@=&>$?L-$B<C=?L $) , =&PLE C<C= M $ ?<$?BB=
ρJi,t(θi,t) ' di,t =
1
M
M∑
l=1
k(zl,t|yn, θi,t−1)ρi,t(θi,t|zl,t)
k(zl,t|yn, θl,t−1).
<@=V,E=ωi,t
V D<'$?P <)U <D< =%, 'Y?< <$ VV &>$? ( T5G=P?<C=.G=%L$F=) $# $#M<$7=%L&<'$"?<C=@&>Y$-<@=
ωi,t =ni,t
di,t
ni,t =1
M
M∑
l=1
g(yn, zi,t|θi,t)π(θi,t)
k(zl,t|yn, θl,t−1).
, $F=GU <'V7<'$-$?*?LE<B<RUUL-- <B <C=A&JD<'L " U#" <G=" < Y!</L-$;U *?L-"#+?LE< O(M2) -?L .D<'$-$U <Z(.<'$@ Y!N&< M1 32 8 5 " 4 ; 4C 5 54 $ 4C &' \ \ " 34 #" " 4 ; 89 4 $ 1 #"
, L7=/$LF=$! B<C=B=A$F=L U"N <*; <$7=A, *$F=AL&<$?<(UL"$?D< ρi,t(θ|z)$F=.T5&L#$
Y& '$' +?LE<, IM 7- M + VV'<$F=*?LE<BU$F=AL-UW$; <DUN<D<O <$F=, Q<@=%@U$-$$' C<G(*U$$G=D=$-UN</ <C=
2E1'1
=&PL#$7=M<@V,E= &.V,$-U <RVLB=.( !! "#$t− 1
<O YN-&K< M <AB<OU$$G=D=$-UN<Q<@=% -$F=@?<C=@"G=A$$E<&<'$ =FV, 7U= 7 <$?B= - = ?&PG=, N<RV- FM,t−1
M
V-V` ? 7( VV,P=D<$?D <U"N-G=ABL$ &- '$Y!<P < D $ ?L) <($F=!$ Qk (1 ≤ k ≤ D) 0U B<$?B= ?LE<O $)' -$/V-
Qk(θ′, A) =
∫
A
qk(θ′, θ)dθ IM
A ⊂ Θ
<θ 7→ qk(θ′, θ)
<@=%/L-$E< <'$F=A, Q C<(:+C=/<(: 'V-B <( YN&< M <*CP<'UD<L- θ′ <@=%B$-UL$.V-&-+B<SV'<&<A"$? <@&<AD<.( 8%L ?<C=EU'"UW! ?=A ?LE<C= <
qk#L$K< <'&.V-<,=A&JV?< <@=% UN<L L$E<
<'$F=!! RY?LF=B=<'$$E<*$?$)UL-G=D<R <C=V` '$-U < <$XL?=$?/ <@=%&>$θi,t−1
qk(θi,t−1, θ) =1√2π
exp
1
2(θ − θi,t−1)
2
.
$F= -
ρi,t(θ|zi,t) =
D∑
k=1
αtk qk(θi,t−1, θ)
IM
0 ≤ αt
k ≤ 1< ∑D
k=1 αtk = 1 M
CL$ <@=@VF=αt
1, . . . , αtk
Y!< =J<'L?<R V$=, %?LE<'$??<< <
ρ(π(θ|yn) M
VV+&>#$%=LN, N<><@=%B L$D&K<'?<LB< ?LE<D<T=$?/Y"$E= M <@=V,E=
=$?Y ?LE<&<$?@$-!G=" @=V-1/D M
# $ /UWLE< <* - E<LY$- <@=G&% N <@=$? ! OVV=, N<@=@VL) "L-$E<*,+'Y?<RE<*&.U'#$) <@=
αtk
L)UL = <C=@, '$F=t M
2 ' 1 *98 VVB ? !" 7( LG=J<'$? L$>$U'#B<L Ki,t <?<
M?L$-+?LE< Y$E<
<O=A&L#$θi,t
θi,t ∼ QKi,t(θi,t−1, .).
<R$-LCP<'LXV,E= <C=AB =@UULE *U&J&<*L$E<*VV,$XL)$&PD<* <θi,t
?L =LC?C!<'$?$#G(J V'<3 + L)V-P=
t <* YN&<
αt+1k =
M∑
i=1
ωi,t Ik(Ki,t)
ωi,t
<C=A@.$&>?=#$ <ωi,t
- CP<'U
ωi,t =g(yn, zt,i|θi,t)π(θi,t)
k(zt,i|yn, θi,t−1)qKi,t(θi,t−1, θi,t).
<'V7<'$-$? - L-U% ? & !" I VLCP<$? ?LE<>U <D<>&% N <;V! C=D<$?B<.L-$I' T3L(&J18<L B=?LE<
MLY&<$?B< - ?<C=EV,E=SE<@L7=E<@=E$ ?LPU$+C!<'YP<$? C!< = 1/D - U < ?L: &J$?D< ?LE< Y!N&<@$ O?<$ -V"#T M <P' T3LO <P &% N <L<@=%* <$F<PV-P= -RE<P!+Y< V"#$ (;V-AR <.LF<<: 'C!<'@JU$F=AL-UW$XLXV,E=
ωi,tM
'($)+*-,.)/&021<$345)7698;:(θ′ 7→ Qk(θ′, .)
($/6<1,.=>02)<@?A4539BC4=>($/&:3D(1(3(<$E <$3($)*$(ν02)10287:<$((-)+02)763D,;1:*$602,.)F1:+*HGA45?0 63(.I
2E1 I −
+ ' 1 *98 $E5 1 $EL 131 " L-U I VV,P=D<$? E<S"! U#" <G=J<' L?=#$ <C=@$- L$F=@?<C=/VE=E<* YN&< - <'$ U"NG=AG=B=$
ωi,t =g(yn, zt,i|θi,t)π(θi,t)
k(zt,i|yn, θi,t−1)D∑
k=1
αtkqKi,t(θi,t−1, θi,t)
. IM
# $)< 0 <(?<C=LB<L =*VLCP<$? ?LE<;U <AD<C!< =$Q < = VD<o()) <$F=, U?<ϕ(θ) = π(θ|yn)<$XU$+CP<Y!<'$?SC!<'B=GL$XV&PL& L%=J<'$F=@ <O.C!<Y!<'$-U <R < . L-U#"" <'-<' M # $) LB<C=GB<&K<@= -
?L-$?,
∫ ∫log
ϕ(θ)ϕ(θ′)
ϕ(θ)D∑
k=1
αkqk(θ, θ′)
(ϕ⊗ ϕ)(dθ, dθ′)
: 'Uf(*U"N ?LE<V-P= < Y!N&< M EG?<L = - L$E<-T3</E<C= Y$-E= $&B<C=<SL$JN' ,+'&K</E<&J!B<U <$?<@!=B=AL-D<'$ RU$+C!<'YP<$-UN<H<$.V-, ,< <'$PL.&% $YP<@C!< =
ϕ(θ)=LF= ?<C=EU$-$F=
=AL-C#$?B<C= - V7<'LXB<C=AUWC!<@= M
0 2 ∀k ∈ 1, . . . , D - ϕ⊗ϕqk(θ, θ′) = 0 = 0?U <AD<RU$-$;$- ?LE< ?LE</(G=AL$ <@= V,E=
&.V,$-U < <@=%>-$-<*VB<C=?LE<>=-D<'&K<'$? M
0 + ∀k ∈ 1, . . . , D - ∫ ∫ | log qk(θ, θ′)|(ϕ ⊗ ϕ)(dθ, dθ′) < ∞ =U <AB<PU$-$ <@=%OL&>#+?LE<'&K<'$?B<&JV< ==LF=@?<C=
qk&J$E<'$?
ϕ - L =J<'$F=@ ϕ/qk <@=%/,$' N< M
< U"N L$t&% $YP<T3!; < L$E< &% N <F<?< - V'<&K<A"$?; D<'$K<'$tY& $3 "FL$$XVV&>#B<L <
π(θ|yn) - <7VUD<><B$UD! LG=?< M # $ < 0 < - .U$+C!<'YP<$-UN<D <* YN&<<C=AX/ N<u( =A, H <@=αt
k
L UL = E<C=), "#$F=t M
3 (U <C=XV,E=i CLE<$?)V7<'L - YN-&K< =" '<7UW$$' *?<*&- '$Y!<*?<*VLF=/V-U"NE<* <
π(θ|yn)L =D<$7=/ < . LU#" <?< M $)?=V=J<($7=Ae L$F<
,+'Y?<$?LC!<JV,L>=AV-V7<'R YN&<PL B<&JVF=T M &J&<.$LF=R F=D<C!<$7=/$7=D?<C= <7 <&JV?<C= -
<*$-&-D<O !! "#$7=T<@=%Q=LCP<$?F!+C=@T3?< 5V-P=VLF=/ L$E<(7$E<OY& $3 "?<&<$? - U < ?L0<@=%,+@=
C#$?Y!<LC+?=- (! C+G= <C=B&% N-+E<C= - NLE<'?<&<'$RU D<L7=J<@= M <V'<$-$? - L-$ U!=B )LF=B<@=αT
k
=A$$+LG=Q=LT L$ ?L C#L/$' 7U <@=D=D<'&<$?1 $- LF< LF<(<*&% '$Y!<OVV,P=, <C=A/$?<&<'$DL$
&>LC#G=(U'$- M VVU#$HU$U!+D<;E< &% N-+E< <C=AO 'C? <'&.&<'$T3U!! N<;V-?<U"N H L$&% $YP<; <; <$7=A, @=G=, &<$?L=&PL?<C= M 3 <$I C#$-U <> < <'$F=!! <;V, $-?< - $V7<'L 8%LYP<A= <O&% $-Y!<O$-!0<C=A@VXL)V-!=P=D=D<R$T5&>#T - </<O&J 7<A<'$XU$F=, %?LE<'$-U < M1 32 1 465789 2 3:; $ 1 \ *98
<QV-;IV! C=D<$-UN< <C= $-$' N<@=>&>$?L-$B<C=;-$F=$P=c 7U"N-$?$F= - $lV'<L<$-$nVV+?LE<'! U#"<'G=$ U&JV/+D<VV=" C<;L § IM 7 M M M + <&>?L$F= ?LE<$F= IM - U&.V-E $H <C=$-$' N<@=P&>$?L-$?D<@=.$?D<'C+?<$?> <T3 9$ =LF=- 8AU <'$?D<VJ L?=#$ <
θi,t−1M LB<&<'$; - <@=
$-$' N<@=S&>$L$?D<@==&PL/ N<C= (P !! "#$t <R YN-&K<DVV,D<'$@L$E<R$T5&>#$ L-,$ <@=%@V?=D<
<$)U&.VD<*V-<O&% $YP< L (. , '$t+ 1 M
Y!N&< -$ VB<$ B=FT5&<>=LC#$B< M
2E1
0 132 5F 5476 8 M 7 1 *98 $E5 1 $EL 131 " Z \ 5+ ##"89$ :; $ 4 #" • :] ^`7a-b=< t = 0
cedm h i g` ]+p s fi # # k` θ1,0, . . . , θM,0 Sdm,i`g
α1k = 1/D
b-f]gk = 1, . . . , D
d
• :] ^`7a-b=< t = 1, . . . , T
ced f]gi = 1, . . . ,M
!"h $ <ia-]-j?`gtb h g j h j?fiQa-]-j^=i7p-f7a i h j?` j?`-<tfg=i )=i7p-`-<Ki,t ∼ MM (αt
1, . . . , αtD)
!Ar&$ <ia-]-j?`gnk`lm h-o fpqi7pk b`pk h p+^`zi,t ∼ k(z|yn, θi,t−1)
`^θi,t ∼ QKi,t(θi,t−1, .)
!s $ sh j s ]j?`gni,t =
1
M
M∑
l=1
g(yn, zi,t|θi,t)π(θi,t)
k(zl,t|yn, θl,t−1)
! k $ sh j s ]j?`gdi,t =
1
M
M∑
l=1
k(zl,t|yn, θi,t−1)
k(zl,t|yn, θl,t−1)
D∑k=1
αtk qKi,t(θi,t−1, θi,t)
! ` $ sh j s ]j?`gωi,t = ni,t/di,t
d
Sd;g s-h p^=ij+j?fp+p`gtj?`-< b h g+^,i s ]j?`< s f]+g h p^`<t<`j?fp j?`-< b-fi#k<Hpfga h ji?<-<ωi,t
Sd sh j s ]j?`g_j`-< p-f]` h ] b-f-i#k< k]taj h p )`αt+1
k =
M∑
i=1
ωi,t Ik(Ki,t)d
! #" $ " &% '() " B(η0, η1, β)
, LF=BT5&>?=$F=DU V-VU$ E<P Y!N&< $F= =C!< =$ !"!<'$E<' ! U#"<'?=, *L&J:+?< (?=?LE<C=@U$-ULB<$? = B(η0, η1, β) M
# $ : 'UL?<F C+E<&J&<$?F VVU#$ < L;&.:+'<@=<7+V,$F<$??< <G < <LF=ALF=- 8AU <$? = - $? <@=E <$F=, C=M=A$ $-, C<C= fE
<fWM L- T3U!B< 7U!LB< -
$LF=@&J [email protected]"&% G=#$XLX&.:+'< M $X$B<
λ = η−10 , µ = η−β
1 .
, $F=yn = (xr, cn−r)
?<C=.$-$' N<@=F=J<'CY N<@= <<n - C!<'U xr = (x1, . . . , xr)
<@=P$$' C<C=P$$UN<$F=L! N<C=. <X?<
r<
cn−r = (c1, . . . , cn−r)E<C=>$$' N<@= : N<C=J < UN<$F=LB<i(ID< M <C=>$$' C<C=
&>$L$?D<@=zL&J:+?< =$?S?<C=GY$F<C=SE<C=F$$3 N<C=SF=D<C& C<C= M # ?<C= =A$F =S <B< r+ 2(n− r)<
k(z|yn, λ, µ, β) = k(zew |yn, λ, µ, β) kE(ze|yn, λ) kW (zw|yn, µ, β)
CP<'U
kE(ze|yn, λ) =n−r∏
i=1
fE(zi|λ) * zi>ciexp(−λci)
,
kW (zw|yn, µ, β) =
n−r∏
i=1
fW (zi|µ, β) * zi>ci
exp(−µcβi
) ,
k(zew |yn, λ, µ, β) =
r∏
i=1
fE(zi|λ) * zi>xi
exp(−λxi)
1−δ(xi)fW (zi|µ, β) * zi>xi
exp(−µxβ
i
)
δ(xi)
2E1 I −
<δ(X) = δ(X |λ, µ, β)
JV, ?LE<X=/ Y$E< <7+V,$F<$??<< M $X
δ(X |λ, µ, β) =hE(X |λ)
hE(X |λ) + hW (X |µ, β)
hE<hW
D<'V! C=D<$?D<'$@?<C=FL <(' T3$-U <OB<C=V7<7UWT=/ <@=@&J+'+?<C=,<V,$E<'$??<0<B < <'L- M# $-$ - C+"G=J<'&$U <; <@=.$$' C<C=.U&.V-E+D<C= g(yn, z|λ, µ, β)
<C=AP?< V-L.E<C=J <$7=A, @=L<7+V,!$E<$?<'?<C=L < <'-L UB<C=V$-$)(J Y$E<* <(U"N-LF<O$$' C<*F=D<C& C<DL)D<7U$7=%!L' N< - <*V- $-: V'<$$-UN< <C=>E<L &.:+'<@= M /$L7=.T3L$-U)D<$->L$F< V-$tUL$?D< (ei,t,wi,t)
<C=$-$' N<@=ED<7U$F=%!L' N<@=H<'$;$$3 N<C=G Y$E< <7V$E<$??<?</L; < <'L- M NUL-$E<B <@= V-A!$F=H<@=%GE<<
n M
8P6 5 n\ )8 1 $ 2 " 4 ; 8 4 $ 1 , LF=OU"NG=?=B=A$F=/UE L?=D<O&J: 'G=$& # # : 7-$?<L)U"N-V-!D< M 3
λ ∼ G(a0, b0),
µ|β ∼ G(a1, b1(a, β)),
β ∼ Be(p, q): -$-< =L
[βl, βr]M
# $)$"$?ei,t = (ei,t[1], . . . , ei,t[n]) - (. !! "#$ (i, t)
E<* YN&< - $ .B=
λ|ei,t ∼ G(a0 + n, b0 +
n∑
l=1
ei,t[l]
),
µ|β,wi,t ∼ G(a1 + n, b1(a, β) +
n∑
l=1
wβi,t[l]
)
<π(β|wi,t)
<C=AD: 7-$?<O(>L$F<U$F=%"$?D<V,+@= M $ VV=J<($-U(E<PU"N?=V,L-D <'$F=A, P$7=%L&<'$"?<UL$?D<
ρi,t(θ|β) = π(λ|ei,t)π(µ|β,wi,t)
β ∼D∑
k=1
αtk Be(pk, qk).
<(U"N ) L$ &% $YP<( <*?= * =L [βl, βr]T3 =D<$F=C+L)?<U"NXE<
π(β) M D<C=ABU"N?=6=L =&K<'$?
Y"$-JV,L-GV'<&<AD<R?<DU"N ; <RVLF=A?<L-B=<C=V` '$-U <@= - <@VL@U"N- ?LE< <C=V` '$-U </?<OU"NJE<DVLF=<'L =C#$-U <@=F' 'UG=B=$B<C= M4, $F= ID = 1, . . . , D M
M 3 ! I1, . . . , Id L$F<V-A!$ ! YLE+'D<; < ID D<'?< ?LE<U"N ?LE<Ik
=P <;< #<L M
$ C
d ≥ 23 )L$)U"N
D = 4L &J$&L& M
M 3 ! d C <'L = (β1, . . . , βd) <
β, 'YL/+B<&<$?@, 'V-?<C=,=L
[βl, βr]M
M 3 ! L C#$-U <@= σ2β,1, . . . , σ
2β,L
U!=B=" C<C= V- C#?<L = : 7UG=D=$?B<C= M V!+C= $D< <7+VB ?<$-UN< - V,L-D ≥ 10
<L ≥ 3 - $-LF=@VV,P=$7=GL-$E<*,+'Y?<RL +V'<
σ2β,1 =
K π[β],
σβ,i+1 = σβ,i(1 − 1/L)VL
i = 1, . . . , L.
2E1
7 M ∀k ∈ 1, . . . , p - $ U"N?= L ?= Be(pk,i, qk,i) C!<7U
i = 1, . . . , L <;&%*'&K<O<C=V` '$-U <
βk<P <
C $U <C=σ2
β,1, . . . , σ2β,lM
% % " , LF= L?=$F= V,LEUN<C= VV-U$F= $+L&% +?LE<@=e?<G&J:+'<@ < <'-L M 3 ! θ = (η, β) M
$PU$F=:+B<?<C=@$$' N<@=
yn
<R?< - U <'$F=AL-, C<G( * - V#C!<'$-$?GL8<LB215
L ?<'L M U"N-VB<F (V M 7 * M
' T3$-U <@=@, C<<@= U <'$F=LD<@=7 - - * - I * 7 - * - I * - 7 F-7 F- 7
IM $$' C<C=B215 M
< # UUL-E /V L$F<B&% N <B < , <@$ + V-NF=A$ <C=A θ = (8752, 1.21) - 'U'B=- +V'< <C=A&% C=(2500, 0.5) M
.C?"?<RC <'L/L)V-&-+B< <@=%θ0 = (8760, 2.15) M
, LF=(V-V=A$F=( L?=D<. &.: 'G=#$ # #G: 7-$?< L U"N-VB< <PB<VG=J<;$F=( =LF=-V-A<XV! 'UR ' <'$B< - V,LJ V-&% G=$ θ = (µ, β) M
$ U"NE<'U"NF<g(QC& F<.<),$tU&.V,D<'&<$? <
ρ(θ)B=?LE<( # # <C=A <$QUU" C!<'U<@=RC?"?<C=BC#?<L =DLQV-&%+B<(&>G=RD<@=%B<B<#CP<&<$?
V'<L $T5&JT M $IU"NG=A($U π(β)U&.&<J SL$T5&K< =L
[0.5, 3.8] - $U.UN<$?! N<<'$ M IM $U$F='+B< ?LE<OJ&- 7$E<O&>Y$-?< =AL
XUB<C=V,$-g(P.&% '$E< =&L/ N<
xe = η0(log 2)1/β0 ' 7380
<D $ U'U" < # # a = 3$$' C<C=G %?LC#?<$?D<@=x(UN<O C?G=B <7+V'< - UN<,?L D<'V, @=J<'$?D<(/ YY+B<&<$?
&J$F= E</(&.!E D </ $T5&>#$;VV,, C<V- ?<C=G$-$' N<@= U <'$F=AL-, C<C= %?LC <'$?D<A<'$?C+$c(* V-VA <* M I $-$' N<@=A=J<'$ $-U#B<L n ' -$ L U"NVB< E- § M IM M, $F= ?LE<P>&% 7$E< # # =Lη<@=%BB= <@=%&% N<l( * M
1 2 4C 1 "@$ 4@ 5+ \ *98 $F=DL$ V-D<'&.?<BD<'&.V7= - $LF=/LG=A$F= D = 12
<'$F=A, @= Be(pk, qk)=L
[0.5, 3.8] - U$F=%L-!B<C=BE<&%*&<T3 9$ L L § IM 7 M M- <'V7<'$-$? - $.&JF<G(U"N- ?LE<, '$J <C=VB "$-UN</ <DUN<C=EG= - (D<'$-$?V'<Lu Y$3 N<> <> <@=%&>#$ < β UL-$B< M LI,L. < ?LE< ?LE<C=O, '$F= 3VL M = 2.105
V-AUL?<C= - CL$) <@=,<C=A&J$F= E< θ <@=%Y!<&<'$ =A-?=, N< UWT M -YLB< IM M CL$ <@=FVE=
αtk
<@=%FT5L$?< =LF -YLD< IM - &>G=FD<@=%B<BV'<LXG=A?< - VLG=L ?=,= 7UAB<$?GV'<L) < 1/D M L,L. L-$ U <$I$&B<; !! "#$F= - LG=#$ E< "8%LD<>VLF=( <L-!,?L <?<;$ VV,D< $T5&>#$QVL>=%"?=D<O?<C=RVF= M, L&% +?LE<'&K<'$? - U <@=RVE=RVC?/ Y?<$?OL$ &- '$Y!<J <J&%*&<<@=AVB "$-UN<
β = 1.904<. 7UB=- ?V'<C=P: 'U?=B=$?B=
(1.32, 0.99, 0.33)3<XV,E=(?<VLF=T5.: @=AY$$?
M Y M <C=A&>#$ -$-<) < θ <C=A θ = (8505, 1.93) C!<7U;?<C=- 'U'B=- +V'<C=
(1340, 0.29) M <C=., @=AL-!"#B=VU"NE<C=@ <@=@C+<@=@C <'L =GLF=AB<$?/?<O$)U&JV,D<&<'$&% $-Y!<'$?@ < DM
; (9*-,.=>?!,.3 6&($=>($)6 ($/6 ?3D,A4B2($=>($)6 1BC4=>,;10 A*-46&02,.) 1($/ ?A45345= 6&3D($/<1(-/ *$,.=>?!,./H45)6(-/ 1: =<$BC45)>.( B2($/ 3(<$/&:B 6&46&/1(@,.:* "!#%$&&')( )(/* 45??B20287:($)769?A45/ 02*$0+(
2 I −
$F=BL-$ =D<'U$-XD<'&JVF= - $LF=B$F<&J $F=DV-!=B<@=DE<$F=!! C=RL &% $-Y!<O(U"N- ?LE<(, '$ M E&J<@= 0U B<$?D<@= <C=VB "$-UN<C=*$?<C=
(1.16, 1.82, 2.48, 3.14) - =J<'L?<C= M * < M =A$?(VC+/ YE C<C=*V?<C=VF= M XE<$F=!! ;LI&% $YP<JE<.V--!! $-?<JXVLF=OT5AB<K<@=% <@=AVB "$-UN< M < 'UA +V'< M * M, LF=@D<'$$7=@L$E<><C=A&>#$ -$?< θ = (8488, 1.92)
7UA +V'<C=(1336, 0.32) M
steps t
alp
ha
0 2 4 6 8 10
0.0
00
.05
0.1
00
.15
0.2
00
.25
IM CL$ <@= αtk- k = 1, . . . , 12 M
steps t
eta
0 2 4 6 8 10
60
00
80
00
11
00
0
steps t
be
ta
0 2 4 6 8 10
0.5
1.5
2.5
3.5
IM CL$ <C= <@=%&>#$F= E<η <$)N-L <
β M
eG=A&<'$ - $ 8%LY!< ?LE<D = 12
<C=A>L$t$&PD<XV ?<C < <$F= C=Q&>$<' M $tVV,P=D< =(L$ & $YP< < 7 <'$F=A @= Be,i
- i = 1, . . . , 4 - D<'?<C= ?LE<X<'LB= <C=V "$-UN<C=L=?<$?.D<@=AV'<'UC!<&<'$e1 = e2 = 1.49
<e3 = e4 = 2.48
=B?<C=DC#?<L =BL <RL < [βl, βr] M $#<LR ,"
<@= 'U'- ?V'<C=σ1 = σ3 = 0.33
<σ2 = σ4 = 0.165 M
$LC!<* =
β ∼ 0.065 Be,3 + 0.935 Be,4
U < L- B<C+?<$? D< ?LE< ( <@=%*L &.?<L VV& (V Be,4M D<'&K<'$? - <C=V '$-U <L<C=A
&>LC#G=D< LlC+Ln <β0&>G=O=LL 'U'- +V7< V->VtT3?< VLB< 'G=%B< M < U"N E<
& $YP<B$! <C=A@$-UD&J V - UN< ?L $+?LE<O 0 B<$-UN<R <RC#?<L@&JVA"$?D< <$?B<D?<C= αTk
3UWT M-YL-D< IM M $ LY&<'$B<( =?<C= 7UA =$ +V7<@= - <'$ U"NG=AG=B=$ σ1 = σ3 = 0.45
<σ2 = σ4 = 0.25 M
$LCP<RB= UWT M -YLB< IM 7(
β ∼ 6.10−3 Be1 + 0.03 Be2 + 0.48 Be,3 + 0.48 Be,4
U < L-6<$?$E<L$E<L<@=AV "$U < #[β] = 2.4
<RL$ 'U'- +V7<E< M 7 M <& $-Y!<($ <C=ADC+G=A?<&<$?RV-!=V&> - VLG= ?LE<D<@=SV,E= B<C=AD<'$?SV7<'L L-- @= M <V'<$-$? - U <D<R <$F= B$F=AL&<$?<BV-"S'<'LULV V-LF=/?=$$?< ?LE<O.V 7U 7 <$?D< M
2 62
steps t
alp
ha
B1B2
B3B4
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
IM CL$) <@= αtk- k = 1, . . . , 4
VLL$X&>LC#?=U"N $ M
steps t
alp
ha
B1B2
B3B4
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
IM 7 CL$XE<C= αtk- k = 1, . . . , 4
V,LU <)&&< U"N - VC=.&J+ U'#$ <@= 'U'-?V'<C= M
L- '&K<'&<$? - =U"N-$ β0 = 2.15 - $LF=OU"NE<"U"N$7=/?<.&<'<'LO& $YP<P <J <$7=A @= X <@=AV "$-UN<
β0<B <*C#$-UN<C=@ 7UG=D=$?D<@=L-eVV-&K<
π(β|yn) M, LF=@T5&J$F=@$-UOL-$X& '$Y!<O$! <> <'$F=! C= Be,i
'U'B=- +V'<C=DD<@=AV'<'UT=(0.66, 0.33, 0.165, 0.1) - V,L i = 1, . . . , 4 M
V@=OL$E<> '$E< "#$F= - ?<C= V,E= < <@= <@=%&>$F= =J<,=A$? =A-?= C= UWT M -YLD< IM I <E $P<'$? ?<G& '$Y!<
β ∼ 0.18 Be1 + 0.26 Be2 + 0.37 Be,3 + 0.19 Be,4
UN< ?L $$F<RL-$ 7UA +V'< -$e < M 7 M
steps t
alp
ha
B1B2
B3B4
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
IM I CL$)E<C= αtk- k = 1, . . . , 4 M
2 )+ I −
1 + 4C 1 "@$ 4C 5 \ *98 *98# $-$ - A$; <BU&JV-B<E Y!N&< C!<7U@<@= UP=D=+?LE<@= <'$ =AL7=S <@= L- <@=S 8 < 0 <'U.
L @=$7= L-U & ! 7- -$ <'<'L 7 !" I < L$ 7 ? & !" I =L/ LD<@=G&J '<@= - $L7=&JV &<'$$F= =L $ 3 M 7 ?<&J ?<- C=<'$ =J<'$ <.U <K=ALC#$? 3C E=%B<$ !" I V,L-L$E<B<C+LE</ <@=EV$UV-?<C= V'<C=S LG=$;LJYU<' M $-!G=$> <@= U"N- $E<@= VV,P= N<UFUD<@=AV,$
β = 2.15<η = 8000
<.D<@=%B< $-U>VU"NE< <@=(C+"?<C=(C#<'LB= M 105 ! "#$7=L=$?L?= N<@=H<G<@= C=L!"# = <C=A&J$; <BO"?< IM 7 =$? &J !<'$$ N<@=H=L *U"N- $E<C= $- V'<$-$?D<C= M E?<L-B= - $LF= 'U'VL-$F=($F=(?<;"?<'L IM I ?LE< ?LE<C=C <'L = <D<&JVF=P < U'ULF <C=.YN-&K<@= § IM 7 M -M < DM
895 \ 1
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j-i<#^ !Ar `^ h Sd ua-] c d u ( u $
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c $ +g h i<` a r j h p s `Ip-fp s `p,< ]g+`
i k !-`i r !Ar `^ hj h a r k h $
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i k !-`i r !Ar `^ hj h a r k h $ !
i $"
r `^ h k r `^ h ! b $ h bg=ifg,i k`-< `?m+m`^,< h j h ^fi g`<
r `^ h () r `^ h )* !#" " c $ # " cr () b-f! ! `
r `^ h $+ ! b-f! !
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j h a r k h() bf! ! a-]! c +r `^ h $ $
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.eO=LB<C=A&>#$ E<η3$7=A?LE<P>C#?<L-RG=VVA$$ C< < 'UA +V'<L=LRUN<V-&B< <C=A
U N< L$E<=L7=$ <C=A&>#$I&JVA"$?D<X <β M.eQU$?CP<Y!<'$-U <> <@=PU"N-/$F<C= =AL-
η - =L.U << <&JV?< - $ <C=A/C?G=?<&<'$DV-!=D<'$+LE<V-@= ; ". "#$F= M .e -YLB< IM 0 &JY$E< - =LD<@= I " <$B<C= V-AUL?<C= - </ $F=A B L$E<DU"N-/ $E<A=L η - ?<$ ?LE< <C=A&J$; < β AB<Y$E<"V <'&K<'$?L$E<)U <'A"$E<%=A - VC= ?LE< ?LE<C=JU <'$?$E<@=. '$F= M LE< ?LE<C= <C=B=?=J CP<'U 106 ! "#$F=$F<&J+ F<'$?/V-!= =LF=A$<'?<&<'$?<C=@ C=L# = M
K G=$ A C+?=O < - ?<C=O& N <@= =$?*?<$ &.$7= <;UUN<C= =L*U <><7#<'&JV<> 'U"N$?$ M LF=A?<L-B=,<@=D=G=,=AL/ LB<C=$$ C<C=@V C=D<$?$?E<C=UUW ?=A ?LE<C=FVU"NE<C= "?< - L <(U <'$F=LD<
2
T0&. P<$$F< & '$E< 'UA +V'< $B<C#< I PI
I η I ! " I - " I
β M " M M " M " - M
η ; 0 0E- "
β M M " M M " IO- M 0
; η
I ! ; ; " 0 - PI β
M M " M M - M PI
I " η 0PI 0 ;" " ! 0 - I
β M M M M " - M 0
IM =%&>$F= &. P<$$ N<C=>=L(?<C= T0 <$B<C=O "#$F=* < U"N-/ $E<@= <
M = 105 V-UL-<@=@VL-!B<C=@V-l$ M/&B<* <*V-AUL<@=
M VF= VF=
IM 104 K= I =105 0 = I K= M 105 *&J$ = O&J$ =106 &J$ = &J$ =
IM I <&JVF= <U'UL-E<C=EYN-&K<@= < 5&J P<$$ @= =AL- ; < 'UL$F= =L <'$L-& M $)U$F=A 'D<OU ;. '$F=@ V$XVL DM
$E <$-UB<G&J$? @DV-LF= Y$- < < DM ?<L-B= - &JY F?</U O(M2) <C= V7<@=E <
" U#" <G=#$;-$F= - U <BYN&<>=J<O C?<D@=U$-ULB<$??< <$XB<&JVF=/ <(U'UL<'<7UW!TC+G=$ A C+?=F <* <7 7UL$X L$F< $F=@L-$XYU<' VLU M
IM 0 OI "P <'$- 'D<@=G '$F= L$F<(U"N-/ $E< =AL-η M
2 I −
IM OI "P <'$- 'D<@=G '$F= L$F<OU"N/ $E< =ALβ M
. <7UWT L$-?=AD<L L$ L"-?=AD<><@=%JV 7UG=J<'BUB<&<$? M = YB B<$ =&.V-<7&<$? <C=L<C=A&>#$7=OV$-UWLE<<@=LIV-&B<
θ - -$F=*?<;UB<>D<@=%D<'$?( <C= &J ?= C=LU"N-V-!D< F- <B$-$ <C= =A&L-$F= H E< θ - 'U"N-$$$-Y!<RV T B<$?<'6=%"#+?LE< <@=%/VT3D<'&K<'$?R-V L-$E<VV,P=$ <E<$F=! $F=AL&<$?<
ρ(θ)<C=ABT3!B<PL § IM M E- B<C=V'<'UW"$?/LB<C=
<@=@U$!$F=FD<?L?=D<C=FV,L@L$F<R,$-$E< <C=A&J$ M ?<OV7<'&< <'$ LB<RE<R&J$D<'G?<*U&.V,D<'&<$?V-T5G=@V &> LF<OE<*> <'$F=! *$F=AL&<$?"?<(U$7=%LD<*V U"N-) 0 ! OJ& N < + M"D$F=L$ U'B<OVLF=Y '$ " - $LF=/U$F=D<$F= LG=J<'B?<(U!B<#"%$ U$F=ALA< L- L)U"NVB<JVL@T5L$L$E<*$-U$)L)<'$ T5$- *L U"N X <*J <'$F=! *$F=AL&<$?"?<
ρ M
E$F=BL$ U'B<(VLF=DY $ " - $QU"NE<"U"NE< ;D<'$BL$E<J <'$F=A P&>$?< <D L?=-< - >V <( ?LE<'?<($ V'<L =&PL?<Q=$F=Q< = - <@=/& N <C=R 'U"N-$$$-Y!<OV T 'D<'$<' V"#!T - <DV-LF=V-UL-B<&<$?;IC!<'B=$&" !"!<$E<'/! U#"<'G= C< < YN&<(' - =J<'&?<$?VVV C<C= M/$7= ?LE<R<@=G T3L =GE<DU <AB<DCP< =A$> <!' <RU"N; L$;& $YP<D <D$ L< =LLG?<DU"N <?<L-B=DC#$-U <@= @P=%D<'Y$E<$?DUN<V'<$--$? =$ L7=Y!< <C=D&J+ ?<C=R <(T3?<P&<$7=A$ - < - <$ ULE<' <* - YP<$?.U'-D<'F $G=$) <* YN-&K<)P.&>$ # M
. =LF<* 'U"N-$?$yn
<C=A/T5AB<&<$?DUN<$F=L - $LF=/ C$F=U$$+L U <'A"$E<C= ;UL-! C= = '<7UW$$E<O?<C=RC#$-U <@=DE<C=O <$7=A @=RL & $YP< M + <'&J?L$F=RU <V'<$$? LF<P L7=Y!<J <.UN<*YN&< <C=AC=RY '$ " E$ <@=%OV-P=DB<C=AB<$?OL &J ?<C= X$-$ N<@=D&>$?L-$?D<@=Q<*EV'<&<( VV&<'DG=%L$ B=?LE<> * $ <@=%PV-!=*$T5&>#!T M,+ $IT3P$F=UN<UP=*L$ U"N $ L$ & $-Y!<J <.E<$F=! C=O <.C#$-U <@=/B<#C!<'&<$?DY!< - <'$ L?=$?OV- <7#<'&JV<.U&.&<. T B<$-UN< <C=A&>#$X <*.C#$-U <O L$ <C=A&JD<'LT5 %?LE<$??=AD< M
Chapitre 6
! "$# &% (' #) +* ,* .-/&0 %+* ) 213*54 ,* &6 ,* .eO&J 'G=$J- C=<'$$E< <O < <LF<=*U'-$.$?SU$F=%!L ?<BU87*L E<BU <@ C#
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π M + LD<JU <'L V C=D<$? C=O$7=O U?< <'AN$ 0 - $P=OV-$-UV-L C=L# =,=A$?<C=A=LC#$?B= M
M /L7= =A&.&<@=FU'V-?<C=F <OVD<'$-B< <'$XU&JVB<DL$F<OU$$G=D=$-UN</T5"U$$-B<Q<A=AL <'UWCP<* <DL N<B <BC??</ L$U&JV=$? V,L-GU$F=ALB<L$E<B&. 'G=#$ T3U?<&<$?GU'-<VL$ $- =AD<O$LF=A<'eLX" C!<'B=F L$ V-& B<DN-&JY$E< .L$E<O"?<( 'U"N-$$XC?LE<'!<C=A,<'$<'</UUN<C=B=A?< PJU&JV NE<'$F=A$ <O <V'< <@V'<&<L$E<*G=ULF=B=A$X,L?=B=$?PL$U$7=J<'$F=AL7= M V'<&<BL7=D= .L$ $- =AD<O$LF=A<' <7 <"U < =JU!+?LE<( <O <7V7<'A?=D< M
M /L7=* C$7=R -$M<(L?= J<;U'D<" $ ?LEV7<'&< <; B<&J$E<'( <.T3KJ$ <7UWC!< =E<@=$T5&>$F= V#C!<'$-$? ?L < <V7<'A?=D< - L-- <C=S$$ N<C= F=D<C N<@= - =$? -LF=AC!<'&<$? Y$ C<C=<&<$-U <$??<D?<$ T5$- *LX @=AL/ $T 'D<'$-U < M VV-/U&.&<RL$E<*D<'$#C!< PL$E<OV!U 'LD<F@= 'UN<$?D< - LV-V C= < RBT3U <&JV&J$-B<C= M .0<UD<@ L?=#$. <UN<LE<@=% +V ?LE<&<$?SUN<L- L$E< L- </VV+?L C< - <M=A$ <'&.V- <C=A V7<'A$E<'$?S$F= ?<C= V- '&<C= <(L"! <B < O$LF=A<'?<C= M
.0< U'D<$ 'UN<C=B=AD<UN<V'<$--$?PL$F< -V"#$ -$F=P <$&B<L UP= V-VU$> U <'?< ?LE<$LF=FVV,P=$7=SV'<LV-AT5?=,=D<D C ?<F& 'UB< = <@= 'U"N-$$F=F <*$-$ N<@=F?=V$<@= =J<R C ?<$?@ <B"?</@=GT3< - E<RUN<$F=LB</T5D<DLO==<R&J ?<BV-& LF<BU"NG=7<@=% <D&<$7=A$ ?<C C< M " <L<'UT= U$ %$ = <GB<'U"NF<"U"NE<=D<"?<$? $-U@ <GD<'$B<E & B< U <D<-V"#$U&J&<J <PV-V=J<'DL$F<P& NY<PV7<'&<A$?O < %LYP<O$7= ?LE<'?<.&K<@=AL-D<. U&.V"G=A$
0I
2 0 −
<'$D<OL$F<(U$-$-G=D=$U < T5D<><L$XT3?<*$&B<* <*$$ N<C=@$-L7=%?<<OT3!Q=D<$7= M
$ =LF=* <JU <@=R @=ALB= - $LF=O C$F=RU$F=U (L$IU"N-V-!D<JL & N <@=R <@=%&>#$Q C=?<$$E< -VL?<C=LF<<@=@$LF=$$$F= Y??<&<$? ?LE< ?LE<C=U$F=D<G=G& NY LF<C= M
" LD<@=@V&<C=@B<$U$? @=G$F=@ $-LF=A?<RV'<L-C!<$?/U <'V7<'$-$Q= VV,P=D<.L$E<(V-VU$X D<7UWB<F <@= LG= <E& N-+E<C= V C=D<$? @= $F= U <AD<FN C=D< M $/$F= - 'Y $B<S& NY<DE<C= $-$ N<@=$LF=A<'?<C=EV-LF= Y $ "?<C= - V-H<7 <&JV?</U <$7=AL N<@= OYL-U"NE</U&J&K<*!B< - $LF=EV-/ B<L$ <$<'L&JV,$ </< =L<BE<OT5LLB<C=/D<7U"NE<"U"NE<C= M ' /?<L-B= - U <'A"$F=@<7UW!T=/E<*D<7U"NE<"U"NE<B$-LF=A=$?/VV-LF=EU&J&<@E<C= U&JV &<$?B= L> <C= Y $ 'G=$F= <$-B<F" C# M 'C!<&<'$ - $LF=H<$JV C=D<$?$F=G=$7=@=L!B<( <*U <(U"N-VB< M
. L$ <C=/VD<'&J<'B=@<'UT= <C=AB < 7-$A<B VV+?LE<'/L$E<(& NY?< =&.D< JL$)$LC!<7L&J+ ?<JE<. T3$U <.D<'$-$?*U&JVB<> <C=OV,P=B=A?<C=O T3$-UN<C=O L$IU&JV,P=$? ∑ X=U!"#$ -)=ALD< <C=(" C L V &.$-B<C=(E< <?<L + $#C & !"" M <>+V'< <;&J ?<;$? B<C=B=D<;L&J$F=L"$?<@=@$LF=A<'?= ?LE<O?<C=@&J ?<C=/E< <L6=L?<C=LF<G=@$LF=@$-LF= =&J&<C=GT5UG= @=$7=U <O C# <ON C=D< M BLF=B$$-$F=/L § 0M L-$E<( $!$)T5&<<( <(UN<*&J+ ?<($F=A L L$F<*G=%B<( <C=V$-UVL<'UT= <*B<'U"NE<'U"NE< M
' /?<L-B= - L)UL-B=F <(C!<'B= <$?D<?<$F=@ C!<7UB?<C=@$-L7=%?<G=L- $?@&.C *U <R C# <RN @=J< <VV-LC O>& N-+Y?<(VV,P= N< - e$LF=A<@=% =L-C!<$?DVV-L ?LE<L=e?<C=RU$$-G=B=$-U < <V'<D<@=Q=$?$&PB<LF=D<C= - <<@= $GV7<'$E< D<A<7V& N<@= <$JB<&K<@=SU&JV NE<'$F=A?<C= *OT5?=EV,L-S?<C= $Y $?<L = <<@==AG=%U<'$F= M <C=AGV,L#?L - -$-FL$E<R&J+ ?=#$ % LJN(+V'<V-"&D<@= T3U?<&<'$@U!"?<C= <@= $-U'#D<'L = U&J&<@L$E<@?<@E<@$$ C<C= C+ALE<<@= $LF=<@=% VV-LJU&J&<@L$.<7UWT$? B<C=B=$ M $BL § 0-M F- $LF=BT3G=A$F= ?LE<'+?LE<@=BVV,P=$F=D <B<'U"NE<'U"NF< <$ UN< =D<$F=6L-$E<Y $ 'G=$ <C= E > $?+L! =PL U"N-V-!D< § M $LF=L=D<&P<O<$ V-AUL<' D<;L-$ L$? B<C=B=$ M
$-$ - $F= DU$?$+L @L.U"NVB< IE- $LF= T3?=$F= L § 0M BL$E<@VV,P=$ <U$F=AL-UW$ <@G=$7=%L&<'$"?<C=BC?G=$?;& D<'B 7U"N-$?$$-YP<*V T B<$?<' $$ -V"#T =A ?LE< M BLF=R C$F=&J?=6<$ C+ <$U < ?<C= T3LB= <S Y!N&< + M $U$F=J<'C#$ C<G L?=D< FT5?=eBU$-$-G=D=$U < <C=X$$ N<@=O<X U$$G=D=$-UN< = ?LE< ?<C=X$$ N<@==$?X$-U&JVB<C= - $LF= VV=A$F=E< $*L-$E<JE$F=AL&<$?"?<PV-OL$E<J&J !<$-$E<.E< =L* <C=*$$ C<C=R$$ UN<$F=L N<C= <D<7U$F=%!L N<@= M /LF=J$-$$F=J$F=@L$ V-D<'&.?<<7#<'&JV<)?L&J$?B<?<)$ U&JVAB<&<$?>L "YP< &JV,$U < M
4 /"! 81/ 1 / )$# 31&% 6 N +* ,* N <&J ?<.;Y$E<'?<&<'$ VV,P= (V- <?<L' + +$#C & !"" $F=DL$E<.U&.&PL$U$
+ < " $ &>G=D$ V-!=DT3!* <* <." C#L L '?<L = M C+G=J<XVD<'$-B<L<'$ U&JVD<.%=L-$=AL-C#$?B< L$U&JV=$ ∑ V'<LD< =AL&J?= <'LJ+V7<@=G <R T3$-UN<C= ?L'<$?D<'$ <$U&JV $> L$E<L<C=ADU'U <$?D<'?<><D LD<L<C=ADLE< =$ C+?<G=D=D<&<$? - =AL7=U! (V-D <C=DU"NUN=J =U$ M = Y!/$-UO L$)&. '<(U$F=ALA=AL?<O&&<*V$-UV'< ?LE< B(η0, η1, β) M
9 2
! " " &<'?<&<$? - $ U$F=A 'D<?LE<P<.U&.V,P=$? ∑ V'<L>=AL-DL$E<J T3$-UN<L ,LO <
NU"NUN= -
<O $)=AL-VV=J< ?L L$HU"NUK<C=AO) 'G=$ L$HVU <@=D=LF=O <' G=D=$I $?D<'$F=! ω M <>V-"&D<<@=%D$7=A ?<*"L VV-$ L$ U"NU M $ $&J&K< S <($-&-D<( < =AU!"#$F=/L &># <' =L
.V ' <[0, T ] - $
ω = S/T.
"R$F=FL$F<*VD<'&J 'D<OVV-U"NE< - $ V'<LU$F= B< ?LE< S <T
<B$Uω =$?/U$-$?L7= M $=LVV,P=D<
'Y?<&<'$ ?LE<(?<C=BU"NUN=A=$?B$- 'V7<'$-$? = <$?B<><L%< ?LE<( VV-$) L-$ U"NUO$E< 'V7<'$-)V-P=/ < ! Y!<L &># '?< - +U'ULB<$-UN< < N U"N-+UC=A<@=%RL$E< =&J&<PE<
NG=Q<7+V,$F<$??<<@=/$- 'V7<'$-$?B<C= <
=L$-U;L$E<> &J&J G(N,ω) < <$F=
fC(t|N,ω) M.0<>"LI < T3$U <; <;UN<;V+UN<C=B=AL7=* <
C+?<?=B=D<&<$?A= 'U
hC(t|N,ω) =(ωt)N−1 exp(−ωt)
∫∞t (ωu)
N−1exp(−ωu) du
.
'E<'L = - ?<OU&JV,P=$? ∑ <@=%V,D<'$??<<'&K<'$? =L-&.G=J <C= T3$U <C=@UU <'$B<<@=@ C!<'UBL$ LhA(t|λ) = λ M + $ <>=LVV,P=D<"($- V'<$$?/ <R ! YP<RL)U&JV=$ </ <@= N U"NUN=?L =V'<LQ=L M " <O&&<@T3KJ$ ?LE<VL ?</&J ?< B(η0, η1, β) - $J&J ?=D</U </U&JV,D<&<'$ V L$F<E<V,$E<'$<'?< <OV-&B<
λ - E<( <$F= *$ N< fA(t|λ) M
$/$F= - =A $X$&J&< X ?<OD<'&.V7=/ <( T3$-UN<*LXU&JV,P=$? ∑ - $)L"
X = min(A,C)
#"
A ∼ E(λ),
C ∼ G(N,ω)
<A<C=$?$ V'<$--$?D<@= M . ! *Y-<OL U&.V,P=$? = 'U
P (X > t) = SA(t|λ)SC(t|N,ω) = exp (−λt)∫ ∞
t
ωN
Γ(N)uN−1 exp (−ωu) du
#"SA
<SC
=$?<@=FT5$-U$F= < =LC+<* <@=@&. '<@= <$)U&JV $ M
$ % " " &% & " ('$ *) " ' " '+ " .0<@= $$ N<C= $-LF=A?<<@= ?=V$<@= =A$? ?<GV-LF= =L-C!<$? U <$7=AL N<@= DD< 5$?<C==LVV,P=D<" C<C=
$- 'V7<'$-&J&<$?* <. <V <'$-U < M "Q<.&&< LF<PV,LR?<P&J ?< B(η0, η1, β) - <?=AD<PL$ILD<?V'< <;$$ C<C=O&>$L$?D<@= $ V'<L$F<;V-!=*U$$/B<. Y$E<> L$E< T3$-U <<'<7UWC!<'&<$?(D<'<'C C< M $ <'UW"$?BV T 'D<'$??<<'&K<'$? > L$ E<C=/&J '<@=A<$ U&JV $ - $ U <$7=ALB<*L$E<P$$ N< <PL N< <OC+?<*V#CP<$-$?F <* LB<O&J+ ?< M
<C=O$$ C<C= =J<.V C=D<$?D<'$O$UL=LF=R>T5&<D = (y1, e1), . . . , (yn, en) #" yi
<C=AOL$ B<&JVF=O < T3$U <L L$ D<&JVF=O <.U <'$F=AL-D< D<PF=D<C - <*U"N-LF<C!<7UWB<L ei ∈ 0, 12 C=Y$E< Y$E< <DP T3$-UN< M .e(V-D<'&.B<BU&JV=$?D<B <OU <DC!<7UWD<'L@ C=Y$E</L$F<RU'LF=D<DUU <'$B<<Q<@P <L&<
2 0 −
L$IC+<'G=B=J<'&K<'$? M $B$F= - <C!<'UD<'L ei = (0, 1)$- LF<>L$E< T3$U <JV-C??<G=D=D<&<'$(UN< N< M
Y$E<RE<yi<@=%/$-U$$+LE<R&>G= LF<
yi$ <@=%/V-P=FL$E<*U <'$F=AL-D< - $X$B<
ei = (1, 1) M $-$ - = yi<@=%L$F<
U <'$F=AL-D< JD< - $X$-D<ei = (0, 0) M
.e?=A$F= =F<@=/ <$7=A @= <C=/ B<$?D<@=$-$ N<@=@D<'$-U$ C<C= M V,LB<@= n1
F=D<C##$7= B<<@= ?LE<ei = (1, 0) - f(yi, ei) = fA(yi|λ)SC(yi|N,ω) -
M V,LB<@= n2F=D<C##$7= B<<@= ?LE<
ei = (0, 1) - f(yi, ei) = fC(yi|N,ω)SA(yi|λ) -M V,Le<@= n12
F=D<C##$7=,D<?<C= ?LE<ei = (1, 1) - f(yi, ei) = fA(yi|λ)SC(yi|N,ω) + fC(yi|N,ω)SA(yi|λ) -
M V,LB<@= n0F=D<C##$7= B<<@= ?LE<
ei = (0, 0) - f(yi, ei) = SA(yi|λ)SC(yi|N,ω) M
$/ [email protected]+"G=J<'&$-UN<D <@=/$$ N<@=@F=D<C C<C= = 'U
L(D;λ,N, ω) = λn1
∏
i/ei=(0,1)
hC(yi|N,ω)∏
j/ej=(1,1)
(λ+ hC(yj |N,ω))n∏
k=1
SA(yk|λ)SC(yk|N,ω).
&' " ) ' " " ) " ) "
"D$F=EL$>UB<F L- <@T5 %?LE<'$?=AD< - <?<L + $-#C !"" $? V-V= <EB<C=A ?<C=EYN-&K<@= < V,L <C=A&<*?<C=RV-& B<C=
(λ,N) M.0<@= ?L-#$F=OLI&J!&PL& <JC+?=D<&P$-U <K=A$?
!=B=J< /V $?<C=( C=L-D<A<FL-$E<B C=L$V-G&.&<$?B=F RVV= C</V,L <C=A&<FVL7= =&.V-<'&K<'$?<@=BV"&D<@= M $+L& ' ?LE<&<$?B?<C= <C=A&>#$F= =D<&P?<$?R$$E<'D <P$F=B C=LB= - L-UL$ @=ALN '+?LE<*$ <@=%Q<'$-UD<*G=V$-< - >$B<(U$$-G=B=$-U < - =ALB L$U .L &>A&PL& E<(C+?=D<&P$-U < -JU$F=?=A$-U < </.V+&J! * <(U <A<C=A&JD<'L M K U L$)VB<&J<'<7UW!TE <OD<7U"NE<"U"NE< M
9 - ?<C=RV- '&K<@=B <C=A&>#$ =$?BT5$-&<'$"?<&<$?R<@=D&'&K<@= ?LE<JU <'L ?LE<P$-LF=R C$F=D<'$-U$? C= V,L- <@=%&< ?<C= V-"&B<C= L&J ?< B(η0, η1, β) M
.eBT3?<S"?<M<EDU <$7=ALB<GE<C= $-$ N<@=$LF=A<'?<C=F%L C<C= ($-#LD<R <DUN<B&J ?< G=LF<C= &>P= ?L @=ET5$?LF<D$F=S(V- LF< - $T5&>!$>&>$L$?D< <@=%SV- 'V$- "$?B<A=ALS $T5&J$>G=AV,$-?< M F<@=%S"G=$$-?</ <BU$F= 'D<'EVLL$E<><@=%&>#$ C=?<$$E<O <@=@V-"&D<@= M
. $T 'D<'$-U <.- C=<'$$E<L<@=%OUEU$F=A '<'&<$? =&JV N<.C+?=- ! C+G=B <@=R?=R < <'L- L$LF<<@=$L7=@$LF=Q=A&.&<@=@$? 'D<@=D= N<@=@L U"N-V-!D< M $ < < - <@=@&J+ ?<C= <7+V,$F<$??<0< &J&>=A$BU$%LYL @= M $)V'<L - L =D<$ L-$E<*V+U 'L-D<* <C=A&>#$ =&JB< >U <?<( CP<VV N<OV- <'AN$ 0 - D<7U$7=%!LE<R Y$E< <JU"N- ?LE<.$-$ N< - $LF=DVLC$F=B&>$<'RE<C=D?= H U$-$$F<<@=,<VU!B<C= M
/$ =J<7U$- <'UTR < B<'U"NF<"U"NE<<@=%;$-U) VD<'>I '&JU"NF<<'$D<'V?=D<XL U"NVB< V,L- &.D<'><*&J ?=D<O U$$G=D=$-UN< G=AV,$-?< - L$F<PT3 J$ T5G=RV"# ?LE<K<(U&.V- 7NE<$7=A?<H$ 'UN<C=B=A =D<"P <RT5L$- <O$LC!<7L;$-UD<L-B=@U&JV 'NE<$7=A?<C=FV-@ <7V7<'A < $-+ =%B<- @=A?<$ M
$-$ - <'<'L + $#C) $?GVV,P= BL$F<R& '"#$>V=D=<RL&J '< - ?L U$F=?=AD<VB<$-B<,<$>U&JVD<?<C= . :5 . GWM( </U"N PD<'C??<$? R?< U <'A"$E<C=E$$ C<C=E </UN<$F=LB<D< - <$.+U'?=$? L$(B<&JVF= N<+ <@ T3$-UN< - &>!=L - <$?B<GE<L(C#?<L-B= <@U <'$F=AL-D< =AL-U'U <@=D=CP<C= M . C+?=D<&P$-U <@ <@= $-$ N<@= <C=AS = &. C< - UN< ?L&J 7< C+ <&J&<$? ?<C= Y!N&<C= <C=A&J$ M/$X" C# U&JV?<B <@=%&>#$ B<C=AD< PT3B< =L/UN<*$LCP<'L&J ?< - VV 'U OV-F?<C=/$-LF=%?<G= M
9 B 2
;14 ) * /% 81/ 1 L # ,* /$F< LD< V-?=AD< E< D<7U"NE<"U"NE< - <'$?B<C+LE< V'<$-$? N C=D< - U$-U <'$E< LF=Y!< <@= H
< E > $?LB=XL_U"N-VB< M ?=$L7=V-?=B=D<$? <C=XU$F=AL-UW$F=>T5&<?<C==LF=U <V<@=@ B<OY $ "?= N<@=<LG= N<C=@V,L-JU"#$ L$)PV- !B<
π(θ|ω)#"
ω<C=A
?<OC!<'UD<'L/ <C=@N(+V'<V-"&D<@=LF<R $)U"NE<"U"NE< < M
" # ' ) $ ' % % ' " ) /LF=T3?=$7=LR X N(+VN @=J< ?LE<(?<(&J+ ?<V-& +?LE< M(θ)
<@=% %LF=% C+?=- A C?G= <@=$$ C<C= M " <C=F& N-+E<C=GV7<'&<$ <DC <'FUN<AB<BN(+VN @=J< =A$?G$&J&<$?FVV= C<C=EV ? <'YP< " M
LV-V=A$F= &>$?D<'$-$?*?LE</(U$$-?=B=$-UN<A=AL <'UWCP<BG=AV,$-?</V'<&<AB<DL>&.$7=S <R -$-GL$E<B<V C=D<$?"#$ D> O <P U$-$-G=D=$U < - L=D<$F=D "<<PD<@=AV'<'UD<L &>A&PL& <@= =V 'UU'A$F=O <J <V'<G=J< M <<7 U <C=A>=LVV,P= N<= <7+V-&<>=AL-RXC#<.7=J<'C#-< X M X <$F= = 7U!f(x|θe)
"θe<C=A/L$E<><@=%&>$) <
θ - Y N<DV- <V7<'A M + "+?< + DY$ & !" 0 V 'U$G=J<'$?@ <@=& N <C=$$ V-"& ?LE<C= - U&J&< <C=%<LH NG=AY&J&K<@= - VLPD<'$P < <V'<PL$ UU"$-D<7UWA=LL$E<><@=%&>#$XLXV"&D< M
LV-V=A$F=PB=LF< $-+ =%B<- C=<'$ V-L?=B=D<%LYP<. <V'<.U&J&K< L$HT5L$-?=B=J<'LPE<X$$ C<C=UC!<@=
xa = x1, . . . , xa ∈ χa M < L-$E< ?< a ∈ IN∗ - <V-V=J< L$F< VB<&J 'D< B<V- @=J<'$"#$ V7<' JLE<QU&J&< L$ &J !<$-$ =L?<C= 7?=#$F=> <
xa- =XU&J&< +:7H >
πa(θ|θe) =
∫
χa
πJ(θ|x1, . . . , xa)
a∏
i=1
f(xi|θe) dx1 . . . dxa,
#"πJ <C=AL$ # $-$X$T5&>T U"N?==VL&J+ ?=D<@ Y$$-U < M ?< a <@=% <'$ E<C=B=ALF=@ <
<q L$ :+@$ . . 3> . - ?LH<@=%( ?L-$?! .&J$&J<;E<;$$ N<
x(l)B<< ?LE<
πJ (θ|x(l)) <@=%VV-D< - $LF= =LYY $7=S <'&JVe L$ +: > E >
πa(θ|θe) =
∫
χq
πJa (θ|x1, . . . , xq)
q∏
i=1
f(xi|θe) dx1 . . . dxq 0M
#"
πJa (θ|x1, . . . , xq) ∝
(q∏
i=1
f(xi|θ))a/q
πJ(θ).
.6<.U"N L$a ∈ IR+ <*$$ VLF=*$F=
IN∗ <@=%( =D&J! >V-O U$-$ ?LE<πJ
a (θ|x1, . . . , xq)=PVV-D< M ?=PVL>U <'A"$F=(&J ?<C= - U&.&< T3&J?<<7+V,$F<$??<< $-#LB<< L ?<&J '<) < <L - $HV'<LJU"N?= a LF=B=A V'<! ?LE< $HC!<'L <$ VUL<'($T <'L O=S $ C=D<;L-$ C=BV# M K D< § M VLBVLF=D <(V 'U?=$ M B$ D<'e C=L!"# - =L/L$ &J '<( V-L U'# LG=J<'GU&J&K< <'-L - ?=B=D<A<C=V 'D<'*?LE<RU <D<D& N <RV'<&K<AB<D <D&. 'G=D<@ <BY!<@=EY&.&<@=S )% 7L # M <L U0=$?V-T5G= <$?B<&<$? <VUD<C= - U&J&<O-$F=@ <7 <&JV?< =ALC#$? M
!#"$&% '($ M*) ?>+N L-VV=A$F=GB<$F L$<V'<GL$E< <C=A&>#$
xe <BPL C<D <DC+<R& '$E<R L$ U&JV=$?F <BL
2 0 −
<* T3$-UN<λ M$BB=F <@=%&>$ < λ <@=% λe = x−1
e log 2 M $F=
a < 1 M$BB=
πa(λ|λe) = aa+1λeλa−1
(λe + aλ)a+1 .
&' " ) ' " " ) " ) "/L7=JVV=A$F=PVL7=A?<L =J <'UW!T=>E<XB<'U"NF<"U"NE< @= L?=#$ <@=%+: > E >
<$ $-+ =J<; C=?<$$E< =L<7UWCP< M G$LF=L=D<&P?<O<$ < < ?LE<XU <AD<XVV-U"NE<;V'<LK=D< C '<'VLF=@U&JV NF<$F=-<DVL@L$ $-+ =%B<B$-LF=A<' M <'V7<'$-$? - ?<RVL7= =L-C!<$? - πa(θ|θe)
<C=AFL$F<R <'$F=! $< FL$E<$? 'Y"?<< DL-$PT3UWB<LE&PL!V-UT,VC= - <S$-UFV'<LJV"# ?LE< LG=#$ M + $. C=B< =@ < ?</CP<'UWB<L
ωE<C=EN(+V7<'V"&D<@=E <
π(θ) = π(θ|ω) - L$E<D <$F= , <T5&</"D<<B <O&>$-<'&K<'$?/VL7=A=A&JV?< M 0<@=%BB=G$-#LB<e <>= ?<'U$$E<'
ω∗ = arg minω∈Ω
D(ω) 0-M
"
D(ω) = KL πa(.|θe) || π(.|ω) .
<D< <$?V<>B<#CP<<C=AU$+CP< <<$)=A$HYL&<$?π(.|ω) M
$B$F= - ?<>&.$&L& <G=%B< <L<@=%(L-$+?LE<=ALF=O?<C=*U$-$F= ?LE<
π<C=A(E<$?-< - =A!?LE<J VVU'#$ Ω : ω 7→ π(.|ω)
<@=%($<7UWC!< - < ?LE<D(ω)
=P <'L T5G=( 'D<'$?< <$ω M
B$F= ?LE<=Aπa(.|θe)
<@=% 7-$?< )L$IT3UWB<LP&LVU'#TVC= - UN<L U $ XL-UL$ <J$7=D?<JUUL- < ω∗ M <C=A>=J<'L<'&<$?*$ 'UN<C=B=B<P <.V,LC =&L?<O <C=Y!<@=
(θ1, . . . , θM ) <J <$7=A
πa(θ|θe)!-$ <C=A&< D(ω)
<>=D<C=R 'C C<C= <$ L-$E<JC <'Lω - VO <C=
& N-+E<C= < $?D<7 M <C= & N <@=H=$?S = +V+?LE<'&<$?SULV N<C= *E<C=E& N <C=S <Y"?<$? -U < ?L V'<L/?<C=BL"@U$F= "?<&<'$ M
.eRVB<&J 'D<@;UL @ C= <$-UF$F= ?<F"D<'&K<'$?EE<FD&.$&J?=#$ 0M M + $PV7<'L V- <7 <&JV?<= VY$E<'/ <L"E=DUUL-?=Q<$ 7U$?DE<PU"N?=
π$7=/L$E<.U!=B=J<(E<P <'$F=A @= ;JT5?=R-V N<C=
U <D+V7<O <O @=AL$ <UN<V'<$--$?,=L =&<$? F<<@=FVLB<DT3U?<&<$? =A&L-<@= M 'E < <&JV?< -=A=$= % =$?F,$ @= - $ NE<'L VV=J<'$?F L?=D< <@= =LD<@=FE<DT3&J<@= <V$E<$?<<@=@VLBVV+&<
πa(.|θe)L =D<$F= <OL--U#" M
<.U"N $ <C=A*U <V'<$$?OV-!=BL%L = %L-U?<L M $L7= =D<&P<.$ B<C=B=$?D <JU"NG=ARL$QUN<$$&PB<G <ST3&J?<C= !$$E<'?<&<$? LG= N<C= < <GT5L-$ <C= YN&<@= <G&J$&J?=#$V @= M $/$F=A - $-LF=/ C$F= >D<'$?L ?LE<'+?LE<@=/ C=L!"# =A=AL-B <C= # > =AL-UWLD<*N- 'U"NG= N<5 L-$) <'$D< <'L<@=%@T5L-$6<$($/$$E<7 <C= § 0M IM M
. LD<S;UL-! *?LE< $LF= V'<"U <'C$F= C= < -$F=e< ! G +: <S VV-&J$ M B$E< T5?= ω∗ <@=%& -L$E< ?LE<@=%$ Y!&<><C=AB < = C,=Ae> <'$F=!
π(θ|ω∗)VV&<R?<$
πa(ω|θe)M /$F<><D<'L A
V7<'L D<.UUL- C< - $+?L-$?DV'<$E<$U <JL U"N- E< π M $/$F= - <'B<L-DB<#CP< ρ <$ $-&K< L1 <'$
C#$;"?< V'<LB<R&> % N<O$F=A UWT M ' " 0E- U"N-V-!D<
ρ ≤√
2D(ω∗)
‖πa(.|θe)‖1
9 F . G 202
#"
‖πa(.|θe)‖1 =
∫
Θ
|πa(θ|θe)| dθ.
-L$eL$.U <$$&PD<G <G C=LB=e-$F= B&&<GCP<$E< - T3G=$? $?D<C!<'$ <@=e$$F= <D<'L U& V NF<$F=-<@=/V-DL$Q$- =AD<$-L7=%?< - $-LF=DV-"/ B<L$Q<7UWTG <PD<7U"NE<"U"NE<*$? B<C=B=$?B = ?LE< $XU"NF<"U"NE<=&JVF</.&J+ ?=#$ M
/$F= ?L L$nLD< V- '&< D<' IIV <'U$l $T5&>$tV- OL-U#"9<C=A <= C =π(θ|ω∗)
$E<R"$7=A&<V-P= $T5&>$=LVV- '&K<'$?B< # -$F=F $T B<$U < .U <?< ?L6<C=A C= N<V-O $- =AD<J- C=<'$>=UN<'U V7<'LL=D<J"LB<.VOL$E<> 'LUW$H"!=A ?LE<J <>XC#$-UN<JE<
θ=D<$
π(θ|ω∗)C?G=- ! C+G= < C#$-U <Q <
θ=J<'$
π(θ|θe)M R$-LF=%=J<'&?<I$-U &JV,$) B<$)L$E<
VU 7LB<R <(U&JV-?=$E<(U <C=E<L C?G=- ! C+G=FE<R $T5&>$)VV, C<RV?<C=$$ N<@= M
;1% # % &* / ) * % 12% # )$# ) # 1 * 81 ( % /% %(* * #
/LF=R C$7=@T5L$ L § IM L$E<.U!+?LE<. < Y!N&< + ?L VV,P=D<U&.&<E<$F=! P$F=AL&<$?"?<
ρ(θ)Q <$7=A <@=L<@=%&>#$F=JL &>!+&L-& E< C+"G=J<'&$-UN<
θ1, . . . , θM 'U"N$?$7=
$$ C<C=*&>$L$?D<@=OD<7U$7=%!L N<@= M L( <7#<'&JV<;E<;)G < <LG <;V"&D<θ = (µ, β) -
L$F<U&.V"G=A$ $+L& LF< C!<7UOL$E< <'$F=! ρR(θ)
<P T B<$-UN<(U$F=%L-!B<L § IM -M F- ?<C=R T3LB=D < + $? *&.G= <'$ 'C? <'$-U <*V,L <C=A&J$X- @=A?<$-$E<* <RT5$-U$F= < θ M
+ $JU$7=%L U <AD</E<$F=! ρR(µ, β)$F= L$>U <$JUB< - " π(β)
<C=A ,$ C< <π(µ|β,yn)<@=%,<VU!B< M "R$F=@L$XUD<RVLF=FY '$ " - <$"&J&<$?,=$7==LVV,P=D< ?LE<R <C=V-UN<RE<C=FV-"&B<C= Θ<@=%,$ - $LF= @=A$F=@VV,P=D<@L$)LD<*U'$- ρJ(θ) M
' LD<&.&<$? - $D< &>"U"NE<*B<C=AD<(VL-D<'&K<'$?Q<&JVG=AD< < = V-VL?<L=AL-B?<(&>$?<&<$?O <C= $?LB=;LtU"NVB< HVL; V"#$tLnU!B<("%$ DM ! πJ L$ ( $$$T5&>#!T <.$$F=
x(1), . . . , x(L)E<C= :+@$ O. . E > . U"NG=?=V-&J
<@=>$-$ N<@=ynM /LF=> C$F=B<&>?L LF<X =LF< Q< <@= <@=%>T3-<)V-."V-VA
n - <@==&J&<C=@N& +?LE<@=/ <! H
L∑
i=1
πJ(θ|x(l)), 0-M
$O <C= ?LE<LF<C=DLE<C=A<RV-C+ Y<'$R <C=DC#?<L =DE<θ=L C<C=R$F=D?<C=DC <'L =/N-LD<@=D <.;C+?=D<&
$U < M " <G/&&<GT3KJ$?LE< + - $LF= VV,P=$F=e <GB<&JVUN< <@= $$ C<C= &J$?L-$?B<C= $F=yn
V <@=E$-$ N<@= =A&L N<@=E&>Y$-?<&<'$ =D<$.O A-$J <VB<$-D<,<$>U&JVD<@LB< $T5&>#$G=V$-< M + $ <@=AVB<;$F= ?LE<>)U&-$-?=$& 0-M VC+ Y<J?<C=*C#<'LB=*E< θ ?$?OL$E<>T5 %?LE<'$-U < '<'C N<O$F=FJG=%L$ # H M + $)U$F=AL!/$-UDL$E<( <'$F=A O$F=AL&<$?<OU$-#B< ρJ(θ) <OPT3KJ$ =ALC#$?D< M
ced#p9<iWa-]j?`θ1, . . . , θM ∼ π(θ)
d
2 0 −
df]gj = 1, . . . ,M
fp <ia]j?`zj ∼ k(z|yn, θj)
`^ fp g` s fp=<#^=i ^]-`M
s-h p^=ij+jfp=<Hk+`n^ h ij+j?`n x
(1)n , . . . ,x
(M)n
d+fi ^x(i)(l)
]+pnk h p,<lj s-h p^=ij+jfpx
(i)nd
+fi ^Lj?`tp-f a r g`tk` b h g s-h p^=ij+j?fp@d
Gd p s fi?<+i ^ h j?fg< s f a+a=`lk`p=<i ^ i7p=<#^+g]a=`p^ h j?`
ρI(θ) =1
ML
M∑
i=1
L∑
l=1
πJ (θ|x(i)(l)). 0-M
0-M I
"D$F= < <'&.V-< =LC#$ - $L7= $LF= V- J$7= -$F= ?<C= &&<@=EU$!$F= <V &<$?"?<C= ?L L § IM -M -<$)L?=$?B?<*&J ?< < <L M /LF=/U&JV-$7= = ρR(θ)<ρJ(θ) - LF<*$LF=/D<'V J$F=,<$ V'< =$
V7<7UWCP<F CP<'U + M L- UN<< <&JV?< - ?<FU"N-*E< ρJ (θ)<@=% ?<$.&K<'?<L?LE< + < $$E<GE<C= @=AL-!"#B=
C=@VU"NE<C=@L U"N X <* T B<$-UN<ρR(θ) M
! "$% &' $ -M#) ?>+N > B DC8E ?/LF=B<VB<$$F=@?<C= 7U"N-$?$F= < <L U <'$F=L @= - =A&L C=BL § IM M M "R$F=/<@=@<7L 0M < 0-M -VL@U"N- ?LE<BU"N->E<R <'$F=! B$F=%L-&K<'$??<Q<,=D<$VL7=A?<L = =UN<$- % - $LF=G$-+?L$F=SL-$;V?<G <BC <'L =EUB<C=V$-$? L <C=A&>#$F= < $?D<7 RE</*L C</ </C+?</&J !<$-$E< 5 <@=AV "$U <>V 'UWCP< =AL
X - $ N< D - E<O=A$ 'UA +V'< σD
<(?<;V,L"U <'$?YP<> < =AL-C+?<>V 7UWTt0 = 130
=( C#<'L(&>A&J<> <> U <'$F=LD< - $ S130M /LF=*$-$$F= <$ $/$$F< <C= § 0-M IM ?<
<C=UVT <ρJ ?L0<@=%A<'$?B<&<$? <VU!B< </.& N <( <>=A&L-$) &JVA"$-UN< M
9 F . G 2
a σβ ρJ ρR
!! #"$% !! #"$# !& ! ! '"% !() ! )*#"$#"$* ! )#"$'"$ ! '"#"+ ! ! ! ! , !( ! ! *% !& ! #-% !( !! #$# !( !! '"+' !& ! !! "$# !() ! #*#"+ ! ), '" !& ! '"., !(
0/ 0M 21 V?< =(D, σD , S130)
V,L/L-$ <$ UUX C!<7UB?<C=D$-$ N<@= xe = 100 M$ + <B?<
U"N ( <@ <'$F=A @ &JV,$-U <ρJ =J<'$ 0M T5L$?=B=D<$? E<G,$$F<C= <C=A&>#$7=eC+G=$ A C+?= <G <C=A&J$
<O T 'D<'$-U < 3E<$ 'D<*U$$E< M
a σβ 34 ρJ 5 ρR
! ! '"*, . ! -'"$*# ! *6"$*% ) ! )"%)"%"" ! ).%$6" ! *' ! "$ - ! *## ! !! #$ ! ! ! ", ! ! '-#$ ! )",.% - ! $2"$,- ) ! -' *#"+- ! '"+-6"$ ! 6"$*#"+ - ! '"$-# " ! )% ! !& ! )"# ! !(
! ! ! "% ., *'"+ ! *'"$ ! ! ! ' .% " . , %) .$-# #)" ) ! ! )#"$* !! *%*# !! .'* ! 0/ 0-M ,1 V?<B=
(D, σD , S130)V,LBL$ PV- 'U' ( C!<'UD?<C=$$ N<@=
xe = 50 M $BB=?LE< +
=L=B<OE< T3LB=B >V'<+JL7=/L § IM -M - <(U"N ρJ E< <$7=A ( &.V,$-U < 0M B<C=AD< =#?=AT3G=$? -C+G=$ A C+?=FL U"N <* T 'D<'$-U < 3E<$ 'D<*U$$E< M
2 0 −
;13' 1 # $ )% "
L-$-π<C=AXU$F=ALX <QT3KJ$lN ""U"N LF< - V-V=A$ 0 V'<&K<X < = V-D<';V-X$C!<7L
N ""U"N+?LE<R?<C= "V'<C= <*.&J$&J?=#$
ω∗ = arg minω∈Ω
D(ω)
"
D(ω) = KL πa(.|θe) || π(.|ω) .
0M (θ1, θ2) ∈ Θ1 × Θ2 ⊂ Θ
(ω1, ω2) ∈ Ω1 × Ω2 ⊂ Ω E E : π(θ|ω) = π(θ1|θ2, ω1)π(θ2|ω2)
9 ω∗ = (ω∗1 , ω
∗2)
ω∗
2 = arg minω2
D2(ω2),
ω∗1 = arg min
ω1
D1(ω1)
D1(ω1) = KL πa(θ|θe) || π(θ1|θ2, ω1) ,D2(ω2) = KL πa(θ2|θe)|| π(θ2|ω2) ,
; BLF=e$$-$F==@VB<LCP< $F= < U'!= #"πa(θ|θe)
<@=% -$?<EV- 0-M M $ C!<7U πJ(θ) = πJ (θ1|θ2)πJ (θ2) !<A<'$ $$xq = x1, . . . , xq !
$)?<$?&J& 'D<&<'$ ?LE<
πJa (θ|x1, . . . , xq) = πJ
a (θ1|xq)πJa (θ2|xq)
/$7=H = −
∫Θ πa(θ|θe) log πa(θ|θe) dθ
<$?V< E<πa(.|θe)
M /$F= Y<'&<$? νe(xq) =q∏
i=1
f(xi|θe)M
$/ =
D(ω) = −H −∫
Θ2
∫
Θ1
∫
χq
πJa (θ1|θ2, xq)πJ
a (θ2|xq) log π(θ1|θ2, ω1)π(θ2|ω2) νe(xq)dxq dθ1dθ2,
= −H −∫
Θ2
∫
χq
∫
Θ1
πJa (θ1|θ2, xq) logπ(θ1|θ2, ω1) dθ1
πJ
a (θ2|xq)νe(xq) dxqdθ2
−∫
Θ2
∫
χq
πJa (θ2|xq) log π(θ2|ω2) νe(xq) dxqdθ2,
=
∫
Θ2
∫
χq
D1(ω1|xq, θ2) νe(xq)πJa (θ2|xq) dxqdθ2 + D2(ω2) + ∆
#" D1(ω1|xq, θ2) = KLπJ
a (θ1|θ2, xq) || π(θ1|θ2)
<
∆ = −H −∫
Θ1
∫
Θ2
∫
χq
πJa (θ1|θ2, xq)πJ
a (θ2|xq) logπJ
a (θ1|θ2, xq)νe(xq)dxq dθ1dθ2
−∫
Θ2
∫
χq
πJa (θ2|xq) log
∫
χq
πJa (θ2|xq)νe(xq) dxq
νe(xq)dxq dθ2.
' L-?L∆$( 'V'$-XV-
ω ! &.$&J D(ω)'C+ '$.&J$-&J'
∫
Θ2
∫
χq
D1(ω1|xq, θ2) νe(xq)πJa (θ2|xq) dxqdθ2 + D2(ω2).
L&$?V,!!T/$- V$-$? M .@&.$&JL$?/
ω∗2 = arg min
ω2∈Ω2
D2(ω2),
ω∗1 = arg min
ω1∈Ω1
−∫
Θ2
∫
Θ1
∫
χq
πJa (θ1|θ2, xq)πJ
a (θ2|xq) νe(xq)dxq
logπ(θ1|θ2) dθ1dθ2
U ?LA ?LC# '$J '$$U **PV-VA$ M
% ' # ! &+) " % ' " " '%$/L!V$-$!@U /U$-$!/( '&.V- ") (U'UL A&J$!, # VU
$!. P- 7L 0-M " 0M $ 'U#$?>)VLC&PL '$'$ Q'$!! X$!AL&$? ρJ(µ, β) 7-$*L § 0M M
/$ V,L %'L-&AL$ ULV- (xi, xj) = x(l)
L!xi 6= xj
xi, xj > 1
L$ ! MB$! ?L Y '&$? ! V@ § M M ! J$! ρJ (µ, β)
AVU!' 'U !
ρJ(µ, β) =1
ML
M∑
i=1
L1∑
i=1
L2∑
j=1
ρij(µ|β)
#"L = L1L2
µ|β ∼ G(2, xβ
i + xβj
),
ρij(β) =(xixj)
β−2
2| logxi/xj |(xβ
i + xβj
)2 .
. *;UL! DBO& ND 7U"N-$?$$-YAG$U(&PLET3U '&)'$?GS 7#$!Eρij(β) M
B$*&JV(& NP U'U#'V$ + ,/ ! U"N-V M V'&R 7UWLDV&$?DU*&PL#$-'$. ' 7UW$-$-$?/ BVL!BVVV M + $)VLR'&>#L! ?L ∀β > 0 !xβ
i + xβj ≥ maxβ(xi, xj)
M + $-$) 7L L!
(xixj)β
(xβ
i + xβj
)2 ≤ exp
(−2β log
max(xi, xj)√xi, xj
),
≤ exp (−β| logxi/xj |) .$B
πij(β) ≤ T (xi, xj) gij(β) "
gijA/.$!! (O.7V$$? E(| log xi/xj |) !
T (xi, xj) =(xixj)
−2
2 (logxi/xj)2 .
$B$! ! $& V-V'XLnY $ ''L 7#/ 0'UL VXL$_YN-&)Q +"L!)HY & +?L G(T−1(xi, xj))
M . AV "$U#OL)$&P* V-V' JU#*Y $ ''LA T (xi, xj)M $B !
max(xi, xj)
min(xi, xj)> exp(1/
√2) ' 2.028 ⇒ T (xi, xj) < 1.
DT3$? ! $ VV,R' 'U$$'*U&JV,@L$ L-$OC 'L@-(B L-$OC 'L@N-L$! 'U"N$?$ V,L-E&.$&JE@'&.VE/UULD& N M /L& L!&$? ! U#& NPL%L-$$ *@ AL@"V ! &&1$!@VV+?L'/U#*Y OV,L/U"N-?L M
Annexe A ; B(η0, η1, β)
. UU J V Y* ! U! Y 7U ! " #$#- ! ! !&% ' ( + ) * VV+ 7U " U " , $, - 0 - & ! ! . #// V V+ V00(& 1/ V 32 G 7 54 36 A 9 H 7#!. $ 9 A : 4 -' =<84?> @ B DCFE > DCFE G94?> -' ' (' G 7 54 > !! H ;: " ! / < 1/ ! =0
T > *1 (0 $ 7 / 3 ) ?
@ A0 X
!B /C"& V $ DE F" ! !B& / @V % EV&$!D YGH ) * ( $ :' E ! ! E)#0 0
y> V0 #/ .
yn:
" % $ Y 7*/ ! / *" ! $" ! " I $ ! "I J( V G 7+V0&(0 " ! " , $ */C"? V+ - 0 - : $ % 7#) JV ! 'K"& L( // V !% V00(0 /!* #M(( 2 G 54 ?6 A 9 H % !" 00:
$ "&$ − , B(η0, η1, β)
+*1 # * &6 1 4 %13* # 4 /;1* *;1 1 / * # (* # # 4 / 1 # 13* F41/ ,* *
"!$#&%&'(*)+%-, . 0/21435 &6+143 78/ :9 8/ 0;=</ >7 8/@?A78/@6CB DBE3F6 0/@?:5 GB5@7 *B78/@H$"786I78J4K &/@?:5@5L</ >7 8/@?A78/@6C/ 5@/ 578/@1%MB :B+5ON 0/ P/ 3 86+3F6< RQ J:/ L</ >7 8/@?A78/@6,&6 03F6*< 78678JS14/ :/@1'1T? 7VU ,WA3F6 X78/2B+5FB)*<YK &/@?:5@5<A/ >7 8/@?78/@6 SZ S146+78/@H+B78/@6[/ S786D7\B+;$]B)&6 X76)R?F6+78JIB)&/2<X7\B5^B :<GB+9+/ 9%_B+/@5 / I5@/ 78/@1%`<:B7\B4B)*B5@Na0/ `b / >7cd78J`1B+/ eJ:B) \B e7 8/ >78/ 6 P78J/ f</ >7 8/@?A78/@6B) Y3 X7< ]Q JI78J6>78/@1B78/26 6)P/O7f3:B) \B1%7 B& '&6 0/2< &<E78J 86:9+J1BWA/216:1g5@/@;$&5@/@J:6X6<B *< hB&N+0/2B)/ `i E3*B& 078/ :52B) 78J6eW/ >7Y6)LB,/kj+&6 0/ >7X73 86X67h6)l78Jm52/@;$52/@J6X6<,&j:B78/@6P/ P3 86H$< on &/ 0/@67>7786 eJ6X6S? 7pU3"B %WA3F6 X78/2B+5_cqK &/@?:5@5B)*<r78J/ &6-143 78/ 9 8/ 0;<A/ >7 8/2?78/@6B) 43 X7< . *<E78J:/ ]B+5O7 :B78/@H$4146< &5/ &6+143*B& <r786"78JYK &/@?:5@5146< &5 86-1 1% 8/ B+5 WA3 8/@1%X7h6[?F6+78J &B+5sB *< 0/216:52B7&<[<:B7\B'7&c03 &/2B525ON[/ GB [/ :<>7 8/2B5&6 X7W7
t[uvlwx '+yo#-z b B+/@5 Q /@1% n / >7 8/@?78/@6l . 9+/ :9K &/@?:5@5 n / >7 8/@?A78/26 s . &/2<X7\B+5 b B+/@5 -L|P6-143 78/ 9 8/ 0;[6< &5_F~B5@9-6 8/O78J1G hB&N+0/kB) i 5 - i 143F6) 07\B 48B+1435@/ 9:s/@;$&5@/@J:6X6< \B78/@6 Q >7
F * # / * 9 ! 1 ( '8 / # ! % " > 
B
" . % H/! ! %L% ! #// ! C % / =@0 ! : " ! */ q % >)!$ d % * / / G * d \M@ "&/C"C /!/ & *" % #/; *"&M % $ / ( / ( d ( #// ! : "& 9>)!
& % #$ : $ % " * #= % * &) * (0 B = min(E,W ) " E
3 * &) *1 (0 $"M *+! 1 * (0 * # E(η0) I
W J * &)# *1 (0 $"= (00
* (0 W(η1, β) " β > 1 ! E &W
( d &&! : ! ! ! " * (0 %B
/C"& C/ M( ="& C η0 ! η1 &
β : L(0 B(η0, η1, β) : % * % 0 /; " 10 % *" 'F+! 1 I (00 % C
hB(x) =1
η0+β
η1
(x
η1
)β−1
. $ :
" % *"0 < *" + ( $8 % ! d B(η0, η1, β) ! % W(η, β)
* (0 % )!$ l d : " 3! ( * /C" / . % * d β > 1 , * ! ( " % /; / #!
$ / % % % (0 I (& 0(0 $7 ! d : /C" / # < / * " & G d @ / 0 $? % E % . C / . 0& 8 & ' //!/1 ! ! /C"I ! + :;: ) % 7C '8 0 ( / #! "0 f :
" ! % / G * d * e?>)! !$:@0 I % % * (0 * #! "I L(>0 C (0 ! ' & ' F 7 ;: " ! % / G d * e 0 (& 3 % A "0 1/ ! *"& " B * (0 * # % F ! * / " ( . & , G ! -A - & "& &/+ ! -AA ;: E" B * (0 * #= e /1 / '< ? %*" 1 G0 ! C/;*/! I $ / G d * e @ !d $" #0 "0 D * ) :
" 7*/ ? ! % #$ : /;* # *" /C"& C/ * */ % *" B(η0, η1, β) * (0 * # r ! : /;* # ! " - * # % *" B(η0, η1, β)
* (0 / #! : J0 " @@ * % " "0 ] C K % " B * (0 % I (0 J "G/ ! ! *M Y ! *"0 #d "" *$" : " ! q / 8 % *" C S ! ! d GI 7C / ( G0 d d /C"> d q *" + * (0 * # %*" C : /; / / '*" "* + " : 1/ & ! " GI 7C #(0 ! % * l (00 * (0 * # ! . B * (0 / ! ! : /; - . *" C = % @0 */! F # (I " 3& ! ! e /;*$ & * I " 7 / ! " /;* # !& *" 4 :
# *1 # * # *;1 B / * # (* . * &)# * (0 : :O B = min(E,W ) !f " " : : E
"& F+! 1 < * (0 * # $" $!
η0 & " : : W "& % (&0$ * (0 * # '*" /! I C η1 & "&
* β ! E &
W( d &+!&! : " H /C"I C/; * 1/ D % *" B ( ( $8 * (0
% #$ : "& % 0 /;* # ! $ ! 1 ( '8 *@$ % & /;* # & ' " ( ( $8 ! $8% 0 /; + % '7 /!/ $ !M(
hB(t) = hE(t) + hW (t) =1
η0+β
η1
(t
η1
)β−1
, t ≥ 0
SB(t) = SE(t) × SW (t) = exp
− t
η0−(t
η1
)β, t ≥ 0
$ "&$ − , B(η0, η1, β)
fB(t) =
[1
η0+β
η1
(t
η1
)β−1]
exp
− t
η0−(t
η1
)β, t ≥ 0.
$ : - + $ : ? e0 q $ G0 % B + % $"M" / * \+I l1F+! * & % (&0$
+ % :
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
exp(2)Weibull(1,3)B(2,1,3)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
B(2,2,0.5)B(2,2,1)B(2,2,2)B(2,2,3)
# $ : G0 % B % :
@ % ! d " # % / S C η0 Iη1
% $ *F ! 'K L % " 4 1/ 0 E/! F % "& D C β = 2 " /! / 0 * #! C "G % : "0 7* d ! " 4 #( ( '8*"&H % * % E
#// ! * % f =(
P (B = E) = P (E ≤W ) =η1η0
√π
2exp
[η21
4η20
]erfc
(η12η0
)
" " % & /;* # erfc
erfc(x) =2√π
∫ +∞
x
exp(−u2) du.
(0#$ : e0 q *" ' = % "0 ( ( $8 % 0 /; % C η0
η1
:
η0/η1 ! ! ! !
P (B = E) .* . &" "+ " ! *
(0 $ : ' ( ( $8 % #// ! C % $ % 0 /;* #J % C* η0/η1 B(η0, η1, 2) * (0 :
" *" * d % 0 / % B % β = 2 / ( *' 7*" % C G(u) = 1 + exp
[1
2η1
(1
η0− u
)2] [η1u
√π
2
% /
1
2η1
(1
η0− u
)].
$ : 0:
B
# "0 * # ! '< f+ (0 * " . & " 1 /
[T ] = η1 exp
[η21
4η20
] √π
2
% /(η12η0
)
&
[T ] = 2η1 exp
[− η2
1
4η20
](− η2
1
2η0
√π
2erfc
(η12η0
)+
1
2η1 exp
[− η2
1
4η20
])−
[η1 exp
[η21
4η20
] √π
2erfc
(η12η0
)]2.
H / # d=( #/ " I! C 7* d ! 0 7 * d ! B(η0, η1, β) ≈ W(η1, β) η0 >> η1
&B(η0, η1, β) ≈ E(η0)
η1 >> η0
: / I 70 (0 / # / $" $ ! " d/! (" $ ! ' (0 7&) "&η0 ≥ η1 !
( /! !η1 > η0
G0 ) % ! / % /!/ * % : *" % #$ d ! *"0 * *& 7&[ +(< " q % C :
* F4 ,* *;1% # 413*1 # # * 1 B / * # (*
+ !#"%$'&( ) *&(,+
- " B * (0 =*"0 C * (0 I% * d ' * /! ( " #d "G % ?> / & *"& * * l " = C F % % (&0$ * (0 * # : " ! ! d *" 30 ! " @) / 0 ( C % " B * (0 * # C ! / 0 * : - " ! ! 0& ! ( * ) &/. 7 7( " ! "&3" " *" J0 *$ 1 ` * ) e / $#d "&/C"9" F ! " @) % 0 /; !& & $8 :0 " !$]d& "ID*" < $" @) ! "& 0 / ! G ! ! ?8 / ! !/! : - "& 0 ! ! " E 30 ! " @)>) " ) & ! /! m ! * !% E ! * / L "& d A:
- " $ / #&#(! */ "& $ / $" G) ! 8 % ! =0! * (& @0 */! 0 : -LY K #0& "0 K / 0 83'B " + (0 C % " % /K"& *" B * (& /! ( + / G0 3 C ) : - "0 (0 9!1 * * # $" $ " 21 k# *'*" % J0 ! " @) 1 . * # "(G! ! . & ,E0( ! A " *> l C *$" $ % q 1 % / - 0 F & d ! : $ ' $B(J& 1/ *" /;* # ! *" * % *"0 /C" / "& *" H " 1 + % " 21 k# *'*" + C $"= e /; " D * % " F+! 1 & (00 * (0 : - "0 *& ! ":>*" <@0 */! E( * > ( C (0 q ( >" ,1 $" : < 1/ *"&B*" 21 k# *'*" "& B(I @ ! !! * ) ' * / G d * e ) $" e C ( " & 0 AG ! A ! #/C"& ! "0 d : ! * &" /C" 7 ! d> *" ;: F! "0 $" * :
$ "&$ − , B(η0, η1, β)
" 2 & *
. y = (y1, . . . , yn)
( G0 % # B * (& "01/C"/ / C 8 "GH/ !! C d / ! * d * F: /C" yi
/ M( *' yi = (ti, δi) !d "
δi =
0
% ti / 7 d )
,1
% ti % $ .
- " ! " .#(! * !9 ! " @) "&/C"
L(η0, η1, β|y) =
n∏
i=1
[fB(ti)]δi [SB(ti)]
1−δi
=
n∏
i=1
[hB(ti)]δi SB(ti),
/ M( $7
L(η0, η1, β|y) =
n∏
i=1
(1
η0+β
η1
(tiη1
)β−1)δi
exp
[− 1
η0
n∑
i=1
ti −n∑
i=1
(tiη1
)β].
$ : A- " B * (& *" J * (& % 9/ G * d e9) '*" dM C : - " 3 d
C =( e I /! /1 ! " % $ : % ti % $ !f &!zi = (zE
i , zWi ) " zE
i = 1 &
zWi = 0
% " % * ti % # " *+! 1 * (0 * # &
zWi = 1
IzE
i = 0 % ti % # *" (00 * (0 : / ! % ti 9/ 7 d
δi = 0 ! zE
i = 0 I
zWi = 0 : - " ! ! " / G0 C /! ( *' !
x = (xi =
(yi, zi), i = 1, . . . , n) = (y, z) : - " ! $8 % / G0 .(! ** #xi
f(xi) = [fE(ti)]
zEi [SE(ti)]
1−zEi [fW (ti)]
zWi [SW (xi)]
1−zWi ,
f(xi) = [hE(ti)]zE
i [hW (ti)]zW
i SE(ti)SW (ti).$<& *" / #[& ?$ $" G) / ( $7
l(θ|x) =
n∑
i=1
zEi ln [hE(ti)] + zW
i ln [hW (ti)] + ln [SE(ti)]) + ln [SW (ti)] .$ : :
- " ,1 $" / #! M % ! d " / I '* # F /** # % *" / G0 8 $" @)@0 l *" .(! * 9 * I /
θ % *" 4 * ) ' C*$ $"
"(G! ! . &%,<0( ! 1 / . /C" I * " A;: - " + /! / 0 * d *"0 / #& $ F /;C ! 0 M(
Q(θ|θ) ! & " 1 E dQ(θ|θ) '*"= e /;
θ :
H/ #! % / / 0 d Q(θ|θ) ! θ ( l " / C :
Q(θ|θ) = E(l(θ|x)|y, θ
)
=
n∑
i=1
[E(zE
i |y, θ)
ln [hE(ti)] +E(zW
i |y, θ)
ln [hW (ti)] + ln [SE(ti)] + ln [SW (ti)]]
=
n∑
i=1
pE(yi) ln [hE(ti)] + pW (yi) ln [hW (ti)] + ln [SE(ti)] + ln [SW (yi)]
B
" pE(yi) \ : pW (yi)
*" / #& $ d ( ( $8."ID*" <( ! % $ yi #!
*" 'F+! 1 m e : (00 * (& : - " Y #( ( '* /! / 0 ! %
pE(yi) =
% yi / !!7 E
δi = 0) !P(zE
i = 1|yi, θ)
= P(Ei ≤Wi|yi, θ
) !
$"
P(Ei ≤Wi|yi, θ
)=
fE(ti)SW (ti)
fE(ti)SW (ti) + fW (ti)SE(ti),
=hE(ti)
hE(ti) + hW (ti).
!
pW (yi) =
0
% δi = 0hW (ti)
hE(ti)+hW (ti)
% δi = 1.
Q(θ|θ) / =( $7 Q(θ|θ) = QE(η0|θ) +QW (η1, β|θ)
$ : "
QE(η0|θ) =
n∑
i=1
[pE(xi) ln (hE(xi)) + ln (SE(xi))]
&QW (η1, β|θ) =
n∑
i=1
[pW (xi) ln (hW (xi)) + ln (SW (xi))] .
- "0 0 '*$ > / #[+ $ % Q(θ|θ) ( ! " / * (0 % *" F+! * & " % (&0$ * (0 ! $ % / 'C *" 1 "01/C"M E0 / * (! : / #! % *$@ d
θ = argmaxθ Q(θ|θ) : # $ : ' ! *$ η0 = arg maxη0 QE(η0|θ), &
(η1, β) = argmax(η1,β)Q(η1, β|θ).- " ! *" % d) ! ' C "G % ) #( * !!
η0 =
n∑i=1
ti
n∑i=1
pE(yi),
1
β+
n∑i=1
pW (yi) ln (ti)
n∑i=1
pW (yi)−
n∑i=1
(ti)β
ln (ti)
n∑i=1
(ti)β
= 0,
&
η1 =
n∑i=1
(ti)β
n∑i=1
pW (yi)
1
β
.
$H& " 0 dθ( / *" / C :
$ "&$ − , B(η0, η1, β)
- " 21 k# *'*" / *" (! * . ! " @)L(η0, η1, β|y)
/C"J' C & F /; / " 1 . > % η0, η1 &
β0& F/ * / & ! "(G! ! . &
,E0( ! 1 / . #/C" I " A A;: r 7*/ 0 ! " '* $I $ % 21 "& J( / 0#d "8 " 1 . * > % η0 ! η1 &
β & *" $ [& $ J 0#d " "! *" / #
! / % " 21 *$" " 0 / ! ! * # % *" ! " @) * #! : # F * ! 21 / " 1 . . % η0 ! η1 &
β % ) C 9/ !!7 G0 '*"@ C ' :
2$ + + (++) & (2 & ++) $ +
J /! ! " & * > % 1 ( $8 * (0 * # % & /;* #! "& ?(E)# % # 7 $ & "& " / !!7 % G0 : /C" $ ! 1 . * !.# ( G / 0 1 (0 % < C / /C"& ! A ;: q 1 % ! / /! (1F /; ! ( % 0 / ! / */ 0$ C / / $I l !d d d0 /!* #! ! " (F 7 @0 !F< *" F0& l % /C"I 7 "01/C"M/! M( C 7 ! [@)E * % * # " % ? * (0 * # C : q 1 % ! / / / l / G d * e ) @ l $(&0$+ * (0 * #! "& ( / !( C " / & d & 0 ! /C"I A : I ;: q 1 % ! / ! * #( ( '8 * (0 * #
π(θ) \ / ! % " G * ) < =() * &M *" DI * * (0
π(θ|y) ∝ L(θ|y)π(θ) % # "01/C"9*" % / ( :$]d q d *" `+ * (& % " Y * ) θ = (η0, η1, β) % B * (0 * #
! C % / G d \ ) / 0 3C e % "01/C" 1 # l q ! d1 1 "& !4 1 1 f *" ) #@@ % / F 1 1 ) " ) ,E#( 7< & ! A . (0(! [& d 0 d 1 A+ k# *'*" /! ( !! < & 0 ! :/0 ! $"I ( 0 1/ ! #/C"& A "& 1 1 *" ) /! ! / #0 "0 ( $ 7 / # ! / $ %e /1 " ! *" 0 % #(! ! % : 1 ! ! d " / / F % 1 M/C"& ! C 9( 1 1 $" / 0 B" ! 1/ /C"& $d & / #! .*" '8 % " + * (0 d q )0( ! : - " ! $ / I ! 7#) * ! [+ * / G0 d /C" ! ! : - "0 *" l /C"=/ ! *" 4 < 1/ : GI 7C / G0 d %,E#( 7= & !$ AA ! /C"& 9( ! " 30 * # % θ′i (i = 1, . . . ,M) % # ! 0 * * (0
ρ(θ) : - " M 0 / M( ! " * (0 * # % π(θ|y)
& *" * 0! C * (0 ρ(θ)
/ * /; !9! d [+ * / "G
ωi =π(θi|y)/ρ(θi)∑M
j=1 π(θj |y)/ρ(θj)
G * "& !)% < % 0 /; % h(θ) !M∑
j=1
ωih(θi) ≈∫h(θ)π(θ|y)dθ.
$ : - " % GI 7C / G0 d /C" @ d @) G+ * / % 0 / ρ : - "0 /C" / C #0 . .!!7 *"&/ # ! / " I * * (0 * #
π /!/ " * "# C ! " 3& !)! ( d "I " 1 / = % *" [+ * / "G ωi
I ' ,E#( 7 &
B
!$ A ! /C"I ;: G+ * / " [& d /C" / ! / % : 1/ 0 *" DI00 1 # d #/C""& /C" / $ ! ( % B 1 1 *$" $ /; @)G A+ & * (0 ! : " ! / # % 7*" / *"& F0 $ ?I / J#C @ ! ' C $" # ? $ * l ) 1/$8 & *" " ! 1/ * !? % 1 /C"& *" e8"&J ! " ! l 1 1 $" : &$ d d 0 ! *" /\ *' % C d C ) " )=- 0 & #d ! - 0 A !,E#( 7 & A ! $ / ( * d % *" d * * /; M % *" #(0 ! )) / [& & % (& GI 7C / % & /;* # ( 3& d d C : = I **$ G+ * / ' G0 d$ /C" *"&H ?0 ! ! : ( d *" H A! ! ! , % M & & /C" ! !!0 ! , *" % #$ d ! 30 / * (! : "(0 d
L (θ|y, z) " / G0 $" @) &k(z|y, θ) = L(θ|y, z)/L(θ|y)
( l " / #& $ L!! '8 % *" 3 d9 * @0 d *" J#(! * ! * ! 'H/! =( / (! % :
• θ = argmaxθ L(θ|y)
• j = 1, . . . ,M
z(j) ! "$# "%& ')()(&*+')($,-$.(/)(&0 ! #+ 1($,,(234' k(z|y, θ) 56 θ(j) ! π
(θ|y, z(j)
) 76 $ r(j) =
L(θ(j)|y,z(j))π(θ(j))k(z(j) |y,θ)π(θ(j) |y,z(j))
' ω(j) = r(j)/∑M
s=1 r(s)
• K $*(-8.(&.,(θ(1), . . . , θ(K)
) ! (θ(1), . . . , θ(M)) .,(39: ;*&.( (&*
';($,-.(2/.(&&:<=(#> /&/=(*()(&, (ω(1), . . . , ω(M))
? % *" 4 A+ * (0 * # * (0 θ, z
y
IM& H * (0 #
θ !
y ! " 0 d' [& θ(1), . . . , θ(K) /! >( l q ! ? ! G0 % *" + * (0 * #
π(θ|y) & * & . * % Eπ(h(θ))
∑Mj=1 ω
(j)h(θ(j)) : $ - 0!
A - ! " " > % "0 $ /C" ! * % ' ( qG d "&π(z|y, θ
) @) * % *" ! / H!! '8
π(z|y) : $ I (0 (0 % *"0 & /C" !) "I ! 8 1/ $ ! q 1 % / /! ( % 0 - $0 G0 d % "0/C" 30 $" G) / r @ F0 (0 ( D /C"I A ;: - " 7 /C" / $. .00( % 0 "I. '* d " $ G0 d- /C" % # θ
#G) /C" / : $ q 1 / *'*/ "&H"& /C" 1/ ) <0 <* #// #0 E" * ! 1/ *1 ( $8
θ)! " 4 * (0 * # ! " GI 7C / % & /;* # / 01M( G / / C ! :
A@CB EDEFHG I B)J B)J LKMG -FHB 2 $ - " %'K00 1 $ /C" !` A+!( d)N &O: ' C $ /C" "& )) / ! /C"$ C* # ! G0 l q ! 30 !% π(θ|y)
I% d q ! 0@( 1 !% ? % + C && "& * (0 : - " 0 8 % " > " ) "IF" $ C ! /! G+ * / [& d9( !# *" / * ! ? [+ C / [& d. G0 $ % 3 A $ ) / % *" ! I % " G0 : # ! / 0 ' 1 /C" / M( % && " / - ;: - " F ! C %
$ "&$ − B(η0, η1, β)
"0 / ) % #$
• j = 1, . . . ,M %#;(%> ! θ(j)0
• ( i = 1, . . . ! j = 1, . . . ,M
& θ(j)i
! qij (θ|θi−1) <# θi−1 =(θ(1)i−1, . . . , θ
(M)i−1
)
56 r(j) =f
“
y|θ(j)i
”
π“
θ(j)i
”
qij
“
θ(j)i |θi−1
” ' ω(j) = r(j)/∑M
s=1 r(s)
M. G+ * / G0 d &)! d)N ! O: ! " /C" / D % qij % $ 0 * 1/; ! & "0 Y I L * (0 /! 8& *" [ @ #! G0J *" " ! / J % G0 J0 % : $ e / /H # % "& / !0 ! 0 ! ' 1 0 " % $ d !A "01/C"* % *" . d * /; ? % *" 4 (0 % #$
• j = 1, . . . ,M %#;(%> ! θ(j)0
• ( i = 1, . . .
j = 1, . . . ,M
& z(j) ! k(z|y, θ(j)i−1
)
56 & θ(j)i
! π(θ|y, z(j)
) 76 r(j) =
L“
θ(j)i |y,z(j)
”
π“
θ(j)i
”
k“
z(j) |y,θ(j)i−1
”
π“
θ(j)i |y,z(j)
” ' ω(j) = r(j)/∑M
s=1 r(s)
/ , ;*$:$# (θ(j)i
).,(3>$#:<(3$#), w(j)
K B B I J I B I I D 2I B;J "0 7*/ !S / # / !! $" q 1 % ! / % *$ / # " ** 1F /; 9( (0 "#@@C % # " C
(η0, η1, β) % *" B * (0 : &$ d? 7 :A ! &@0 & #/C"&
A : & d > "F * ! / ! $E 0! "& " ' "& D C β
dd+! M( *
[β`, βr]: / ! " !o d ! $F 70 "&
β` = 1 &
8 /! $ % βr βr = 5 : - " ] * $8>/C" % β & % * (0 * # *"0 * :
< ! ? / l . * * (0 * #! % " / I C η0
&η1 ! " /C" * * (0 % #$ : 'D 7 d λ0 = 1/η0
&λ1 = 1/η1 !F "&
π(β, λ1, λ0) = π(β)π(λ1|β)π(λ0|β, λ1), $ : q
B
" π(β) *" .0 % * (0
[β`, βr] !
π(λ1|β) =λcβ−1
1 dcβ exp(−dλβ
1
)
Γ(c)I(0,∞)(λ1)
&π(λ0|β, λ1) ∝
λa−10 ba exp (−bλ0)
Γ(a)I(0,λ1](λ0).
! "IB" * (0 % λ0 . * (0 G(a, b)
0 / λ1
: $<& ! ' /! (4 ! "&E" / I '* # * (& % λ1
@0 d β /C"="&<" 4 * (& % λβ
1
@0 l β . * (0 * # G(c, d) : - "0 /C" / % * * (0 * #! "I .(! +!M( < & @0 A O: % <" (00 ) / #& $ $ *" * ! % β ! J & @& A % ' # % (&0$E * (0 ! : . * (0 * #! "& M(/C" ! r&! "" *"0 #d " : 0 7 0 ! " . * (0 * # * (& * # & "0 /C" 1/ ) L0 +A @) % F @0 !F % L" / S C :
0 C a, b, c Id % F ! * / /C" *" % #$ d q : ?m* \!
$ % I (0 $! % η0 &η1
: "(0 d [η0`, η0r] e /; !
[η1`, η1r]" * !$
a = α(1/η0` + 1/η0r)
2
(1/η0` − 1/η0r)2
b = 2α(1/η0` + 1/η0r)
(1/η0` − 1/η0r)2
" α /C" !" 0#d "M 7 *1 / F % *" . * (0 ! &
c = τ
(1/ηβ`
1` + 1/ηβr
1r
)2
(1/ηβ`
1` − 1/ηβr
1r
)2
$ :qA
d = 2τ
(1/ηβ`
1` + 1/ηβr
1r
)
(1/ηβ`
1` − 1/ηβr
1r
)2
$ :
" τ /! M( /C" ! *" ' )F 0! E α : - "& /C" / &! * l *" D * (0 ! (
/ 9# *" . % F 7< * & " 4I ( '8 % 0 d *" 4 * E * / :
EF$ $J G 2I B;J B G J>@ K B B I I D 2I B;J - " G0 * * #= % " (+ / (! G+ * / 1 G0 l /C" !<) E0 < # ? / & * I E& * ! " : 0 / >(F : ! $ [+ * >(0 1 G+ * / % 0 /; ρ '*" " @ * *"& " ]I * ( d q !,<( & !$ A ! /C"& ;: *"& 0 I ! $J (! / .0 *" 9 d8 * e / : - " ! ! " M d8 * / ! ! =0 !) / ! ( " ( /;
zi & 1/* l *" * (0 * # FI# * E
(00 % "&/C" % $ // : - " / & " % $ )? % " * (& FI# * (00 & H #! *" .(! * % ) ! & " % $ E( &*" / 7 d * % (+ " " "FI# 1 & " (00H * (0 * #! : - " ! *" / G0 ! $" G) * *" %
L(η0, η1, β|y, z) = L(η0|e1, ..., en) × 2(η1, β|w1, ..., wn)
$ "&$ − B(η0, η1, β)
" ei \ : wi
$" D*" H#(! ! % $ * ti *" F+! 1 \ : (00 * (0 * # =*" 30 ! % $ /!/ d " F+! 1 . * (0 * # E(η0) \ :
W(η1, β) ;:. * d
λ(j)0
&λ
(j)1
/ ( / # d A ! " / #& $ ) * (0 J % λ(j)0
G(a+n, b+
∑ni=1 e
(j)i )
& *" / & $ I * (& % (λ(j)1 )β(j) G(c+n, d+
∑ni=1(w
(j)i )β(j)
) : / ! 0 / d 4 * % <" * "I C % (00 * (0 ! ! C* d β(j) / * # '*" // + /;< $"
: . ! C β % π(β) :): . ! C u % 0 % * (0 * #=#
[0, 1] : : 'D β(j) = β
% u ≤ π(β|y,z(j),η(j−1)0 ,η
(j−1)1 )
π(β)maxβ L(β,η(j−1)0 ,η
(j−1)1 |y,z(j))
! " # #:
1 * 1 # # 1/ * # (* - " 3#(+ / % "0 /; r $ / ) % F/C" @ l " J ) #d " "0 % #$ l8! ! ! " / d % / G0 $8 *" /F+! * ! % (&0$ & B@ : > ( $89/ # ! " '*+! 1 ) % *" A+!= * (0 * # : ! % d !e /; ! ! " % (0&$ @ /! (1F / ! ( l * : $ % E*" B >)! !'<"I (/ "&<'< = / G0 *"& *" % (&0$ >)! :
- "& & ! " C & ! " @)%, %- 0 ,*- G@)) % > ! G 1/ % C : 1 / ! ! " *"0 >)! "J(0! ! " + % #$ l[ / ) A+! $ *" 1 C & F+! 1 ! (00 < l0 ! % G0 : ! d :
O*;: $( ! "0 & (0 J /;< l : <& " F@0$ "* I " H0
β = 1 ! *" , - C* 1/
2 ln(
LW (η,β)LE(η0)
) / χ2
1
* (0 * # : / &0 ! I % " < " /! " *" 4 @ ! /C" G (00 ) ! " , - B& d0 / E( ! (00 & B @ : *" ! " ! " ! I@0 " *" <$H "M* d /!/ & H + (0 /!/ * / !F %K% $ : - (
/ ! % F*"0 ' / & ! & d λ0 = 1/η0 ! " @0 ": + *" G / * eI#& d *" % (&0$@ ? &!!
H0 λ0 = 0 (η0 = +∞)
"01/C" ! "IF" /!/ * / G+! ? " ;: / " Y C
λ0 / ! C J( ≥ 0 ! *" F * (& M0&
H0 % *" , - C* 1/ '*"0 C* #! % $ : :OA
2 ln(
LB(λ0 , λ1 , β)
LW (λ , β)
)= 2
[n∑
i=1
δi ln
(λ0 + λ1
βxβ−1
i
)−
n∑i=1
(λ0 xi + λ1
βxβ
i
)]−
2
[n∑
i=1
δi
(β ln λ+ ln β
)+(β − 1
) n∑i=1
δi lnxi −n∑
i=1
(λβ xβ
i
)]
∼ 1
2χ2
1 +1
2δ0
" δ0 !0 *" " C/ * (0 * # 0 : $ ! C L ! C* #8 % "& 0 3/ ( % 0&
.F * F & 1 % 7 ! /C"I q;: I" * & +0 *$8 / I '* #! *"&H % 0 &
( *" B * (0 * # ! *" ! % @ % ! *"&.*" - q $ F ! % lnLB (+
λ0 = 0
+ (0 !F "01/C"= * ! / lnLB
I! " (I " @) % λ0 = 0 :- " ! *" l0 1/ % " , - * * "# % : ' / ! d "0 & ! $9 I+ (0 C ) M" / G0 '8 % @ : H 1/ "I?'. E+ (0 M 3> % *" / $ ! /; % " "& ) : - " / $ /1 @ X $"3 C % ! # k & " 3& ! " @)
θA q * /! ( *$7 !
lnLX (θ)− k2 lnn > 0 " n " H % " G0 : - " ! /; !) " "0/C" ! "0 " / $ :
41 # M "13' % 1 # !41 * "%$ &( *&(2+ $ +! & $'&($ ,+ $ & ! ! *&
"0 $ /;* # ! !> % *+! 1 ! (00 I B * (0 * #! J0 * % B(200, 100, 2)
* (0 * # / #! : H *"& % /C" * (& ! " ( ( $8 % % 9( #// ! C
0.34 : G0H % / d$ "& .(! 3& ! G0 % /C" & "& ( / !!7 $"J" d !X "01/C" d q *$ / ! !J C :,7& % # / !! !$ G0` % - & - (70 * ! - (& $ : &$ : : ? * # 7& 0 '*" " H! 1 & *"
[25%−75%] */ &
*" G0 : - " ! 4 % * ! " $ .70 * " 1 # $@0 *1/ F * ! H( /! ! $ / "& l! "IF" 3 % η0 " B * (& I "0 "= : J K* (0 ) 1 !0 * #*J(I) *" ! % ? 1 *$" % *" B * (& ! # , " 9 / *$" $" C / d) l0 ! " (00 * (& &*" = $ / $" G) % >" "F+! * E * (0 : *" q 1 7* l ! * * (0 ! "& 3( % " ) * #!` $ / $ : : % " % #$ dr * % * #! : - " * "& 4 C β
[βl, βr] = [1, 5] ! " * /! 4 C η0 % *" 1F+! 1 / GI# / # / ! C F(η0l, η0r) = (1, 300)
& " / L C η1
% *" (00 / #GI# / # / ! * * (η1l, η1r) = (1, 200) : " C (0 *" ' * #M % " .) ) 7 @
MRST (t0) =
∫∞tS(u) du
S(t)
"& ( &! $" ' G */! * I M @ [ " ! S !0 * d *" 1 @ % 0 / * # &t0
d! $ % * : " ! 1/;=" ( "& @ % " )F( #& *" #(! ! * (0 !P /C" @ t0 = (η0 + η1)/2 = 150 "01/C" " *1/ G0
SB(t0) = 4.97% &MRSTB(t0) = 25.25 : G '*" " 1 , (- ( C ! $" " (00E * (0 * # ! " G % " 4 ! / \ / $ "0 " "G *"0 d " *" % 50− !M C :
F 9(3& 1/ "&H*" * * * # % * η0 $ (3 *! '*$ / !!7 d * &$ G0 "& " 3 " : 0 ! " , - I / $ /C" G *" / * /; * (0 9 /! : " $0 G0( & / !! ! / ! " ) /C" / % d *" (00L * (0 <0 d$ / " 1 % η0 B ) d &! "0 " !
$ "&$ − B(η0, η1, β)
7 *"&E*" B * (0 !$ d q ! '*" (00 * (0 :
B
η !"$#%'&'( η β ) *!"+#,%-&.( η0 η1 β !"+#,%-&.(G * *$ * ! " $ ! ! " + ! !
"0/-%*1 #203'4*1 %'( "03*1 5'3.( "0/-&61 %-203,51 3.(7"$#.1 8-8-2$#1 4'9'( "$#51 :./'( "+#,%-861 5'2$#,9-%61 /'(7"03,51 :-2$#'#961 /'(7"$#.1 9.#20:61 :'9'( "$#-#1 :'%.(; I|=< * *$ * " ! $ . !! .$ ! .$ "0/-%*1 #203./1 #,( "09*1 &'( "0/-861 :-203-961 9.(7"$#.1 8-3-2$#1 /-:'( "$#51 8'3.( "+#,3-%61 8-20:':-461 9.(7"03-461 %-2$#,:-&61 #-(7"$#.1 9-&-20:61 :':'( "$#&61 3'3.(
- (0?> : 1 & q 1 * !K % * $" & * I 1 , - '*" C & 9 G1* #! % - ! B(200, 100, 2)
G04 % / !!7 9 * ;:
B
η !"$#%'&'( η β ) *!"+#,%-&.( η0 η1 β !"+#,%-&.(G * " *$ " * ! *$ )"++ ! * ! * " * *)"
"@4-561 #209'%*1 5.( "0:-861 /'( "@%'361 %-209-%61 3.(7"$#.1 8'/,20:61 #5.( "0:-:61 5( "+#-/,961 5'20%'4'/*1 3.(7"0%-361 9-2$#'#51 /'(7"$#.1 5':-20:61 :./-( "0:-:61 4.(; I|=< * * * * * ! ! )" )$+ ! *$ ! *$ ! "@4'9*1 %,209'&*1 /-( "$#%61 /'( "@4':61 %-209.#1 5(7"$#.1 %-8-20:61 #,&'( "$#,/*1 4.( "+#,3.#1 %-20:'4-:61 /'( "04-861 %-209-361 /'( "$#.1 4'/,20:61 &'9'( "$#:61 %.(
- (0?> : 1 & q 1 * !K % * $" & * I 1 , - '*" C & 9 G1* #! % - B(200, 100, 2)
G0D % / ! !M C :
A* +BC DEAF GIHKJ LM B
N ! OP+Q N L! OP+Q N ! ORP$Q . " . . . . . ) ! . . ) " ) ) * ! "
- (0S> : O 1 @ /C" 1/ % G0` % *" H1 = B * (& % - ! & - - ( ,*-UT ! = G0 :
- " ` % / E % (I "9 d #/C" K0I " G&$ ": + *" % (00+ * (& I - (& V> : - : ! " S A+ ? % G0m l $ !$ % # W(100, 2)
* (0 * # % "01/C" " ,*-g - & * */ #// " F@0$ "* I " : - " *" ) d ( d A * " " A+ % / !!7 C / ! ( / ! " G 1/ d q % *" , - E C* l : "XW & &( 2+ +! & YW &MW 2+ (VW ) &( %2! *&
- " % $ d> CJ G0 % [Z)\]R^'Z : -_` * $8* (0Y $ M( ? ?g ? ? q % 100, 000
0 (I A : ! *"&J" I $@ ) & *G I Ia+ /;* #! " % :- "0 * $8C (0 4 ! ! " #$#T -: "0 *" ? 1 % " B * (0 * # C η0 = 1662 ! η1 = 97.4
! & β = 12.4 : - " % "&
DEAF GIHKJ Q i | D6AF6 IHKJ Q i | . . * . .* . . . .$ &" * .* * * ) *
- (0S> : - A+ !K %$ / A !* G0D % - A % & * "G " H0 = W(100, 2) * (0 :
* β
.0@! : & 1/ "&J l /!/ ! * d % 7# * ! "01/C" 0 $ d / # / d " 0 M % : - " B != G&) 1/ $ 9/ G0 ! % #
E(X) =
∫ ∞
0
exp
− x
η0−(x
η1
)β
dx ' 90.8 :
H "& /!/ d *" 0 *" 3 % )) " 01=( % " 0 ! #01 0 (0(0 $7 l % d ?#0 * ! /!/ & $3 D * $ " / $" (00 * (0 " % ! / % A& (+ : X> : *G % 0 / ! % " % (0&$ & B = * (& ! . e0 q ! :
*" > * % ! & e /;*$ * &)#> [& !+% d*" * ) E % " *"0 / ! != % * J>)! & d " 0 E % "* I " :>H$ G0 / ! ! & / !!7 : /! !? % *$ [& 1 4
n ≥ 100 ! 30 $" G) !! ) " * % " ?) : - " 1 & " 1 - *
t0 = 97 f - (0S> : ! ,*- 70 - (0S> : &% (&0$
0 % *" / > :O):
A* +BC B
η (η, β) MRTS(97) (η0, η1, β) MRTS(97)
c * * c + " ( * c . c ( * ( *)" * * * c ( c ** * c " "
- (0 > : 1 & 1 - . * t0 = 97
% # G0F % ! & / l % " 4 e /;*$ 7C '8 C (0 :
A* +BC B KD D6
c a )"$ "$ a " + ) a " B ( a )" . a . )" a $ .$ B
- (0 > : $" @) &>)! /C" 1/ % G0H % ! & % # *" e /;*$ 7C '8 C (0I - !
(00P&$ > : ! $F/ d "&H*" B * (0 @ ( 7 " > C"I " (00K * (& : # /; d *" #// ! * / !. % @ lE @0 / & (0 < % e /;*$ " & % ) : ?L ! $ J / < % . @ % & /;* #!% " (00 I *" B * (0 * #! d E > : :
$ " > − B(η0, η1, β)
>< * l /C"& *#/; 1/ % " B * (& / ( #(! ! # (000$ ! > :O): - " !G $ *1/ + * (0 * # $@ ! ` " ] q $8 % log t = 4.1 ! / eI#& l \ / " 80& @ d F+! * H * (0 * # % log t ∈ [0, 4.1] & *" (00 * (0 * # : *" % F % B * (& ! " (0040
log(− logSW (t)) =
log(t/η0 + (t/η1)β \0 q @ " " (I# " G) % log t = 4.1 : - " ! ! d ( / "
! </ ? %L% $ % t > to ' exp(4.1) = 60.34 :
2 3 4 5 6 7
−6−4
−20
2
Weibull plot
log t
log(
−log
(S(t)
))
2 3 4 5 6 7
−6−4
−20
2
2 3 4 5 6 7
−6−4
−20
2
Weibull plot
log t
log(
−log
(S(t)
))
real dataweibull dist.B dist.
# > : (000 % E C &0 ! [& % *" 4 e /; F * $8 C & " .) " % - (& > : :
+ W W %2 ) YW (YW
- % * " * ) [0 − 1)
[11 − 12) 4 ! % - H "0/ ! l *"
- 1 7 / 8 % * * !eI 7C % / C / : *"0 d0 1/* # $D / #! *"&B*" % $ // D*" !& % #/C" & % : @ *" / ! ! % $ (1, . . .︸ ︷︷ ︸n1
, . . . , 12, . . .︸ ︷︷ ︸n12
) "
(n1, . . . , n12) = (2, 3, 3, 2, 2, 4, 5, 4, 5, 6, 5, 9) : ? * * #! E ! - (0K> : & / * \+I l% (&0$R0$ . \& q ! # > : O&:
B
G η = 8.83 β = 2.52 η0 = 16.86 η1 = 11.01 β = 8.10; I|=< η = 10.35 β = 2.41 η0 = 20.75 η1 = 10.50 β = 5.10
- (0 > : 1 & q 1 * ! % *" "01/ % G0 :
0 20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
years
perce
ntage
of su
rvivo
rs
> : ) @ % 0 / ! % *" ! M % C (0− F6 D*A*F LI D B
A* -[A6IF LID )A* L-*[A*F LID L
$ ' / ( 0 */ ! #G) #// / ( ! 1 & q 1 % / : - " B * (& / > ! *"& G0 % (0&$ * (0 * # ! & q 1 ) I @ ( 7 *"& 1 * # : 0 D * 0 ! /C" !
[1, 6] % β I[1, 30] % (+ *"
η0 &
η1:
- " B * #. " d d ( "& @ "0/C" $<0 d % H !! "&/ d q *$ % % / ! / #G d $" "& " * % #// ! : H */ "& " 9! %*" B * (0 * # / #0()F /; ! 9( ! *" /! % "01/ % *"& q ( * !9( /C" 1/ /! !4 ! d <// * /! ! :
0 1 2 3 4
−4−3
−2−1
01
0 1 2 3 4
−4−3
−2−1
01
0 1 2 3 4
−4−3
−2−1
01
0 1 2 3 4
−4−3
−2−1
01
log t
log(
−log
(1−F
(t)))
0 1 2 3 4
−4−3
−2−1
01
log t
log(
−log
(1−F
(t)))
FailureWeibull (ML)B dist. (ML)B dist.(Bayesian)
> : O (00 ( ( $8 0$ E % "01/ % $ E * :
$ " > − B(η0, η1, β)
- " B * (0 * &* G0 >)! * #! / G * d * e % " % $ : </ =(.)! " *"& *" )! "& ( A+! : ? % " " $ - 1 ) % "& & 1 A "01/C" / & > C : " ! G0 ( " C (0 q : " ` D*" B * (& "I D(!/ #! *$ " (00K * (& % >)!$ d d : /$ C/ 0$ * / '</! 9( " #d "G % 1/ ) > / ( d *"& " (00 * (0 : ><& / C % d "& (!J /C" @ ( !**+! 1 ! % (&0$ & B * (0 * #! :- " (00+ * (& 0 * (0 l "I < / 0 8J C l // 0 $ e /; % % * > C : >H ! * * "0 " "G M( 9" % * (0 J C : [> : " "I<*" ( (00& * (0 K0 (0 B" &!G $ *1/ *@$ % 0 /; G $ 3( .*" *$ C (0 : *" / # C ! " B * (0 % /H Ra ! ! E "0 / : " + (0. C ( / % *"0 <>)! "&<'< # "0 D * : &! 'H/! ( / 0 H *" * % & T H/ ! !- G0 : 0 <"0 9 7# ! q 1 % ! / < C (0 I H (0 * ! : "& D I - / H GI 7C / G0 dM) " ) d q *" GI * * #8 % *" B * (& C : J % B */C" e /;*$ #$ G0 & ! 1 (& ! 1/$* G / !) % BC l #// #0 * % * # d *" B * (0 * # q :< "&?/ 1 / & ( * /!/ #0 [& q $"8" B * (0 : 01/ #! % $ d " % / [A! % *+! 1 % &# % % (&0$% $ I $F0 *"0 / / '*"M*" / 1 "0 d "= ! A 1/F + :- " B * (0 / & $ 3 #/C" 0 % % G@))0#d " % &0! * 1 d& 1/ ! & q ( / ! !9( 9' C/;*/! ! > % / . * (0 * # % E ! 1 ( '8 I :
%;%1 ;/ ' 13' *1 .1 # *1 * # I,* # *;1 1 I /1 ,*
- " % *"0 /;* # = *"&.*" $" @)% * #! % ?*" B * (0 "& = G "01/C" / I G */! 0 * (0 : - " ] @ % ! "I & " ! "& /! : $ $&% OI: K G J1G 2
f(x|θ C Z\] C C _ _ H > ^_ H G+E : '_,] θ = (θ1, ..., θk) C C_ ` : .]I\ Z \ 4 L\ C ] _ .] @ Z \ 4 L\ ^'Z : Θ > x1, ..., xn
C K_ > .Z > ] C ^L\` -_,] ^ ] G \ > ] 4 ` \_ C X _ @ > ^_ H f(x|θ) @ _ < _ @ ]] > E -_,] ^ \ _` C8H ∂ ln L
∂θ = 0 @ L\ lnL =
∑ni=1 ln f(xi|θ).
2 θ0> ] @ -\ E ` E ] G θ "! ?_^ ^ ^ E)4 > @ θ0 _ C^ ?^C] 4 SZ ]_ _
Θ # @ $ _ G&% ] > _ -_,] ^' )( C ] @ ] >* @ L\ _^ -^ (E _+ E : ] ^ _^ ^ '_ 4 .]I\ θn ^L] E -_,] ] G @
_ < _ @ ]] > , E '_,] ^ .- E \ @ L\ 4 ]I\ √n(θn − θ0)_^ ^ H4 Z] '_ : H ]\ 4 H(> _^ '\_ CFE > _ @ 4 #/ C\]
> : ]` \_ : 4 -\_ I−1θ0
@ C\ Iθ0
_^ @ - _^ @ C\[_ ?G ]\ 4 '_,] 4 -\_
• ' - ]\ 4 ]R^ x > G ]I\ θ ∈ Θ ∂ ln f∂θr
, ∂2 ln f∂θr∂θs
∂3 ln f∂θr∂θs∂θt
_^ G ]\ r, s, t =
1, ..., k
• - ]I\ 4 ]R^ x > G ]I\ θ ∈ Θ∣∣∣ ∂f∂θr
∣∣∣ < Fr(x),∣∣∣ ∂2f∂θr∂θs
∣∣∣ < Frs(x) >
∣∣∣ ∂3f∂θr∂θs∂θt
∣∣∣ < Hrst(x) @ L\ Hrst
_^^ E :+@ @ ∫ +∞−∞ Hrst(x)f(x)dx ≤ M < ∞ > Fr(x)
>Frs(x)
\ C ] E > > G ]I\ r, s, t = 1, ..., k
• ( - ]I\ θ ∈ Θ @ 4 '\_ I(θ) =
∫ +∞−∞
(∂ ln f
∂θ
)(∂ ln f
∂θ
)′fdx
_^ ZM]^ _ '_` > _
- " *"0 / #& $ !? % *" "& I M" ! 0 /C" / ! % ?" " @@% !? % *" B * (0 : L H( "& " S @ % ( / $& " I ( $8 % / 7 ?(! * : -7 /C" /! ! & d c 7 /C"= / ! !9(! * ! *" 'F #9 % " 4+ % " !
f(c) = exp
− c
η0−(c
η1
)β
! 9 % -> : - ;:,0 / % 7*" ! x q !0 >/ ! !9(! ** # "I % : $ ! *" 4 1 *$ % " 4 % f F % " % $ d %
∂ ln f
∂θr=
∂f
∂θr
1
f
∂2 ln f
∂θr∂θs=
1
f
∂2f
∂θr∂θs− ∂f
∂θr
∂f
∂θs
1
f2
∂3 ln f
∂θr∂θs∂θt= 2
∂f
∂θr
∂f
∂θs
∂f
∂θt
1
f3− ∂2f
∂θr∂θt
∂f
∂θs
1
f2− ∂f
∂θr
∂2f
∂θs∂θt
1
f2− ∂f
∂θt
∂2f
∂θr∂θs
1
f2+
∂3f
∂θr∂θs∂θt
1
f,
" θ " / C θ = (η0, η1, β) : 9 &) / ! $ /! =(4 *"&
f & $ 7*
*$ % & =(!$ ( g /! =( *' ! " % #$ d q
exp
− x
η0−(x
η1
)β[
P (1
η0,
1
η1, β) +
M1∑
k1=0
M2∑
k2=1
1∑
k3=0
Qk1k2k3(1
η0,
1
η1, β)
(ln
(x
η1
))k1(x
η1
)k2β−k3]
" P ( 1η0, 1
η1, β)
&Qk1k2k3(
1η0, 1
η1, β)
I# 0# 1 1η0, 1
η1
&β : #! ! I '* #
! : / I) ! 7* & * g ./ @0 % 0 /
x I
θ : - " ! g D(+0&! % θ ∈ Ω &x
/ ! : - " % /C" / & '* # ! $ 7 / / #! $ F( "& @ % $ % x : E "& *" ( + $ H@0J( A
&B /C" *"&
g % *
e−Bx × xA % 7 / ! x &
θ ∈ Ω : / e−Bx × xA <(I#0& ! I '* # !! :>( % & $ ! I(θ) !R "01/C"8 M/ 1 / * ! Y+ '*$ > & $ 0& $
a, b, c0
$ 7 /C" "&a∂ ln f
∂η0+ b∂ ln f
∂η1+ c∂ ln f
∂β = 0 : > G0- * # % " " C "G % ) F"& " E0 / : - " ! " *"0 / & $ D % "& & F*" ! * ! :
Appendix B
- "0"!> 1/ $# ! 0 # !.*" #M I %! % "I # : 1 # # .$ I ! # !3 &# % & / ( C '# *" ) % D" # !7"& (# C '# # β % " # # (00+ "!7 * (0 :) +*G. # ,# G0-# $!D # #! *@/ # ! &# * d " # .# " ))#$ 1/ /! (# /;0! 1# #!$d(# "& # : *" # # 2*43 # ! ,# # % # %!D *" # / #[&-# # " # !7"! ) / &.#! : !o#&%&'(65-%-, Q J87:9&6<;X7=7W76?>h6<@/ACB>7=@*<NI/DB/D;*<+@+B>7EA8/2B+5FA=7&5@/2B+?:/@5@/O7pNXclUmJ87%A=7Y52/G>$778/@1H74<:B7\B BIA=7J@8BE@:B+5@5ONK9%70;8B06?A=7<B?;*</D;LB01B5@5M;4@:1Y?17NAPORQhB?9e;9?A86<@8;:<I/S;/> 6?A81B78/@6<;r/DBBH-B/252B?:5D7T>-A86-1U7eW317NA07=B0O Z @+A3+A8/@62ARBE@:?4VW70978/@HX7`;4;:6Um5D7<927Y/DBf62;:5ONB+?F62@7S78J875@/G>"778/@1H7`62>RB2;I/D;:<+@8B>7EA8/2B5M9&6-143F62;870;X7SB?;*<';:67fB?F6<@A7M78J+73*B?A\B1H77=7%A=BS62>BYKY7/2?+@:5@5$<A/DB>7EA8/2?+@78/@6<;GUmJ/D9eJA=7&3+A=7%BE70;X7=Bf78J:/DB5@/D>$778/@1H7<O4KY7Y3+A86-3F62BE7D786.> 6Z9%@8B78J874<A/DBE90@8BEB0/@6<;?177pU7070;78J+7J7WA317%A07=B]B?;*<r78J+7D/D;*</@8B>7EA8/2B+5B2;:B+5ON/B>7B+?F62@7]78J87HB0/D[07"62> F6E <:B7\B.A=73+A=70BE70;X78/D;:9G78J+74H-BIA8/2B+?:/@5@/O7pN62>m78J87H7WA317%A07`6+3:/D;:/@6<;,O i ;:<+707&<lco78J/DBTB0/D[07/DB62;8762>78J87\BE9B?A=9%7D/D;*</D9B7862A=Bf78J:B7f?F678J9B2;L@8;:<+7%A=B>7\B?;*<,O Q J87Y3/A8/@62AC9B5@/2?/A\B78/@6<;/DB1B-<+7\7B?B>NXcoB?;*<B06-1H7`1H778J6<+BfB?;*</D;*</D9B7862A=Bm/D;89&5D@:</D;:94B4</7%>_B?@:5O7]9B+5@/@?+A\B78/@6<;I1H778J6<[B?A=7]3/A86-3F6<BE7&< 786DJ87523"78J87^QhB&NZ7%B0/kB?;EB2;:B+5ON/B>7 78J+7N_9B2;[?17C7Wa7=70;*</7<"786Y/S;/>"7%A=7%;89070BS6<;[6+78J87NA<A/DB>7EA8/2?+@78/@6<;+BL78J*B2;"KY7&/@?+@:5@5 OQ7%B0/k</70B&c:78J+73F6<B>7=7%A8/@6?A`9&6+1438@A7\B78/@6<; ?N"/@143F62A07\B?;8907B8B+14352/D;9M/DBFB0/@143:5D7hB?;*<^B8B78/DBW> NA/D;:9/O b /D;*B+5@5ONXc78J+A86<@9-JB]A=7B547WB+143:5D7-c78J+7a67eW/@?/25@/O7pN]6?>*78J877&5@/D9&/O7\B78/@6<;D/DB^/@5@5D@8B>7EA\B7=7&<,Ot[uvsw`x 'y$#-z A=7&5@/kB?:/@5@/O7pNXcd</@+A\B+?/@52/O7VNXc1QhB&NZ70B0/2B?;B?;*B5@N/B0/DB&cFKY7/2?+@:5@5$<A/DB>7EA8/2?+@78/@6<;lc67WA317%A07M6-3/D;:/@6<;lc1BE@:?4VW709e78/2H<7`3/A8/@62AcH/DA07=@:B+5l<:B7\BAc+9070;+B062A=7<I<B7\BAc+bC@:5@5@?*B?9e;XaV(7/2?5D7%APO
− !"# %$ '&()" -* )"+ ,-/.10 3254%687# 92 6
;:=<?> ##@BA <*C AB@DADE GF#H : EN#E *IH : # H*J EK&#ML H8! E.##LG@ HX #ONG@BA J #EABL'#(NA ! E > A CP EA H : ! A ! EK&# #1A CP @D@'NQA ! E > A CP EA H : < @-# !N! *RF K < TS O: S-'KA !'U # > ! < EAB@$# W(η, β)
NQA ! E > A CP EA H : * ADEKVN1# : ! ADE 8J P: F EA H :
fW (t|η, β) =β
η
(t
η
)β−1
exp
−(t
η
)β
< : NWK < <*> N >X< EN#hW (t|η, β) = β
η ( tη )β−1 K&# > # η, β > 0
*%F < :YC # P ! #1N JZH > L H N&##@B@DA : ?A : J < : E[L H > E < @BA\E N1# J # F E ! K&# : EK&# ! K < (#I <*>< L'#EN# >
β < 1* < *A : K&# : β > 1
H >]< F1F AN1# : E < @ J < AD@ P> #C ADEK < F#H : ! E <*: E>< EN# K,# : β = 1
S ;: EKA !^F < ! # * ADE > #ON P F # ! E H <*: #* H : # : EA < @TNQA ! E > A CP EA H : A\EK !F < @$# <*>< L'#EN# > η S;:?:RP L'# > H P ! A : N P ! E > A < @ ! E P NQA$# !2* > ##@BA < C AB@DADE `_ Xab cd eCZ C\_ #f c g0_ _-_h ) ? N < E <
tn = t1, . . . , tn< > # H J E%# : L H N&##@B@-#ON A\EKiEK&# #1A CP @D@ NQA ! E > A CP EA H :j<*: NkEK,#C < >X< L'##EN# > ! < > #Y# ! EABL < EN#ON C L < QABL P L@DA /#1@DABK HRH NVA : J # > # : F # P ! A : ^# E H : ml < K !H : l H > ? 1 < @ H > ADEKL ! S 0 H # U # > * EK,# HH N C ##K < U A H > H J@DA /#1@DABK HRH N[L < RABL`A < EA H : A ! # : ! P> #1N H : @ K&# : EK,# ! < LG@-# ! A #
nA ! @ < > /#`# : H P *K < : N K,# : EK&#!@DA J #EADL'# !
< > #^@BA\EE@$# F # : !H > #1N= < F K < S EK&# > A ! # * P ! A : < # ! A < : A : J # > # : F #^E%# F KA P # ! A ! > #1@-# U < : E K&# : > A H > : H @-#ONF/#nA ! < U < AB@ <*C @$#G l H C # > E q S2-'KA ! A ! EK&# FH : E%#RE H JIH P> ! E P N +* K&# > # !H L'#J#F# > E ! > H"U ABNNA : JZH > L < EA H : ! < C H P E^EK&#[@BA J #EABL'#T
H*J <*: A : N P ! E > A < @ FH LG H : # : E ∑ S-'K&#]L < A : NQADo F P @DE A ! E H E ><*: ! @ < E%#EKA ! #F# > E H A : A H : A : E H A : JZH > L < EA H : H :
(η, β)* EK > H P KpEK,# F K H A F # H J < > A H > NQA ! E > A CP EA H : ADEKVN1# : ! A\E
π(η, β)S2-'K&# : EK H+! EN# > A H > N1# : ! ADE
π(η, β|tn) =L(tn; η, β) π(η, β)∫
IR
∫
IR
L(tn; η, β) π(η, β) dηdβ
< @B@ H ! E HpH C E < A : # ! EABL < E%# !qH*J > ##@BA < C AB@DADE rJ P: F EA H : !h(η, β) KA F K < > # H J A : E%# > # ! E JZH > EK&#[A : N P ! E > A < @ <*:<
@ 1! E * < ! EK&# ! P> U A U < @ J P: F EA H : S 0 # > # L(tn; η, β)N1# : H E%# ! EK&#n@DA /#1@DABK HRH N H*J EK&#]N < E < S
-'KA ! <*> EA F @-#rA !JZHQF P ! #ON H : EK,# F K H A F # H*Jπ(η, β)
S # FH : ! ABN1# > EK < E`EK&#Y#F(# > Eq ! FH : ! AN1# > #ON < !L < @$# ! JZH > ! ABLG@BA F A\E F < : #F > # !%! KA ! ZEKA > C ##@BA-# J ![H : @ <*C H P EsEK&#rL < > *A :< @9NQA ! E > A CP EA H : M H J
T
A\EKVN1# : ! ADE
m(t) =
∫
IR
∫
IR
fW (t|η, β)π(η, β) dηdβ
< : N : H E'NQA > # F E@ <*C H P E(η, β)
S #^E < +#` > H E H J EKA !tJ # < E P> #^E H CP AB@N < > A H > JZH > EK&# !F < @-# < >X< L'#E > A < EA H : H*J EK&# ##A CP @B@%L H N1#1@M ! #2# > H TS A S- A F < @B@ +* K&#rZEK&# F <*: *A U #4 > A H > # ! EADL < E%# !!H J EK&#[L.# < : *EK&#qL'#ONQA <*: * EK&#^L H N1# H > # > F # : EAB@-# !H J M S 0 H # U # > * K&# EK,# F <*: K < U #qNQADo F P @DEA$# ! E H # ! EABL < EN#qKA ! ! ##@ JP: F # > E < A : E vu A : N&#?#1N * < : #F(# > EsA ! H J EN# :G: H E < ! E < ! EA ! EA F A < :G< : N : H EA H : ! < !J! E < : N < > N?N1# U A < EA H :w> ##L < A :J P + S - H(H C E < A : HRH N # ! EADL < EA H : !xH J EKA ! > A H > P: F # > E < A : E /* NA !F P !%! A H : ! C #E #?# : EK&#'#F(# > E < : NVEK,# < # ! A < :M<*:< @ 1! E K H < EK&# > ! EK > A H > A : JZH > L < EA H : * H8!N! A C @ ##A *KEqADE < : NMEK&# : A : J # >^C H EK(EK&#]N < E << : N8EK&# > A H > <*> #!N&# ! A >X< C @$#*S 0 # : F #!EK&# : #?#ON H J A : NQA F < E H > ! KA F K < > # P: N1# > ! E <*: N <*C @-# JZH >tC H EK H*J EKL(S
> J EN# > !H L'# > # F < @B@ ! <*: N > # F A ! A H : ! <*C H P E3EK&#VL'# <*: A : H*J EK,# #1A CP @D@] < >X< L'#E%# > ! A : 1# F EA H :iyQy * #" > H H8! #sA : 6# F EA H : S <*: ADLG > H"U #1L.# : E H J EK,# < l > H < F K > H H+! #1N C # > +# >^< : N P: A <*: N < F K < gDz A : E H #1@DA F ADE4 > A H > NQA ! E > A CP EA H : ! S -'K&#`L < A : H A : E H*J EKA ! ABL[ > H"U ##L'# : ExA ! E HFH : ! AN1# >EK&#` > A H > NQA ! E > A CP EA H :p< ! <*:`< d > H QADL < EA H : H J EK&#` H8! E%# > A H > NQA ! E > A CP EA H : FH L`A : J > H L <s: H : A : JZH > L < EA U # > A H >'<*: N `_\ | g%a K H8! # ! A # a A ! <> ##@$# U < : EqA : NQA F < E H > H J P: F # > E < A : E S
> F#F#H > NQA : `EK&# ! # F A F < EA H : !qH*J EK&# > A H > : H @$#1Nd/# * # H C E < A :?< #QA C @-#3KA$# >X< > F KA F < @^ > A H > J < LAB@ JZH > KA F K #[ > H H+! # !H L'# ! E >X< E%#A-# !!H*J F < @DA C>X< EA H : A : 6# F EA H : S O S? ! (# F A < @B@ +* < N&# J < P @\E F < @DA C>X< EA H :L'#EK H N?A !J! P /# ! E%#1N S ;: N&## : N1# : E@ VJ > H L EK&#'#*# > E * EK&# < # ! A <*:w< : < @ 6! ExL P ! E H*J EN# : > HQF #?#ONE H < : < @ > # F < @BA C>X< EA H : ! EN#+TS ;: EKA ! < ABL * < : A : NQA F < E H > L'# < ! P> A : EK&#_# # F EA U # ! A # H*J EK&# F # : !H > #1N(N < E < ! #Etn
A ! A U # : E H @ HQF < EN#8EK&# ! E > # : KE H J EK&# ! PC a=# F EA U #pA : JZH > L < EA H : ADEK > # ! (# F E]E H EK&# ! E > # : *KE H*J EK&#H C a=# F EA U #pN < E < A : JZH > L < EA H : S # ! AN1# !2* C A C @BA H >X< KA F < @ > # ! P @\E ! <*> # > # F < @D@$#1N <*: N F <*: C #E > HQF #2#1N1#ONGA :=<> ##@BA <*C @$# <*: N ! ABLG@-# < S -'K&# : EK&# FH : ! # : ! P ! C ##E #2# : ! # U # >< @(#F(# > E ! A ! C> A$# K! E P NQA$#1N S
) A : < @B@ +* # ! K H A : 1# F EA H : S[EK < E EK&# FH LG P E < EA H : H J EK&#D H+! EN# > A H > NQA ! E > A CP EA H : A !C! ABLG@$# P ! A : <*: ABL[ H > E <*: F # ! < LG@BA : < @k H > A\EKLS ;: 1# F EA H : S * < J P @B@ < # ! A < :w<*:< @ 1! A ! A ! @$#1N JZH >n> # < @N < E <(< : NE H #F(# > E H A : A H : ! E H AB@D@ P ! E >X< EN# EK&#sL < A : H A : E !'H*J%H P> ##@BA F A\E < EA H : L'#EK H N S
- . 2 (7 4 2 4
W & &(W'&( 2+,& &
) H > n ∈ IN∗ * @-##E Tn = T1, . . . , Tn ∼ W(η, β)C #GA : N1#+(# : N1# : E@ <*: NiAN1# : EA F < @B@ NQA ! E > A CP E%#1N > # < @
H > U # F E H > U < @ P #1N ><*: N H L U < > A < C @$# ! A : EK&# ! < LG@$# ! < F #IRn
+ A\EK > H C<*C AD@BADE N1# : ! ADE J P: F EA H : N J fW (t|η, β)
< : N ! P> U A U < @m > ##@BA < C AB@DADE J P: F EA H : SW (t|η, β)S
-'K&# < U < AD@ < C @-#]N < E <p<*> # < !qJZH @D@ H ! @$#E tn = (t1, . . . , tn)C # < : H C ! # > U #1N ! < LG@$# H*J
nN < E < S ! P< @B@
tnF#H : E < A : !
rP: F # : !H > #ON _ zG_ zaz N < E <
xr = (x1, . . . , xr)JZH @D@ H A : W(η, β)
< : Nn− r
d1#1N > H > # !%! A U #E # > A *KE F # : !H > #1N U < @ P # !2* N1# : H E%#1N
cn−r = (c1, . . . , cn−r)S%-'K P !2* EK&# H C ! # > U #1NG@DA /#1@DABK HRH N F < :YC #
> A\EEN# :(< !
L(tn; η, β) =βr
ηr
r∏
i=1
(xi
η
)β−1
exp
−
n∑
j=1
(tjη
)β .
+ + )! #" %$ ( '& & W
-'K&# P ! P< @ #1A CP @D@)( < >X< L'#E%# > !(η, β)
K < U # NQA # > # : E ! # : ! # ! S-'K&# !F < @$#*( < >X< L'##EN# >ηA ! EK,#
63 +-, (# > F # : EAD@$#H*J EK,#]NQA ! E > A CP EA H :V<*: NMA ! K H L H/. # : # H P ! E HTS'0 P E'EK&# ! K < ((#1( <*>< L'#EN# >
βK < ! : H NQABL'# : ! A H : S
> : #324(# > E U A-#657( H A : E < C H P Eβ< ()((# < > ! E H C # |98 g0_;:<8=:-_`?> EK&# U < @ P # H*J β > # # F E ! <A@R: H 5q@$#1N . # <*C H P E
EK&# Z)\]R^'Z)> c :-_`?>CBZH > J P E P> #D C ##K < U A H > H*J EK&# F#H LE( H : # : E ∑ S -'KA ! @R: H 5q@-#ON . #A ! A : E > A : ! A F E H EK&# F#H LE( H : # : E *A : N1#6((# : N1# : E@GF H J EK&# |98 f9:'_;:<8=:-_`?> @: H 5q@-#ON . # H*J EK&# J < AB@ P> #!EABL.#H5qKA F KpA ! E ><*: ! @ < E%#1N C F <@R: H 5q@$#1N . # H :J P: F EA H : !tH*J
(η, β)< ! EK&#xL'# < : * EK&# L'#1NQA < : * ##E F S-'KA !JI P< @DADE < EA U #!L'# < : A : .[H*J
βF < :MC # KA . K@BA . KE%#1N < !
JZH @D@ H 5 ! 5qK&# : EK&# FH LE( H : # : E ∑ A !C! PC LADEE%#1NrE H < . A : .,* < . A : . >< EN# A !7. A U # :(C F
∂hW (t|η, β)
∂t=
β(β − 1)
η2
(t
η
)β−2
<*: N < . A : . < F#F ##@$# >X< EA H : F <*:rC #sL'# < ! P> #1N C F
∂2hW (t|η, β)
∂t2=
β(β − 1)(β − 2)
η3
(t
η
)β−3
.
-'K P !2* < ! 5`# ! < AN C # JZH > # * EK,#H#K2( H : # : EA < @TNQA ! E > A CP EA H :(β = 1)
F <*:rC # P ! #1N JZH > L H N1##@B@DA : . < FH LA( H : # : E5qKA F KpA ! H : @LF ! PC LADEE%#1NpE H < F#F ABN&# : E < @ J < AD@ P> # * 5qA\EK H P E < . A : . >X< EN#S ) < F #^E H < . A : .,* # : . A : #2# > ! P ! P< @D@LFE > F
0 − !"# %$ '&()" * )"+ ,E H! @ H 5 N H 5 :(< . A : . C F ( > # U # : EA U # F < > # B ! P F K(EK < E EK&#H# ! EABL < E%#1N
β> #1L < A : ! C ##E%5 #2# : <*: N DS
;: H P> ! E P N)F * EK&# < U < AD@ < C @$#Y#K2(# > E H (A : A H : ! <*> #L < A : @LF I P< : EA\E < EA U # <*: N . A U #@DADEE@-#wA : JZH > L < EA H :< C H P E
βS -'KA ! A ! < ! A . : A F < : E3NA # > # : F # 5qADEKYEK&# FH : NQA\EA H : ! > # I P A > #1N JZH > A : ! E < : F # C F QA : . ( P> 5 < @D@ <
< : N H : . B D <*: N RA : . ( P> 5 < @B@ < B D * 5qK H ( > H ( H8! #ON KA$# >X< > F KA F < @ ( > A H >8CP AB@NQA : .+! 5qADEK :RP L'# >H P ! K4F4((# > ( < >X< L'#E%# > ! S H E%#!EK < E0`# > . # > <*: N P: B /D < !%! P L.#ON8EK&# ! < L.#xK4F ( H EK&# ! A ! H*J < ( > H L`A : # : E( > A H > @R: H 5q@$#1N . # H :
βS 0 # > # * 5 # ! K < @D@ ! ABLE(@LF F#H : ! AN1# > EK < E P ()(
(π(β))A ! C H P: N1#1N JZH > H C a=# F EA U # > # < !H : ! S
;: EK&# JZH @D@ H 5qA : .,* 5`# P ! # JZH > F#H : U # : A-# : F # EK&#( <*>< L.##E > A < EA H :µ = η−β S 1 H > # . # : # >< @D@LF * 5`#]N1# : H E%#
θEK,#( <*>< L'#EN# > !^H J EK&#[L H N1##@ * # ! ((# F A < @B@GF 5qK&# : 5`#3N1# : # H >!P ! #[EN# F K : A I P # ! 5qKA F K F < :?C # < ()(@BA$#1NVA :
L H > # . # : # >< @ ! ##EEA : .8! EK <*: ##A CP @B@TA : J # > # : F #*S
-/. r4 6s4 687
! 6W & %$ &(,+ &) W4 & &
# < !%! P L'#tEK < E < : H C a=# F EA U # @R: H 5q@-#ON . # H JβA ! < U < AD@ < C @$# ! P F KsEK < EIEK&# ( > A H > N H L < A : F < :3C # C H P: NN H :
[βl, βr] ⊂ (0,∞)S ) H @B@ H 5qA : . <*> . P L'# : E !9J > H L ? > E H B /D * 0`# > . # > < : N P: B /D <*: N'# ! ((# F A < @B@GF 0 < F K <
>K:18"gBz B D * EKA ! A ! <> # < !H :<*C @$# < !N! P LE(QEA H : ! A : F #β
NQA > # F E ! EK&# @ A : #EA F2!]H*J < . A : . S : L'# F K < : A F < @! F ! EN#1L !2*RJZH > (K F ! A F < @ > # < !H : !?*
β > 5A ! : # U # > # ! EABL < EN#ON H :r> # < @ #1A CP @D@%N < E < B #K2 F #3(QE5qK&# : EKF FH L'#
J > H L < ( < >X< L'#E%# > #1A CP @D@ NQA ! E > A CP EA H :< : NML P ! E C # > #ON P F #ON J > H L <3CP>: A : EADL'# DS! ! P< @D@LF *β! E < F !
A :[1, 2.5]
S 6#?# < :: H F < : N > HQF < F1F A < B D JZH >[< # : . A : #?# > A : .U A-#657( H A : E1S 1 H > # H"U # > * 5qK&# :Y< . A : . A !< !%! P L.#ON * 5`# K < U # F @$# < > @LF
βl = 1S
) H @B@ H 5qA : . 0 < F K < B /D * 5`# ( > H ( H+! #xE H P ! # < 0`#E < Be(p, q)NQA ! E > A CP EA H : JZH > EK( > A H > H :
β*
π(β) =Γ(p+ q)
Γ(p)Γ(q)
(β − βl)p−1(βr − β)q−1
(βr − βl)p+q−1 [βl,βr](β)
C # F <*P ! # H*J ADE ! #K2QA C AD@BA\E%FS E F < :(C # F < @BA C>< EN#1NrA : E%# > L !qH J%U <*> A < : F # < : N F # : E >X< @ U < @ P #sA : N1#6((# : N1# : E@GF H J(βl, βr)
S 1 H > # H"U # > * ?[β]
N(# : N !tH : @LF H*Jp/q
<*: N F <*:C # F K H8! # : A : N(# : N1# : E@LF H*J <*>[β]
B < : N ! ADL`AB@ <*> @LFJZH > EK&#nL H N1#15qKA F KN1#3(# : N !qH : @GF H*J
(p− 1)/(q − 1)DS 0 < F K < >3:C8"gBz B D!( > H"U AN1#ON < L'#EK H N H @ H . F`E H
F < @DA C>X< EN#π(β)
5qKA F KV@ < F !qH : EK&# ! (# F A F < EA H : H*J <*: H C ! # > U < EA H :βm
S! +# :)@ A : !H : B //D . A U # ! <`> # U A$#35H J EK&# F < @BA C>< EA H : L'#EK H N ![H*J < 0`##E < ( > A H > NQA ! E > A CP EA H : A JT!H L'# U < @ P # !3H J
βF < :=C # CP AB@\E J > H L ( < ! E
#K2(# > ADL'# : E !3H > #K2((# > E I P # ! EA H : A : . S 0 H 5 # U # > * ! P F KkL.##EK H N H @ H/. A$# ! > #1L < A : 8*a#"M] c S 1 H > # H"U # > * ! A : F ##K2(# > E ! < > # : H E ! E < EA ! EA F A <*: !?* EK&# I P # ! EA H : A : . K < ! E H C #sA : NA > # F E1S
#1( > H ( H+! # EK&# JZH @D@ H 5qA : . L'##EK H NrE H8H C E < A : ( > A H > # ! EABL < EA H : !'H*JβS > C H P EqEK&# FH LA( H : # : E ∑ * < ! @
E H EK&#.#K2(# > E ! EK&#A( > H C<*C AB@DADEA$# !p0
< : Np1
E HpJ < @D@N H 5 :wC # JZH > #]EK,#3EADL'# !t0
< : Nt1 > t0
* > # ! (# F EA U #1@GFS-'K&# :
1 − p1
1 − p0= exp
−µxβ
0
(x1
x0
β
− 1
)= (1− p0)
“
x1x0
”β−1
< : N(N1#ON P F #nEK( > A H > # ! EABL < EA H :
βm = log
log(1 − p1)
log(1 − p0)
log−1(x1/x0).
B 0 S D
)
;:?: #K2RE ! PC ! # F EA H : * 5`# JZHQF P !!H P> ! E P NF H : EK&#E( > A H > NQA ! E > A CP EA H : H*J EK&# !F < @$#( < >X< L'#E%# >ηS[> K4F (# >
( <*>< L'#EN# >a5qKA F K F < :WC #rA : EN# > ( > ##EN#1N < ! < U A > E P < @ ! < LE(@$# ! A #8A ! N1# : #1N S iK&# : )21#1N * EKA !3U A > E P< @
! A #5qAB@D@ C # P ! #1N < J EN# > 5 < > N !B A : § 0 S O S S DqE H ( > H ( H8! # < F < @DA C>X< EA H : L'##EK H N H*Jπ(β)
5qK&# :w: H c \>XaI_ bgG>A : JZH > L < EA H :r< C H P EqEK&# P: F # > E < A : E%F H*J
βA ! < U < AD@ < C @-#S
!W'& $ & 2+ &) W & &
J I J G B)G B G J DEJ #J#K2REN# : N(K&# > # EK&# < ()( > H < F K H J 0 # > . # >q< : N P: B /DS H : NQADEA H :< @GFpE H
β* EKF F K HRH+! # JZH > EK&# !F < @-#
( <*>< L'#EN# >ηEK&# . # : # >< @DA #ONpA : U # > ! # . < L`L < NA ! E > A CP EA H :
η|β ∼ GIG(a, b, β)
5qADEKVN1# : ! ADE%F
f(η|a, b, β) =baβ
Γ(a)
1
ηaβ+1exp
(− b
ηβ
)1[0;+∞[(η)
B 0 S /D
5qK&# > #(a, b, β) > 0
* 5qADEKL H L'# : E?[ηk] =
bk/βΓ(a− k/β)
Γ(a)∀ aβ > k > 0,
<*: N L H N1# 1d[η] = (bβ/aβ + 1)1/β S*-'KA ! J < L`AD@LF A !TF @ H8! #ON C F !F < @$#%E >X< : !JZH > L < EA H : B AS #*S
X ∼ GIG(a, b, β) ⇒∀c > 0
*cX ∼ GIG(a, bcβ , β)
D 5qKA F KWL <=@ # ! ADE]A : EN# > # ! EA : . E H > #3( > # ! # : E ( > A H > A : JZH > L < EA H : !nH :Y< !F < @-#( <*>< L'#EN# > S 1 H > # H"U # > * A J 5 # F#H : ! ABN&# > EK&# > #3( < >X< L'#E > A < EA H :
µ = η−β * 5 # H C E < A : ( > A H >'<*: N ( H8! E%# > A H >
µ|β ∼ G(a, b),
µ|β, tn ∼ G(a+ r, b+
n∑
i=1
ti
).
-'K P ! EK&# ( H+! EN# > A H > F#H LE( P E < EA H : B ! #2# § 0 S DA ! L < N1# # < ! A$# > S 0 H 5`# U # > * 0`# > . # >p< : N P: B /DN H: H E ( > H"U ABN&# < L'# <*: A : . E H EK&# K4F4((# > ( <*>< L'#EN# > !
a<*: N
bS -'KA !rF K H A F # > ##L < A : !(FH : E > H"U # > ! A < @ ! A : F #
π(µ|β) = π(µ)S-'K&# : 5`#H( > H ( H8! # < :< @DEN# >: < EA U # F K H A F #*S
P (9( H8! #wEK < EMEK,# I P<*: EA\E < EA U # #K2(# > E H (A : A H : F < :iC # > #6( > # ! # : EN#1N C F < `_\ : |98 g ##A CP @B@ ! < LA(@-#xm = (x1, . . . , xm)
S #EπJ (η|β) ∝ η−1 C #!EK&# FH : NQA\EA H :< @ +# > #6F ! ( > A H > S J%H P> FH : NQADEA H :< @ ( > A H > H :
ηF < :(C # > #3( > # ! # : E%#1N C F8EK( H+! EN# > A H >πJ (η|β, xm)
* 5 # H C E < A :
a = m,
b = b(a, β) =a∑
i=1
xβi .
-'K P ! EK&# F K H A F # H*Jb< ! < J P: F EA H : H J
β< ()((# < > ! E H C # :< E P>X< @S 0`# ! AN1# !2*
aE < @ # ! EK,#!L'# <*: A : .]H J EK&# ! A #
H*J < U A > E P< @ ! < LE(@$#HFA$##@NQA : . EK&# ! < L'# A : JZH > L < EA H :r< ! EK&#H#324(# > E H (A : A H : S 0 # : F # a A ! : H 5 < c 8"g0_ bL\8?:'_,]"fK4F4((# > ( < >X< L'#E%# > ! A : F # EK,#*0 < F+# ! A <*:[< :< @GF ! E F < : L H N P @ < EN# EK,# ! E > # : . EK H J EK( > A H > |98 f9:'_;:<8=:-_`?>IH (A : A H :EK > H P . K(EKA !T! ABLE(@$#[K4F4((#6( < >X< L'#E%# > S -'KA !T! < LA(@-# ! A #]A ! < : # < ! FMA : EN# > ( > #E < C @$#1( < >X< L'##EN# >x< : N < .HRH NJZHQF P ! ( H A : E JZH >'< NQA !F P !N! A H :pC #E%5`#?# : EK,# <*:< @LF ! E < : NpEK&#:#K2(# > E1S #H( > H ( H+! # !H L'#xN&# J < P @\E F < @DA C>X< EA H :U < @ P # ! JZH >
a<*: N I P # ! EA H : A : .! P ./. # ! EA H : ! A : § 0 S O S
0 − !"# %$ '&()" * )"+ , N1# < @D@LF *
π(µ|β)K < ! <:< E P>X< @IKA$# >X< > F KA F < @ ! E >P F E P> #*S > !%! P L`A : .
xmi.i.d.∼ W(ηe, βe)
5qK&# > #(ηe, βe)< > # ( > A H > # ! EADL < EA H : !2* 5 # . ##E
b|β ∼ G(a, η−βe ),
µ|β, b ∼ G(a, b).
0 H 5`# U # > *vJZH > ! ADLE(@BA F A\E%F > # < !H : ! 5`#E( > # J # > E H?F K HRH+! # < N1##EN# > L`A : A ! E #K2( > # !%! A H : H JbA : J P: F EA H : H J EK,#
< U < AB@ <*C @-#H( > A H > ! (# F A F < EA H : ! S -'KA ! A ! N H : #sA ! EK&# : #K2RE( <*>< . >X< (K5S GJ FHG I J DEG J 2I G I B I J I B;J-'K&#Y#K2(# > E H (A : A H : A ! #|98"f :-_;: 8?:'_`?> 5qK&# : K&#E < @ @ ! A : EN# > L !8H*J @BA J ##EABL.#S 1 <*: F < P EK H > ! A : 0 < F/# ! A < :@DADEE%# >X< E P> # P ! # #324(# > E H (A : A H : !xH : EK&# U < @ P # !xH J EK&#( < >X< L'#E%# > U # F E H > B ! #?#E0 < F K < * TA a H A H >iA !%! # >3: 8"gDz / DS E F <*:pC # P: N1# > ! E < : N < C @$# 5qK&# : ADE F#H : F # >: ! EK&#:#K2( H : # : EA < @ L H N1##@ *1! A : F #!EK&#xL'# < :@DA J #EADL'#A ! EK&#( < >X< L'#E%# > SH0 P E P ! P< @B@GF(EK&#'#K2((# > EnN H # ! : H E @R: H 5 EK,# ! E < EA ! EA F < @%L H N&##@ < : NVEK&#E( > A H >A : JZH > L < EA H : H : ( <*>< L'#EN# > !nF#H >> # ! ( H : N ! L H > #`E H < 3 E ><*: !JZH > L < EA H : H J EK&##32((# > E @R: H 5q@$#1N . # C F?EK,#0 < F/# ! A < :M<*:< @LF ! EOS
-'K P !?* QA : ( P> 5 < @B@ <!< : N H : .EB DvEK,# : RA : ( P . ( P> 5 < @B@ < B D FH : ! AN1# > #1NsEK < E <*: #32((# > E 5 < ! < C @$#E H'! (# < @8< C H P E EK&# L'#1NQA < : @DA J #EADL'#
m = η(log 2)1/β K&# A !`! P ()( H+! #1N8E HA. A U # <*: # ! EADL < EA H : H J EK&# ( > A H >L'# < : ?
[m]S ) AG2QA : . π(β)
* EKF H C E < A :M< F#H LE(@$#E%# ( > A H > H :pC H EK ( < >X< L'#E%# > ! S ;: H P> A : N P ! E > A < @ FH : E%#K2RE *E H C #^L H > # . # : # >< @ * 5`# F#H : ! ABN&# > EK < E <*: #K2(# > E F <*: . A U # < : # ! EADL < EA H : H J < I P<*: EADE%F F K <*>< F EN# > A A : . EK,#hE8I\ _f 8"g%a_^ :-\_ b#|:'_,]"f M H J @DA J #EADL'#
T5qA\EKN&# : ! A\E%F
mW (t) =
∫
IR
∫
IR
fW (t|µ, β)π(µ|β)π(β) dµdβ.B 0 S /D
> : A : EN# : ! A U #]@BADEEN# >< E P> #]N1# < @ ! 5qADEKEK,# I P # ! EA H : A : .MH J #K2(# > E !B ! #2# < : # ! @ K < K O JZH >x<8> # U A$#35 DS H L'#NQA !F P !N! A H : E%# F K : A I P # ! < ! EK&# bC_^3> c :-_,] f L'##EK H N B . < > EK45 < ADEN# >K:8 gDz //DnK < U # C #?# : ( > H ( H+! #1N5qA\EK ! P F1F # !%! E HH C E < A : J > H L #K2(# > E !*I P< : EA\E < EA U #xA : JZH > L < EA H :(< C H P E EK&# C ##K < U A H > H J < ! E P NQA$#1N ! F ! EN#1LS;: H P> F < ! # * E%5 H I P # ! EA H : ! < > ## !%! # : EA < @ 3 <*: F H P . A U # <V> #3( > # ! # : E < EA U # U < @ P #
teH*J EK&#p@BA J ##EABL'#
TH J'FH LE( H : # : E ∑ < : N 3 iK < E]A ! EK&# ( > H C< C AB@BA\E%Fα
JZH > EK,# FH LE( H : # : E ∑ E H(J < AB@ C # JZH > #EABL'#te
B EK&# FH : E%#K2RE H J C H EK I P # ! EA H : ! NQA # > EK&# > ! E H : # FH : F # >: ! \ >g0_ 8bC_g+_;: <*: NpEK&# ! # FH : N H : #]N1# < @ ! 5qA\EKa"|\8bC_g+_;:puF! #?# JZH > A : ! E < : F # < 5q@-# !N! /DS -'K P !2* 5 # < !%! P L'#nEK < ExEK&# #K2((# > E F <*:VC # !H @DA F ADEN#ONE H#. A U #< @BA J ##EABL.# U < @ P #
te<*: N(EK,# ^'Z)> c _ c 8?:'_,]"f(H J
te5qADEK > # ! (# F E!E H M S-'KA !!U < @ P # F < :?C #(# > F #1A U #1N < ! EK,#
# ! EABL < EA H : S H*J EK&# α H > N1# > (# > F # : EAB@-# * :< L'##@LF
P (T < te) = α ∈]0, 1[u
QS H*J #324(# F E < EA H : ?[T ]
uS H*J L H N1# 1
d[T ]B K H 5 # U # > *(! # U # >X< @ ( > A U < EN#3NQA !F P !N! A H : ! 5qADEK > #1@DA < C AB@BA\E%F ! (# F A < @DA ! E ! FH : F @ P N1#sE H EK,#
U # > F#5 # <=@ ( > H C < C AB@DADE%F8EK < E <*: A : N P ! E > A < @F#K2(# > E F < :> # < @D@LF ! (# F A J F < L H N1#DS-'K > H P . KWNA !F P !%! A H : *! (# F A J FRA : . ( > # F A ! ##@LF?EK,# :< E P> # H J < : #32((# > E H (A : A H : A ! # !%! # : EA < @S ) H > A : ! E <*: F # *!H L'# <*P EK H > ! K < U # : H EA F #1NMEK < E^EK&#H#K2(# > E ! E%# : NrE H . A U # < L'#ONQA <*: U < @ P # < @DEK H P . KrEKF P: N1# > ! E < : E
te< ! < L'# <*: B F KA$# > # : * < :: H F < : N > HQF < F#F A < DS-'K&# ! (# F A F < EA H : F < :C # > #1A : JZH > F #1N C F !H L'#H (# : I P # ! EA H : ! < !
S iK&# : EK&# < . # H J < . > H P ( H JFH LE( H : # : E ! > # < F K,# !te* 5qKA F K ( > H ( H > EA H :
1 − αA !^! EAB@B@TA :P ! #
)
S ! EK&# U < @ P #te
< L.# < : H J ( < ! E^# U # : E !2* E < @ A : . A : E H < F1FH P: ER#K2RE > #1L.# U < @ P # !'H >q: H E
S ! EK&# U < @ P #te
EK&# J < AD@ P> # EABL.#15qKA F KK < ! C #2# : H C ! # > U #ON8EK&#sL H+! E J > # I P # : E@LFpA : EK( < ! E
-'K&# : *5F#H : NQADEA H : < @LFE H EK&# F K H A F # H Ja* EK&# F K H A F # H J
bA ! . A U # : A : EK&# JZH @B@ H 5qA : . ( > H ( H+! A\EA H : J > H L EK&#
@R: H 5q@$#1N . # H*J EK&#sL < > . A :< @ NQA ! E > A CP EA H : M S
S >3: te ∈ IR+∗ a > 08 f a
α ∈]0, 1]z A>#f)]?: >
b1(a, β) =((1 − α)−1/a − 1
)−1
tβe ,
b2(a, β) =
(Γ(a)
Γ(1 + 1/β)Γ(a− 1/β)
)β
tβe ,
b3(a, β) =aβ + 1
β − 1tβe .
A>f]?: >Bi i = 1, 2, 3
: "9> _f a | c >Xa Z)\_,]I\ h ]1a/>gg+_f ^z ")>#f _C]\*8"gg c "M]_ c >S]_ π(β)^X| c ": "98?: |ZEZ (π(β)) ⊂
[βl, βr]
(i) P (T ≤ te|B1) = α,
(ii) [T |B2] = te_ _a > β−1
l
(iii) d[T |B3] = te_ _βl > 1
z "9>C\>: ")>A>eCZ9> c : 8?:'_,]"f _^1: 8 d >f _;: " \>C^-Z9> c : :.]#: "9>]h 8\ _f 8"g aI_^:'\_ b#|:-_,] f M _;: "wa/>f ^ _;: !z ( z
@ -B B 6#2# >()(# : NA 2WS S
H EA F # EK < E < : #K2(# > E F < : . A U #nL H > #!EK < : H : # ! (# F A F < EA H : S ) H > A : ! E < : F # * K,# F < : . A U # < c \>XaI_ bC_g0_;: _f :<>L\`8 g
[te1, te2]< : N ( > H C < C AB@DADEA$# !
(α1, α2)! P F K EK < E
P (T ≤ tei) = αiJZH >
i = 1, 2S # ( > H ( H+! #
E H > #1N P F # <*: F #K2((# > E H (A : A H : E HL3 NA !F > #E%# ! ((# F A F < EA H : !i = 1, . . . , P
EK&# : E H ! #1@-# F E FH P (@$# !(ai, bi)
S) A :< @B@GF * 5 # ! P L < @D@ ! (# F A F < EA H : ! EK > H P . KMEK( > A H >
µ|β ∼ G(
P∑
i=1
Pγiai,
P∑
i=1
bi(Pγiai, β)
)B 0 S O D
5qKA F KMA ! EK&# ( H+! EN# > A H > FH LA : .3J > H L ! P F#F # !N! A U # 0 < F+# ! A <*: A : J # > # : F # ! H : U A > E P< @ ! < LE(@$# ! S #1A . KE !γi > 0B ! P F K EK < E ∑P
i=1 γi = 1D <*> # )21#1NkA : J P: F EA H : H*J EK&# > #1@ < EA U #rE >P ! EpA : EK&# ! (# F A F < EA H : ! S < EN# > A :
EK&# < > EA F @$# * <*: < ()(@BA F < EA H : 5qAB@D@#321##LE(@BA J F B < : N F @ <*> A J F)D!EKA !]F K H A F #*S E]A ! > ##A : JZH > F #1N C F?EK&# F#H : U #32RADE%F( > H (# > EA$# ! H J < ()(@BA F < EA H : !
β 7→ b(a, β)S ? ! (# F A < @D@LF * ADE < ()(# < > ! H C U A H P ! @LF3EK < E
b1(a, β)B EK P ! <*: F F#H : U #32
! P L H*J ! # U # >< @b1(a, β)
D 3 L`ABL`A F2! [EK&# . # H L'#E > A F < @ C ##K < U A H > H Jβ 7→ ∑a
i=1 xβi
S E : #?#1N !H!H L.#A5`##A . KE%FF < @ F P @ < EA H : ! B : H E . A U # : K&# > #D CP E F <*: # < ! AD@LF C #?# ::RP L'# > A F < @B@LF F K&# F @ #1N S
;:]: #32RE . P> # ! 5`#tNQA ! (@ < F !H L'#tN1# : ! ADEA$# !m(t)
H J EK&#!( > A H > L <*> . A : < @*NQA ! E > A CP EA H : M * 5qK&# > # te = 100*
[βl, βr] = [1.1, 5]*(p, q) = (1.5, 1.5)
<*: Na = 2
S ) A . P> # 0 S ! K H 5 ! m(t)5qK&# :
teA ! ! P F#F # !%! A U #1@GF FH : ! AN1# > #1N
< ! EK&# ( > A H > L'#ONQA <*: T L'# < : T"L H N1#S ) A . P> # 0 S ! K H 5 ! EK&#Y# U*H @ P EA H : H Jm(t)
5qK&# :te
A ! EK,#α
H > N&# >(# > F # : EAD@$# **JZH > ! # U # >X< @ U < @ P # !9H J
αS -'K&# F#H >> #1@ < EA H :3C ##E%5 #2# : ( <*>< L.##EN# > ! A : N P F #1N C F]EK&#( > A H > L H N&##@B@DA : .
A ! > # U # < @-#ON C F EK,# c ]"f `>e "R|Qgg H J EK&# ( > A H > ! < LA(@DA : .=u A : ) A . P> # 0 S * H J ( > A H > ! ADL P @ < EN#ON(η, β)U < @ P # ! < > # P ! #1N JZH > N&##@BADL`ADEA : . EK&# F#H : U #32 K P @B@ S -'K&# JZH > L H J EK&# F#H >> #1@ < EA H : > #1L < A : ! : # < > EK&# JZH > L
H*J EK&#[EK&# H > #EA F < @ a H A : E FH : N1# : F # <*> # < ! K H 5 #ON C F P B D u EK&#[A : U # > ! #[NQA ! E > A CP EA H : H J ( < >X< L'#E%# >U < @ P # ! 5qKA F KVA ! P ! P < @B@GF ((# > F ##A U #ONrA !!F#H K&# > # : E5qA\EKEK&# F @ < !%! A F < @ C #1K < U A H > H JIJ > # I P # : EA ! E\# ! EABL < EA H : !qH JEK&# ( <*>< L'#EN# > ! B P ! A : . P:)@R: H 5 : N < E < D A J β A ! H"U # > # ! EADL < E%#1N3EK,# :
ηA ! P: N&# > # ! EABL < EN#ON * <*: N FH : U # > ! ##@LF*S
0 − !"# %$ '&()" * )"+ ,
t
m(t
)
0 50 100 150 200 250 300
meanmodemedian
te=100
) A . P> # 0 S T# : ! ADEA$# !m(t)
H J EK&# ( > A H >L < > . A :< @NQA ! E > A CP EA H : M B A : N1#326#ON C F[EK&# ! ((# F A F < EA H : H*J
te = 100DS
t
m(t
)
x
m(x
)
x
m(x
)
0 50 100 150 200 250
te=100
10%
25%
33%
50%
66%
75%
90%
) A . P> # 0 S \# : ! A\EA-# !m(t)
H*J EK&# ( > A H >L < > . A :< @NQA ! E > A CP EA H : M B A : N&#K21#1N C F[EK&# ! (# F A F < EA H : H J
αDS
eta
beta
0 20 40 60 80 100
020
4060
8010
012
014
0
) A . P> #10xS H : U #K2pK P @B@ H J < ( > A H > ! < LE(@BA : .`H J ( < >X< L'#E%# > !(η, β)
S
-/. 4 6s4 4 2 6
H L'# <*P EK H > ! < ! RA : . ( P> 5 < @D@ <'< : N H : .B /D CP AB@BN( > A H > L H N1#1@D@BA : .+! 5qADEK :RP L'# > H P ! K4F4((# > ( <*>< L'#EN# > !2*5qKA F K < @B@ H 5GE H L H NQA J F!EK&#]#32((# > E5A : JZH > L < EA H : A : @ HQF < EA H :]< : N P: F # > E < A : E%FSY> ! TA : NQ@$#3F < : N QA : . ( P> 5 < @D@ <B D : H EA F #ON * A\EsK < ! C #2# : H C ! # > U #ONEK < E:#32((# > E ! E%# : NwE H ( > H N P F # 3 @ HQF < EA H :G< : N !F < @$# C A < ! S ;: EKA !5 H >@ * H P> H (A : A H : A ! EK < E[EK&# J P @B@ #32((# > E @R: H 5q@$#1N . #8A !. A U # : EK > H P . K
teB < : N
αA Jte
A ! < (# > F # : EAD@$#U < @ P # DS H3H C a=# F EA U # F > A\E%# > A H :M< @B@ H 5 ! EK,#H0 < F/# ! A < :p<*:< @LF ! E E H L H NA J F8EKA ! #32((# > E @R: H 5q@-#ON . #S*-'K,# H : @GF( H8!N! A C AD@BADE%FpA ! E H L H N P @ < EN# EK&#H#32((# > E P: F # > E < A : E%F8EK > H P . KMEK&# F K H A F # H J
aS ;: H P> ! # : ! # * EK > #?# ! E%#3( !'H*J
F < @DA C>X< EA H : ! K H P @N C # FH : ! ABN1# > #ON S*S > a > _K8"|Qg : c 8"g0_ bL\8?:'_,]"f! EN#6( u < F1FH > NQA : . E H EK&# :RP L C # >q< : N`EK&# :< E P> # H J! (# F A F < EA H : !?* 5`# ( > H ( H+! #!H L'#xN&# J < P @\E U < @ P # !tJZH >
aS -'KA ! A ! N H : #!A : § 0 S O SS 0`# ! AN1# !2* 5qK&# : a K < ! C #2# : F K H+! # : * 5 #H( > H ( H+! #
A : § 0 S O S*S < N1# J < P @DEsL.##EK H N JZH > F < @BA C>X< EA : .π(β)
5qK&# : : H A : JZH > L < EA H : A ! < U < AB@ < C @$# <*C H P EnEK&#P: F # > E < A : E%F H*J
βS
S > c ] h`bL_f >Xa c 8"g+_ bC\8=:-_,] f! E%#3( u C # F < P ! # a F < :MC # P: N1# > ! E HRH N < ! < U A > E P< @ ! < LA(@-# ! A ?# * A\E'A ! < FK.HH N( < >X< L'#E%# > JZH >r< NA !F P !%! A H :kC #E%5`#?# : EK,# #K2(# > E < : NkEK� < F+# ! A <*:k<*:< @LF ! EOS ) A > ! E U < @ P # !pH*J
a( > H ( H8! #ON C F3EK&#nN1# J <*P @\E F < @DA C>X< EA H : ! EN#6( F <*:pC #xKA . K 5qK&# : EK&#:#K2((# > E A ! U # > FA( > # F A ! # u EK&# A : E%# : EA ! E H L < @ # EK,#H#K2((# > E > # < F E'E H KA !^! ##@ J > # F @ @$# !%! : # !N! < : N F#H >> # F E
aS
S > tf 8 g \> c 8"g+_ bC\8=:-_,] f ! E%#3( u EK&#E0 < F/# ! A < :?< : < @LF ! E F < : K < U # !H L.#A( > A H > @R: H 5q@$#1N . # <*C H P ExEK&# > #1@DA < C AB@DADE%F H J EK&##32((# > E H (A : A H : S > K P . #8@BA\EEN# >< E P> #pA ! N1#1NA F < EN#ON E H EKA ! < : < @LF ! A ! B ! #?# HRH @ # < : N. HH8!N! # : ! JZH >!<p> # U A$#35 DS-'KA ! < :< @GF ! A !qF < :VC # C< ! #ON H : H EK&# > ! PC a=# F EA U # C #1@DA$# J !AB 5qKA F K <*> #: H E#K2(@ H > #1N=K&# > # D < : NYEK,# FH LE( <*> A !H : 5qADEK H C a=# F EA U #MA : NQA F < E H > ! S 0 # > # * A J a < n
< : N A J EK&#N < E <
tn<*> # P: F # : !H > #ON * EK&# ( H+! EN# > A H > NQA ! E > A CP EA H : A ! # : ! P> #ON E H C #ML H > #pN1#6((# : N1# : E J > H L EK&#
H C a=# F EA U #3N < E < @R: H 5q@-#ON . #S ;: § 0 S O S * < ! ABLAB@ < >!P ()((# >!C H P: N JZH >aA ! N1# : #ON 5qK&# : EKN < E <
tn< > # F # : !H > #1N SE0`# ! ABN&# !2* !H L'# C A C @BA H . >X< (KA F < @ > # ! P @\E ! < > # . A U # : E H L H N&# >X< EN#EK&##K2(# > E H (A : A H : !< : NrEK,# F#H : ! # : ! P !'H*J! # U # >X< @#K2(# > E H (A : A H : ! A !C! E P NQA$#1N S
& $ "! # G D1F%$ G F I G$ I B)J0B&$(' *) G F ,+ G-G $ -'KA !^F < @BA C>X< EA H : F < :(C # ! #6( < >X< E%#1N < F#F#H > NQA : . E H EK&# @ A : N H J! ((# F A F < EA H : S"-'K,# U < @ P # H J
aEK < E5`#1( > H
( H+! #1N3A ! EK&#qL`A : ADL < @ :RP L C # >amin
H*J ( < ! E U A > E P< @N < E < EK < E ! #?#1L : # F # !%! <*> FnE H < !%! # !N! EKA !]! (# F A F < EA H : S. G J0/21 I G J B 31 B y > FH : J P ! A H : C #E%5`#?# : EK H+! #3EK > #?# ! (# F A F < EA H : A ! # < ! F B # ! ((# F A < @B@GFVEK&#E%5 H > ! E H : # ! DS-'K&# > # JZH > # < N1# J <*P @\E F K H A F #^L P ! E C # N H : #S # ! P ./. # ! E E H3F K HH8! #
amin = 1JZH > EK&#!L'# < :
<*: Namin = 2
JZH > EK&#^L'#ONQA <*: S4((# F A J FRA : . EK&#^L H N1# > # I P A > # ! A : E P A\EA U ##@LF[L H > #7( < ! E U A > E P< @N < E < B < : NEK&#F # > E < A : @R: H 5q@$#1N . # H J < . A : . DS # ! P . . # ! E
amin = 3! A : F #^ADE'A ! EK&#xLA : ABL < @ :P L C # > H*J N < E < E H N1# : #^EK&#
L H N < @ F @ < !N! H*J < FH : EA :RP H P ! NQA ! E > A CP EA H : S ;: - < C @-#10 S * 5 # . A U #T# ! EADL < EA H : !'H J EK&# ! E <*: N < > N(N1# U A < EA H :σ
< : NwEK&# ^ d > f >^ ^ γ H J M * P ! A : . EK&#E( > A H > F K H A F # ! N H : # JZH > NQA ! (@ < FRA : . ) A . P> # ! 0 S <*: N 0 S * P ! A : .! # U # >X< @ U < @ P # !H*J
aS-'K&#qL H N1# ! ((# F A F < EA H :8< ()(# <*> ! < !'. A U A : . K,# < U A-# > E < AD@ ! < : N J < U*H > A #A : . KA . K&# > U < @ P # !
H*JX
S # > # FH L`L.# : NME H P ! #nEK&#sL'#1NA <*: ! (# F A F < EA H :(C FrN1# J < P @DE ! A : F #sA\E^# : ! P> # ! < @ <*> . # >'P: F # > E < A : E%FEK < : EK&#sL'# < : ! (# F A F < EA H : S
a 4 752; 47698 7 4 7 8;: 52;σ γ σ γ σ γ
< =?> O @ @/O A B =DC O > B O B EF@+O < A4O AB A(B/O G < O @ G?A4O C < O A =(= O C < O >@ A C O C O E A O = E4O G AFG/O < O E(@A @?G/O O @HA B = O > > O A A(A4O G O @ =C @D@+O C O <(= C O > = O B A C O B O B
- <*C @$# 0 S ? ! EABL < EA H : !qH J! E < : N <*> N(N1# U A < EA H :σ< : N ! @ #65 : # !%!
γH J NA ! E > A CP EA H : M S
IKJMLN$DO%P Q)F+ ROS ;LT$DO%UVN P> 5 H >@ A : . K4F4( H EK&# ! A ! A ! EK < E`EK&# #K2(# > EK < ! ((# > F ##A U #ON < ! L P F K ( < ! EU A > E P< @tN < E <w< ! ADE3A ! : # F # !N! < > FVE HwH C E < A : EK&##( > # F A ! A H :
α5qK&# : K&# ! (# F A F A$# !
te< ! < (# > F # : EAD@$#*S ) H >
A : ! E < : F # * EK&# 0 < F+# ! A <*:?< :< @GF ! E F < : ( > H ( H+! #a = 10
5qK&# :α = 10%
SW-'KA ! A !J! ABL`AD@ < > E H EK&# :RP L'# > H P !"_^:] \8 h L'#EK H N ! EK < EsK < U # C #2# : N1# U ##@ H (#1N JZH > EK&# < ! @ A : .(H J #K2((# > E ! SX-'K P !2*ZY < :\[ HRH > E%5qA a @ >K: 8"gDzB D ( > H ( H+! #nE HL! # . L'# : E
IR+A : ! #3( <*>< EN#sA : E%# > U < @ !?*F K HH8! A : . C H P: N < > A-# ! : # < > E H > # < @ H C ! # > U #1NN < E <
B H > F # : !H > #1N8N < E < DS]RA : F #^EK H+! #^@BA J ##EABL'# ! F < :8C # > #3( > # ! # : E < EA U # JZH > EK&#\#324(# > E B # U # : K&#!A !]! P ()( H+! #1N : H E
0 − !"# %$ '&()" * )"+ ,E H @R: H 5 EK&#sN < E < D * K&# F <*:MC # < ! @ #1N <*C H P E KA ! ( > H C < C AB@DADE%F H JTJ < AD@ P> #*S [ #32RER#32 < LE(@$#!AB@D@ P ! E >X< EN# ! H P> 5 < FH J EKA :)@ A : . S
^ S >K:(50, 80, 90, 170)
b >K^L] hE> $Z ]R^ ^_ bg c >#fM^L]\>Xa ]*bC^3>C\`?>Xa`_K8_g|\> :-_h >C^z [_`_ a > : ")>g0_ _K>3:-_h > ^ c 8"gG> _f D1 = [0, 90)
8 f aD2 = [90,∞)
z >3:(n1, n2)
b>H: "9>nf|Qh8b>C\ ^ ]_ `_\ :|98"gTa 8=:<8 _fD1
8"fvaD2
\>C^'Z)> c :-_`?>#g *z |ZZ ]R^3>: ")8?:*: ")>E>eZ9>L\ : _`?>^ c ")8 f c > ]+_3bL\>8 d _f aE] f _fD1
z ")>#f n1 = 2n2z
[_`_ a >f] D1_fD1,a = [0, 50)
8"f aD1,b = [50, 90)
z ! f aI_ c 8?: >:]#: ")>E>eCZ)>L\ : : ")8=:*: "9>A> > c :?>3g _K>K:h > c f9: 8f >Xaf
D1z | K> : ")8=: ")>!f a c 8=:<>" # c ")8"f c > _rb%$>8 d f'&Ya ( f)f
D1,b * : 8+h,gG8$- |9>" : f f 8.)8$ :;: f _ D2 / D1,a
$ D1,b / | 0 3>E: ")8=: ")>|Qf 8*b#gG>A: 8"f1 ( >%$#z32>43| 0 3>Xa : ")8>65 9>"$ c >%>Xa 7H8?: gG>88 : n1 = 4
$ : |98 g a 8=:<8z:9;f 8"gg / ")> c 8"fWb ><&0>#fn1 + n2 = 6
=$ :|98"g%a 8?: 8*z
> ? OA@O%P O$B O N$ LP )(+ ;O%SRLT$(O UN ] P ()( H+! #8E HwH C E < A : J > H L < : #324(# > E]EK&#MA : EN# > U < @[te1, te2]
<*: N( > H C<*C AD@BADEA$# !
(α1, α2)! P F KWEK < E
P (T ≤ tei) = αiJZH >
i = 1, 2SDCqNNQA : . E%5 H (# > F # : EAD@$# ! ((# F A F < EA H : !
: #?#ON ! E H : H > L < @BA ?#a1, a2
! P F KwEK < Ea = γ1a1 + γ2a2
A ! EKL < 2RABL < @ ! A 2# < EE < A : < C @$# C FEK&# ! #3( <*>< EN#! (# F A F < EA H : ! S -'K P !?*X! ((# F A F < EA H : !
(50%, 90%)5qAD@B@ . A U #%EK&# ! < L'#9N1# J < P @DE ! A 2#
a = 10EK <*: ! (# F A F < EA H : !
(10%, 90%)SFE H > # H"U # > * A J EK,#!( > A H > N H L < A : JZH >
βK < ! C #2# : #1@DA F ADEN#ON * EK&# c ")>%$>#f c > H J EK&# F > # C AD@BADE%FxA : EN# > U < @
5qA\EK(EK&#HG##A CP @B@TL H N1##@TL P ! E C # F K&# F @ #1N P ! A : .3JZH > L P @ < B 0 SDSJI Jβe = log
log(1 − α2)
log(1 − α1)
log−1(te2/te1)
B 0 S /D
A ! : H EqA :[βl, βr]
EK&# H > N1# > !`! K H P @N C #nL H NQA,#ON ! P F KrEK < Eβe = β0
BZJZH > A : ! E < : F #xEK&#sL`ANNQ@$# H >q<3C H P: NH J
[βl, βr]DS ] H @ U A : .
(yQy
) = β05qADEK < [ #35'E H : l < (K !H :Y< @ .*H > ADEKL A !_! ABLA(@-# B ! #2##0 H P ! I P #E 0K JZH >
N1#E < AB@ ! DS6L > N1# > ! < > #E5 #1A . KE%#1N 5qA\EKGEK&# FH : ! E < : E FH : U # > . # : F # >< EN#γ1/(1 − γ1)
SK-'K P !2* A Jγ1 −→ 1
*α1> #1L < A : !R! E < C @$#*S
M PNU HO$(' 1 S,O ON'$DO%NPN $D' '? OA@O%P O$Bα
QRTSVUW0 < α0
1 < α02 < 1 X0Y Z α0 = (α0
1, α02)
R\[VU Y1] W U `0 = (xe,2/xe,1)β0 ^ `(α) =
log(1 − α2)
log(1 − α1)X0Y Z`_8a1]] b U Xdce U _Vf'bf]0Y εRhg f8i 0 < ρ 1
R
jkR6lW U c k = 0, ...K m• _0] n1co W UpW a U)q U _ W ]0e δk =
((1 − αk
1) log(1 − αk1) (`(αk) − `0) /`(αk)
−(1 − αk2) log(1 − αk
1) (`(αk) − `0)
) r
• _0] n1co W U hk = − (1 − αk1)
(1 − αk2)`(αk)
R
• _0] n1co W U αk+1 =
αk
1 − ρhk(αk2 − αk
1),
αk2 − ρhk(αk
2 − αk1) γ1
1−γ1
• b W ]0ctsa U Y ‖δk‖ ≤ ε
^ Svu " 3> β0 = 38"fva
(te,1, te,2) = (200, 300)zT9;\ef'&
(α01, α
02) = (0.05, 0.95) / ( > b3:<8f
βe ' 10.03z3w "9>1>e"9>"$ : xf1 fyfva"| c >" 8"f(|Qf1$>8"gz: c 98$8 hE>3:<>%$")8)>Oz `a > _K8"|Qg : /V| e γ = (1/2, 1/2)
zw ")>#f ( > bK: 8f α = (0.3, 0.7)
z: _ ( > c " = 3>f ( α01 = 0.25 / ( > b3:<8f βe ' 5.78
z6w ")>4fva"| c >Xa 8=&f&$>#hE8f n|Qf1$>88 f 8*b#gG>Oz!~!;: " > - |98 g ( >%z&": / ( > | f a α = (0.4, 0.8)z<|Qh4f'&8h| c "(h $> c $>Xa b%g;: Tf
: ")>[g ( >%$3b |Qfva / ( > | e γ = (0.95, 0.05)z,w ")>#f
α = (0.26, 0.64)z u f>%$3>g / ( ;: " γ = (0.05, 0.95) / ( > bK: 8f α = (0.55, 0.93)
S
# LJMP%$ ;LP OA@?L$(O%UVNU $(' *) ' L + ,+ LHL1 $ L : F #
aK < ! C #2# : F K H+! # : * A\E F < :pC # P ! #ON8E HF < @BA C>X< E%#
π(β)A J : H > # < @ A : JZH > L < EA H :r<*C H P E EK&# P: F # > E < A : E%F
H*Jβ
A ! < U < AB@ <*C @$#*S I+N&# < @B@LF *M! A : F #5 # ! #2#π(η|β)
< ! <8> # J # > # : F #( H+! EN# > A H > N&# : ! A\E%F 5qADEK > # ! (# F E!E HrU A > E P< @N < E <
xa
*π(β)
! K H P @N C # F K H8! # :r< !
πJ(β|xa) ∝ βa−1
(a∏
i=1
xi
)β
(a∑
j=1
xβj
)a ,B 0 S K/D
5qADEKπJ (β)
EK&# : H : A : JZH > L < EA U # > # J # > # : F #( > A H > S 0`# F < P ! #sEKA ! ( > A H > NA ! E > A CP EA H : A ! : H ExE >X< F E < C @-# B A\EnA !5`##@B@ @: H 5 : EK < E'EK&# G##A CP @B@5NQA ! E > A CP EA H : N H # ! : H E < NLADE < : F FH : EA :P H P ! FH :+P . < EN# *RFJ S"] H @ < : N K D *5`#xL < N1# < :r<*>C A\E ><*> F F K H A F # H*J
π(β)S!0 P E'EK
`#E < ( > A H >
π(β|p, q) F <*:MC #H##@BA F A\E%#1N < ! EK&#nL`A : ADL`A ?# > H JEK&# > #1@ < EA U #:# : E > H (4F
∫ βr
βl
πJ (β|xa) logπJ(β|xa)
π(β|p, q) dβB 0 S D
P: N1# > < FH : ! E >X< A : E H : EK,#[L'# <*: H > EKL H N1# * 5qKA F K?A ! E >X< F E < C @$# P ! A : . B 0 S DS H 5`# U # > * EK,# U < @ P # !^H JEK&# U A > E P< @ ! < LA(@-#
xa
> #1L < A :YP:)@R: H 5 : S*-'K&# > # JZH > # 5 # ( > H ( H8! #`E H ! < LE(@$# U A > E P < @ ! < LE(@$# !sJ > H L EK&#W(ηe, βe)
NQA ! E > A CP EA H : * 5qK&# > #ηe
<*: Nβe
< > # ( > A H > # ! EADL < EA H : ! S"-'K P ! 5`# > #6(@ < F #πJ(β|xa)
A : B 0 S D C FEK&# >e"9> c : >Xa :<>%$+ $ x$+ $
πJa (β) =
∫
IRa+
πJ (β|xa)
a∏
i=1
fW (xi|ηe, βe) dx1 . . . dxa.
-'K&# : L`A : ABLA #A : . B 0xS=D A :(p, q)
A !C! ABLAB@ < > E H L`A : ADL`A ?#
logΓ(p)Γ(q)
Γ(p+ q)+ (p+ q) log(βr − βl) − (p+ q)
[log
βr − β
β − βl
] B 0 S D
5qK&# > #xEK,#sL.# < : [.]A ! 5qADEK > # ! ((# F E'E H
πJa (β)
SkGiK,# : EK( > A H > L'# <*: B H > L H N1# D H*Jπ(β)
A !qF K H+! # : * EK&#!H @ P EA H : A ! P: A I P #*S-'KA ! L`A : ADL`A < EA H : F <*:C #^N H : # P ! A : . E H : EN# <*> @ H # ! EABL < EA H : !9H J EK&#^#32RE > ##L'# > A . KEE%# > L H*J B 0 S DSI : H P> < ()(@BA F < EA H : !?* EKA ! L`A : ADL`A < EA H : . < U # ( > H L`A ! A : . > # ! P @DE ! S 0 P E ADE : #?#ON ! L H > # 5 H > @ E H# : ! P> # EK < EEK&# P @B@ C< F @ ##A C @$# > ( > H # F EA H : N H # ! : H EI@$# < N]E H < : H"U # > ! EABL < EA H : H*J EK( > A H > A : JZH > L < EA H : S
%$M"! ! 0! ! > UV1 + LHO%NPN ) JF@ 9$(O LN ? U@ 9$DO ! N U"XP ? N I : EKA ! ! PC ! # F EA H : 5`# . A U #pEK&#N1# : A\EA H : H J <*: A : NQA F < E H >
n5qKA F K=L'# < ! P> # ! EK&#L#$## F EA U # ! A ?# H J EK&#
< U < AB@ < C @$# F # : !H > #1NpN < E <tn
S-'K&# U A > E P < @ ! A ?#a! K H P @N C # F#H LE( <*> #ONE H
nE H @ HQF < EN#xEK,# ! PC% # F EA U # A : JZH >
L < EA H : 5qADEK > # ! ((# F ExE H EK&# H C% # F EA U #N < E < A : JZH > L < EA H : S<C ! ABL`AD@ < > A : NQA F < E H > F < :?C #.# < ! AB@GFN1# : #1N JZH >H EK&# > L H N1#1@ ! EK < : EK&#HG#1A CP @D@ NQA ! E > A CP EA H : S
0 − !"# %$ '&()" * )"+ ,\# : H E%#
xm
< :i .i .d .
G#1A CP @D@ ! < LE(@$# H*J ! A ?#mFRA$##@NQA : . EK&# ! < L'#xN < E < A : JZH > L < EA H : H*J
tnS ) H > L < @B@GF *
N1# : H EA : .
DJc,J(tn,xm) =
∫
Θ
πJc (θ|tn) log
πJc (θ|tn)
πJ(θ|xm)dθ
B 0xS DEK&# P @B@ C< F @ ##A C @-# > NQA U # > . # : F # C #E%5`#?# : E%5 H > # J # > # : F #7( H8! E%# > A H > NQA ! E > A CP EA H : !2* 5`#xN1# : # H P> A : NQA F < E H >H J # #M# F EA U # ! A ?# C F
n = arg minm
Xm DJc,J(tn,xm).B 0xS D
) > H L A : >3:8"gBz B K * > H (TS D * EK&#K#32QA ! EN# : F # < : N P: A F ADE%F H Jn
A ! # : ! P> #ON S I : B 0xS D * πJ < : NπJ
c< > # : H : A : JZH > L < EA U # ( > A H > ! 5qADEK .*HRH N J > # I P # : EA ! E FH"U # >< . # H J EK( H+! EN# > A H > 0 < F+# ! A : E%# > U < @ ! JZH >xm
<*: Ntn* > # ! ((# F EA U #1@GFS I : NQA F < E H >
nA !^H*J A : EN# > # ! E ! A : F #3A\E!ABLE( > H"U # ! EK&# FH : ! # > U < EA U # F K H A F #
rB EK&# :P L C # > H*J
P: F # : !H > #1N(N < E < DC5qKA F K P: N1# > # ! EADL < EN# ! EK&# #$## F EA U #]N < E < A : JZH > L < EA H : S ! (# F A < @D@LF * 5qK&# :?< @D@ < U < AD@ < C @$#N < E <`< > # F # : !H > #1N B
r = 0DC5`#nK < U #
n > 0S;L JFH P> ! # * 5qK&# :
tnA ! P: F # : !H > #1N * 5`#nK < U #
n = nS
! P< @B@GFπJ <*: N
πJc
< > # ! < AN c >%$8%& >(h 8?: c "Vf'&6 $+ $8u ! #2# < EE < B K D < : N K H+! < @ B D JZH >> # U A-#65 ! < : N ( > # F A ! A H : ! S-'K&# N1# : A\EA H : JZH @B@ H 5 ! IN1# : H EA : .
θn(α)EK&#H( H8! E%# > A H >
α− I P < : EAB@-# H J θ C < ! #1N H :H C ! # > U < EA H : !
tnB AS #S
P Jc (θ ≤ θn(α)|tn) = α
D * ADE^L'# < : ! EK < E
Pθ (θ ≤ θn(α)) = P Jc (θ ≤ θn(α)|tn) + O(n−i/2)
5qK&# > #^EK&#!@$# J E K < : N ! AN1#!A ! EK&# J > # I P # : EA ! E ( > H C<*C AB@DADE%F * 5qK&# > #θA ! )21#1NpA :
Θ<*: N
θn(α)A ! >X< : N H L * < : N
iA !B AN1# < @D@LF)D EK&#sKA . K&# ! E < EE < A :<*C @-# U < @ P # B EK&# F#H"U # >X< . # H > N1# > DS I : H P>^< ()(@BA F < EA H : !?*
πJ 5 < !qF K H8! # :(< !EK&# $> _K>%$>#f c >k $+ $B 0`# > . # >9< : NE0`# >:< > N H D 5qKA F K8A ! < E9@$# < ! E H*JvH > N&# > E%5 H < F#F#H > NQA : . E H ] P: B DS]RA : F #
tnA !sF # : !H > #1N * 5`# F K H8! #
πJc
< ! EK&# ! (# F A < @ /#$# > #3F ! ( > A H > B A : F @ P NA : .VF # : !H > #1NEADL'# ! D 5qKA F K F < :C #3N1# > A U #1N J > H L \# ] < : EA !>3:78 gDzB DS0-'K&# < P EK H > !^! K H 5 ADE!K < ! C #EE%# > ( H8! E%# > A H > FH"U # >X< . #xEK < : EK,#! E < : N < > N +# # > #3F ! ( > A H > S ) > H L 0 H P ! I P #E B K * K < (QEN# > D * ADE!A ! N1# : #1N < !'JZH @B@ H 5 !?* 5qK,# :
tnF#H : E < A : !
n− r> A . KE F # : !H > #ON )21#1N U < @ P # !
c1, . . . , cn−rS
>3:γ
b>#: ")> |QgG>%$ c f : 8"f : γ ' 0.57722 z >K:γ1 = π2/6 + γ2 − 2γ > 0
8"fvaγ2 = −2(1 − γ)
z>#f :<>
δ(µ, β) = δ2(µ, β) + [δ(µ, β) − 1](γ1 + γ2 logµ+ log2 µ
)+ π2/6 − 1,
δ(µ, β) = n−n−r∑
i=1
exp(−µcβi ).
w ")>#f : ")>.9> c 8 g>$> 8x$+ $t_ $1: "9>D~E>" b#|Qgg98$8 hE>3:$+ 68=: f (µ, β)
πJc (µ, β) ∝ (µβ)−1
√δ(µ, β).
N$DO%P U ? ) ;U ;$(O UN] P ()( H+! # EK < EIEK&#^NQA !F P !%! A H :3C ##E%5 #2# : EK&#R#324(# > E < : N[EK,# <*:< @LF ! E F < L'# E H < :`< . > #?#1L'# : E <*C H P EIEK&# H > N1# >α
H J < ((# > F # : EAB@-# U < @ P #teS -'K&# : * A : N1#6((# : N1# : E@GF J > H L EK&##32((# > E * EK&# <*:< @LF ! E F <*: F#H >> # F E EKA !sH > N1# >
P ! A : . EK,# > # ! P @DE ! < U # >X< . #ON C F :P L.# > H P ! < P EK H > ! < C H P E EK,# > # < @ ! # : ! # H*J EK&# (# > F # : EAB@-#A( > A H > # ! EADL <EA H : ! S ) < F #sE H EK&# > # ! P @DE !qH*JU <*> A H P ! #K2(# > ADL'# : E ! A :V> ##@BA < C AB@DADE%F * EK H8! #H# ! EADL < EA H : ! < > # . A U # : F > #ONQA C AD@BA\E%F
H > N1# > ! EK < E P ! P< @B@LF < > # U # > F H"U # > # ! EABL < EN#ON SI;@B@ P ! E >< EN#ON C F6E #6F/# >^< : N 0 HH @ # > B =D < : N < :: H F < : N > HQF < F#F A < B D * < E < F ADE >P @$#MA !'! P LL <*> A ?#ONYA :k: #32EE <*C @$# * L H NQA J FA : . EK&# H > N1# > H*J ((# > F # : EAB@$#MEK&# :. A U A : . P ! P ()(# >^C H P: N !
a∗JZH >
aS
717NA 6 : ; : 6 ; A=75 : a∗
A < A @<FC B(B B< A @ C <> A =(C <G C =(= BEDA > A @
- <*C @-#10 S - < F ADE > #ON P F EA H : H J (# > F # : EAB@-# H > N1# >qC ##E%5 #2# : #K2((# > E H (A : A H :(<*: N > # < @BA\E%FS
> UVN ) N ) J ) @ $ " N + ?$ )GiK&# : ( > A H > !
π1(θ), . . . , πM (θ)<*> #xL H N&##@B@DA : . ! # U # >X< @ < U < AD@ < C @-# B A : N1#3(# : N1# : ED #324(# > E H (A : A H : !2* EK&# c f?>e ( >%z&":<>Xa c h`b%f 8=: f
π(θ) =
M∏i=1
π$i
i (θ)
∫
Θ
M∏
i=1
πϕi
i (θ) dθ
,
5qK&# > # ∑Mi=1 ϕi = 1
* LA : ABL`A 2# ! EK,# P @D@ C< F @ #1A C @-# > A : JZH > L < EA H : @ H+!%!
KL(π;π1, . . . , πM |β) =
M∑
i=1
$i
∫
IR
π(θ) logπ(θ)
πi(θ)dθ
<*: N F < >> A-# !'H P E <*: H (QEABL < @ c f 3>f1| H J EK&# H (A : A H : ! S0]1#?# ADA ! C # > . B D H P C^@DE P>< F < >3:8"gDz B DJZH > L H > #A( > # F A ! A H : ! <*C H P ExEKA ! #1@DA F ADE < EA H : S -'K&# > ##@ < EA U #[ABLE( H > E < : F # H*J EK&#.#K2(# > E ! A ! +P N . #ON?EK > H P . KEK&# F K H A F # ! H*J 5`##A . KE !
$iS HH @ # >K:*8"gDz B D H > 0 P N1# !F P< : N l <*: EAB@D@ < B D < L H : .H EK,# > ! ( > H ( H8! #
! # U # >X< @ F > A\E%# > A < * @BA @ # EK&# ( < ! ER# >> H > >X< E%# * E H )2MEK&# 5`##A . KE ! EK > H P . KrL.##EK H N H @ H/. A$# ! 5qK H+! # L H8! E J < L H P !A ! ( > H C<*C @GF`EK&# T##@L(KA5L'##EK H N B ! #?# A : ! E H : # < : N,- P> H ## JZH >^<> # U A-#65 DS ) H > E P:< E%##@LF#5`# H C E < A :
µ|β ∼ G(
M∑
i=1
$iai,M∑
i=1
$ibi(ai, β)
),
β ∼ Be
(M∑
i=1
$ipi,
M∑
i=1
$iqi
)
5qK&# :πi(β)
A ! < 0`##E < N1# : ! ADE%F Be(pi, qi)N1# : #ON H :(< F#H L`L H : N H L < A :
[βl, βr]S -'K&# FH : ! # : ! P ! NA ! E > A CP
EA H : H :µ< ()((# < > ! < ! EK( H8! E%# > A H > NQA ! E > A CP EA H : F#H L`A : .`J > H L < FH : ! # : ! P !'U A > E P < @ ! < LE(@$#15qK H+! # ! A ?#sA !
EK&# 5`##A . KE%#1N ! P L H J < @D@ ! A 2# ! SI J #32((# > E ! F < :p: H E C # FH : ! AN1# > #1N < ! A : N(# : N1# : E B P ! P < @B@GFE5qK&# :M > 2
D *F#H >> #1@ < EA H : ! L P ! E C # < NN(E H EK&#[L H N1#1@D@BA : . S ) < F #nE H EKA ! A !%! P # * L < . < : B * //D . A U # ! :P L.# > H P !<*> . P L'# : E ! E H N&# : # < F#H : ! # : ! P !HB C F8NQA !F P !%! A H : L'# < : ! D C # JZH > #xEK&#xL H N&##@B@DA : . S'0 P E*5 # H C E < A : EK&# ! < L'#@ A : N H*J ( > A H > L H N1##@B@DA : . S
T 0 − !"# %$ '&()" * )"+ ,-/. 6 2 46s4 6 M2 26
0`# > . # >< : N ] P: B D1( > H"U ABNN ! EN#6( !`H J A CC !'! < LE(@BA : . 5qK&# :π(β)
5 < ! @ H . F#H : F < U #S L P> F K H A F #FH P @BN C # ! ABL`AD@ < > SJC^@DEN# >:< EA U ##@LF *6! A : F # ] P ()(
(π(β))A ! C H P: N1#ON * A\E'A ! # < ! FE H P ! # h $ :<8 f c > 38 h:vgf'& E H
# ! EABL < EN# EK( H8! E%# > A H > L.# < : H J < J P: F EA H : H J A : EN# > # ! Eh(θ)
I =
∫
Θ
h(θ)π(θ|tn) dθ,
=
∫
Θ
h(θ)π(θ|tn)
ρ(θ)ρ(θ) dθ
5qK&# > #ρ(θ)
A ! < : F J P: F EA H : ! P F KnEK < E] P ()((π(θ|tn)) ⊂ ] P ()(
(ρ(θ))B l H C # > E < : N < ! ##@B@ < DS K HH8! A : .
ρ< ! < N1# : ! ADE%F * A\E^A ! # < ! F8E H # ! EABL < EN#
IC F
IM =M∑
i=1
ωih(θi)
5qK&# > # EK&#θi
<*> # ! ADL P @ < EN#ON J > H Lρ(θ)
< : NMEK`##A . KE !ωi = ω′
i/∑M
j=1 ω′j
5qA\EK
ω′i =
π(θi)L(tn; θi)
ρ(θi).
: N&# > L`AD@N FH : NQADEA H : !2* < @BADL`ADE F # : E >< @IEK&# H > ##L # : ! P> # ! EK&# F#H : U # > . # : F # H*JIM
E HI5qK&# :
M → ∞ S-'K&#[L < A : NA\o F P @\E%FA ! EK < E EK&#[E < AD@ !!H*J
πL P ! E C #3K&# < U A$# > EK < : EK&#[E < AB@ !!H*J
π(θ|tn)B < : NVAN1# < @B@GF *
ρ(θ)! K H P @N C # F @ H+! # E Hπ(θ|tn)
DSkI : H P> F < ! # * 5qADEKθ = (µ, β)
* 5 # H C E < A :
µ|β,B, tn ∼ G(a+ r, b(a, β) +
n∑
i=1
tβi
),
π(β|B, tn) ∝ βr
r∏
j=1
xj
β
(β − βl)p−1(βr − β)q−1ba(a, β)
b(a, β) +
n∑
i=1
tβi
−(a+r)
[βl,βr](β).
-'K P !?* F K HRH+! A : .ρ(µ, β) = π(µ|B, β, tn) βl≤β≤βr/(βr − βl)
* 5`# > # ! (# F EqEK&# F#H : NQADEA H : !qH J < ! < EA !+J FRA : .ADLE( H > E <*: F # ! < LE(@BA : . <*: N H C E < A : EK,#nEK&# H > #EA F < @ B P:: H > L < @BA 2#1N9D 5`##A . KE
ω′(β|B, tn) = βr
(r∏
i=1
xi
)β
(β − βl)p−1(βr − β)q−1 ba(a, β)
(b(a, β) +
n∑i=1
tβi
)a+r βl≤β≤βr.
! ((# F A < @B@GF * < : A : N P ! E > A < @ > ##@BA <*C AB@DA ! EqA ! A : E%# > # ! EN#1NMA : EK,# FH LE( P E < EA H : H*J EK,# JZH @D@ H 5qA : ..x$>Xa c :?>I P<*: EA EA$# !
*S EK&# hE>8 fwg _K>3:h >
[T |B, tn] =
∫
IR
Γ(1 + 1/β)
b(a, β) +
n∑
i=1
tβi
1/βΓ(a+ r − 1/β)
Γ(a+ r)π(β|B, tn) dβ;
QS EK&# |V$+=?8"g 8?:C:h >t0
S(t0|B, tn) = P (T > t0|B, tn) ,
=
∫
IR
1 +
tβ0b(a, β) +
n∑i=1
tβi
−(a+r)
π(β|B, tn) dβ;
% T
S EK&# $>"+ a"|98 g5g _K>3:h >E8+_: >"$1:h > t0 B ! #2# JZH > A : ! E < : F # ) A :)@ ##@ ! E%##A : 0K DE l - ) (t0|B, tn) = [T − t0|T > t0, B, tn] ,
=1
S(t0|B, tn)
∫ ∞
t0
S(x|B, tn) dx.
-'K P ! E H : EN# < > @ H # ! EABL < EA H : !]H*J EK&# ! # I P<*: EADEA$# ! : #?#1N ! ABLA(@GF ( H+! EN# > A H > ! < LE(@BA : .wH Jβ*H >[P: A JZH > L
ABLE( H > E < : F # ! < LE(@BA : .3H :(βl, βr)
S
- . 4 # KG# FH : ! AN1# > EK&# > A . KE F # : !H > #ON > # < @@BA J ##EABL'#pN < E <
tnBn = 18
D J > H L - < C @-##0 S SK-'KF FH >> # ! ( H : NVE HJ < AD@ P> #^EABL'# !'H > ! E H ()(A : . EADL'# ! F#H @B@-# F E%#1N H : !H L'# ! ABL`AD@ < > N1# U A F # ! ∑ C ##@ H : . A : . E H EK,# ! # F#H : N < > F5 < E%# >F A > F P ADE H*J :RP F @$# <*> (@ < : E ! S A J #EABL'# ! < > # . A U # : A : L H : EK ! S ) H > (K4F ! A F < @ > # < !H : ! <*: N < F1FH > NQA : . E H < @ < > . #F#H : ! # : ! P !2* EK H+! #!N < E <]<*> # < !N! P L'#1NE H < > A ! # J > H L < G##A CP @B@ NA ! E > A CP EA H : W(η, β)
S -'K&#!L < 2QADL P L @DA @ #1@DA K HRH N # ! EABL < E H > B E D^A ! (ηn, βn) = (140.8, 4.51)
5qADEK # ! EADL < E%#1N ! E < : N < > NN1# U A < EA H : !σn = (7.3, 1.8)
S-'K&#nKA . K U < @ P # H J
βnA ! P: #K2(# F EN#1N <*: N ! P ./. # ! E ! < 0 < F/# ! A < : # ! EABL < EA H : S
A=75 &> 5 : D@+A=7 : 4 70B B(@/O E A < O B(B/O > @/O G C O C < E4O C > G/O > >D< O G B < O < TE <O G
A : 907%;8B 6 A=7 8 : 4 70B >(C O C AFE/O ATEDG/O A =D> O < =D= O GE?A4O BTG C O ETG(B/O <
- < C @$#0 S A J #EABL'# !B L H : EK ! D H J :RP F @$# <*> FH LE( H : # : E !1BJ > H L ! # FH : N < > F 5 < EN# > F A > F P A\E ! DS- 5 H ( > A H > H (A : A H : !9H : EK&#!@DA J #EADL'# < > # < U < AD@ < C @-# H : N1# U A F # ∑ */. A U # :8C FA : N(# : N1# : E #K2(# > E ! E1
< : NE2
S&-'K,#3F < > # ! P LL <*> A ?#ONrA : - < C @$#0 S S2-'K&# E1H (A : A H : A ! L P F KwL H > #sA : JZH > L < EA U #sEK < : E2
<*: N C H EK<*> # > A . KE ! KA J EN#1N 5qADEK > # ! (# F EnE H EK&#pN < E < S E H > # H"U # > EK&##324(# > E ! <*> # : H E < ! @ #1N < EnEK,# ! < L.#`@$# U ##@ H J( > # F A ! A H : S E1
A ! <V:RP F @$# <*> H (# >X< E H >]<*: N ! (# < @ !nJZH >< ( < > EA F P @ <*> FH LE( H : # : E5qKAB@-# E2F < :WC # ! #?# :=< !
< FH LE( H : # : E( > H N P F # > 5qK H+! # H (A : A H : E <=@ # ! A : E H < F#F#H P: E < U < > A-##E%F H J >P:: A : .?FH : NQADEA H : ! S -'K P ! EK&##324(# > E H (A : A H : !qF < :(C # F#H : ! AN1# > #1N < ! A : N1#3(# : N1# : E1S
9%A=7 8;: : : : ; =7%A(5 DB%A EDA"! 4 7 8;: 5?;#(5 D@+77 17%A E1 $ <(C(C B C(C&% < A C7 17%A E2 $ C(C A C(C&% < A C- < C @-#10 S 2((# > E H (A : A H : ! < C H P E EK&#s@BA J ##EABL.#
TS
C . A : . A ! < !%! P L'#1N H5 # F K HRH8! #βl = 1
S ) H > EN# F K : A F < @ > # < !H : ! 5 # F K HRH8! #βr = 5
SE ! A : . B 0xS D * EK&#P: N1# > @GFRA : . ( > A H > # ! EABL < EN# !!H*J
β( > H ( H8! #ON C FEK&#.#K2(# > E ! E < @ # U < @ P # ! A : 8.02, 10.0, 11.7 B #324(# > E E1
D<*: N 2.11, 2.53, 2.84 B #K2((# > E E2
DS -'K P !2* EK&# > ! E #K2(# > E H (A : A H : ! #2##L ! N PC A H P ! ! A : F #8A\E3A : N P F # ! < :P:> # < !H :<*C @-# < . A : . S [ H E%#[EK < EH5qK&# : 5 # > #6(@ < F # H > N&# > !
(5%, 95%)C F
(33%, 66%)BJZH P: N P ! A : . C!@ .H
> A\EKL DH5 # H C E < A : ( > A H > U < @ P # !nH Jβ
: # < > E H2.5
S ) H > EKA ! > # < !H : * 5 #E( > # J # > EK&# ! # F#H >> # F EN#ON H > N1# > ! S-'K&# : *RJZH > # < F K ( > A H > N1# : ! ADE%F
π(β|(p, q)Ei )Bi = 1, 2
D * 5 # ! ((# F A J F JZH > EK&# h a/> EK,# ! < L.# U < @ P # β∗ = 2.5S
I : N&#?#1N * 5`# : # F # !N! < > AB@GFVK < U #(p, q)Ei > 1
5qKA F K < @D@ H 5 ! : H EsE HVH C E < A :(' < E( > A H > ! S -'KA ! A ! EK&# ! E < > EA : .( H A : E H*J EK&#H##@BA F ADE < EA H : L'##EK H N(A : 0 < F K < >3:*8 gDz B DS
T 0 − !"# %$ '&()" * )"+ ,[ H 5 @$#E
(a1, a2)C #3EK&# U A > E P< @ ! A ?# JZH > # < F K #K2(# > E1S3C N1# J < P @DE1( > A H > JZH > EK&# F#H L C A : < EA H : H J C H EK
#K2(# > E H (A : A H : ! A !
µ|β ∼ G(a, b(β)),
β ∼ Be(p, q)
5qA\EK
a = $1aE1 + (1 −$1)aE2 ,b(β) = $1bE1(β) + (1 −$1)bE2(β),p = $1pE1 + (1 −$1)pE2 ,q = $1qE1 + (1 −$1)qE2 ,
aE1 = 3(γ1a1 + γ2a2 + γ3a3),aE2 = 3(γ′1a
′1 + γ′2a
′2 + γ′3a
′3),
qEi = 1 + (δ − 1) (pEi − 1)JZH >
i = 1, 2,δ = (βr − βl)/(β
∗ − βl),
< : N
bE1(β) =
(1
0.661/3γ1a1− 1
)−1
200β +
(1
0.51/3γ2a2− 1
)−1
250β +
(1
0.331/3γ3a3− 1
)−1
300β,
bE2(β) =
(1
0.951/3γ′1a′
1
− 1
)−1
100β +
(1
0.51/3γ′2a′
2
− 1
)−1
250β +
(1
0.051/3γ′3a′
3
− 1
)−1
500β.
G#MK < U # : H ( > # F A ! A H :Y<*C H P E]EK,# > ##@ < EA U #8@$# . A\EADL < F F H*J EK&# ((# > F # : EAB@-# ! S -'K P ! 5 # 92γi = γ′i = 1/3
S]RABLAB@ < > @GF * 5 #nK < U # : H8H C% # F EA U # F > ADEN# > A H :r< @D@ H 5qA : . P ! E HJ < U*H >q< : F#K2(# > E1S-'K P ! 5`# )2
$1 = 1/2S
-'K&#pL`A : ADL < @ :RP L C # > H*JqU A > E P< @tN < E < E HVH C E < A : EK&##(# > F # : EAD@$# !(25%, 50%, 75%)
A ! S - H ! ((# F A J F((# > F # : EAB@$# !
(25%, 75%)* 5 # : #2#1N
a1 = a3 = 2a2S 0CF : H > L < @BA < EA H : * 5`# H C E < A : EK&#?N1# J < P @DE U < @ P # !
a1 = a3 = 1.6< : N
a2 = 0.8S ) H > EK&# ! # F#H : N#32((# > E H (A : A H : * C F EK&# ! < L'#wL'#EK H N * 5 # H C E < A :
a′1 = a′3 = 10a′2<*: N
a′1 + a′2 + a′3 = 20S-'K&# :
a′2 = 20/21 ' 0.95< : N
a′1 = a′2 ' 9.5S [ H EN#]EK < ExEK&#
F # : !H > #ONMN < E < ! ##Etn
FRA$##@N ! < (9( > H 2QABL < EA U #1@GF < ! L P F K(A : JZH > L < EA H :< !n ' 11
z za*z N < E < B ! #?# § 0 S S SDS-'K P !?* EK&# ! # F#H : N ( H8! E%# > A H > NQA ! E > A CP EA H : 5qAB@D@ C #`L H > #N1#3(# : N1# : E J > H L/EK&#A( > A H > EK < : EK&#`N < E < B A :W< :< ()( > H 2QABL < EN# >< EA H3H*J KKAD A J
a2A ! : H E^L H NQA(#1N BJZH > A : ! E <*: F # P ! A : . - <*C @-#0 S DS
-'K&# : *6! A : F #a1
<*: Na2
< > #)21#1N * EK,# ( > A H >9P: F # > E < A : E%F H :βA ! ##@BA F ADEN#1N < ! #324(@ < A : #ON`A : § 0 S SS SFG#
P ! #(ηe, βe) = (290, 2.5)
E HY! < LE(@$# U A > E P< @9N < E < * ! A : F #EK,#L'#ONQA <*: H J W(ηe, βe)A ! S4G# H C E < A : #1N
( > A H > U < > A <*: F # !σ2
1 = 1.07< : N
σ22 = 0.67
* > # ! ((# F EA U #1@GFS G# H C E < A : #1N : < @B@GF(p, q)E1 = (1.19, 1.31)
JZH >EK&# > ! EC#K2((# > E * < : N
(p, q)E2 = (3.13, 4.56)JZH > EK,# ! # FH : NL#324(# > E1S
) A :< @D@LF * EK&# FH LE(@-##EN# ( > A H > F#H >> # ! ( H : N ! E H < U A > E P< @ ! < LE(@$# H J:! A ?#a = 12
* 5qKA F K ABLA(@DA$# ! EK < EEK&# ( H+! EN# > A H > NQA ! E > A CP EA H : A ! < (9( > H 2QABL < EA U #1@GF < ! L P F KN1#3(# : N&# : E J > H L ! PC% # F EA U # @R: H 5q@-#ON . #xEK < : J > # I P # : EA ! E @: H 5q@-#ON . #S ]6#6( < >X< E%#H( > A H > N1# : ! A\EA-# !^H : ( <*>< L'#EN# > !
(η, β)<*: NrEK&#[L <*> . A : < @TN1# : ! ADE%F
m(t)<*> #
NQA ! (@ < F/#ON H : ) A . P> # ! 0xS * 0xS < : N 0xS KS4E < > . A :< @%N1# : ! A\EA-# !m(t)
. #E .*HRH N FH LE( > H L`A ! # ! C ##E%5 #2# : EK&#! (# F A F < EA H : ! S ) H > EK,#.#32((# > E E1
* ##LE(A > A F < @'((# > F # : EAB@-# ! H J'H > N1# >(33%, 50%, 66%)
< > #(197, 250, 298)
S) H > EK&##K2(# > E E2
* ##LE(A > A F < @ ((# > F # : EAB@$# !nH J H > N1# >(5%, 50%, 95%)
< > #(98, 255, 494)
B EK&# ! # > # ! P @\E ! <*> #FH LA( P EN#ON H :
105 ! < LE(@$#1N ( < > EA F @$# ! DS IH+! EN# > A H > ! P> U A U < @ J P: F EA H : ! S(t)< > #sNQA ! (@ < F/#ONpA : ) A . P> #10 SQS
% T
eta
densi
ty
100 200 300 400 500 600
expert E1expert E2consensus
) A . P> # 0 S \# J < P @DE ( > A H > N1# : ! A\EA-# !π(η)
JZH > ! #3( <*>< EN# < : N FH : ! # : ! P ! #K2(# > E H (A : A H : ! S
betadensi
ty
1 2 3 4 5
expert E1expert E2consensus
) A . P> # 0 S \# J < P @DE ( > A H > N1# : ! ADEA$# !π(β)
JZH >! #6( < >X< E%# <*: N F#H : ! # : ! P ! #32((# > E H (A : A H : ! S
t
densi
ty
0 200 400 600 800
expert E1expert E2consensus
) A . P> # 0 S K \# J < P @DEVL <*> . A : < @H( > A H > N1# : ! ADEA$# !m(t)
JZH > ! #6( < >X< E%# < : N FH : ! # : ! P ! #K2((# > E H (A : A H : ! S
t
S(t
)
0 200 400 600 800 1000
0.0
0.2
0.4
0.6
0.8
1.0
post. (expert E1)post. (expert E2)post. (consensus)
) A . P> # 0 S IH+! EN# > A H > ! P> U A U < @ J P: F EA H : !S(t)JZH > ! #3( <*>< EN# < : N FH : ! # : ! P ! #K2(# > E H (A : A H : ! S
T − !"# %$ '&()" * )"+ ,-/. 6 6 I : EKA ! <*> EA F @-# * 5`#8K < U # ( > H"U AN1#1N < ( > A H > L H N1##@B@BA : .wH J EKG#1A CP @D@C( < >X< L'##EN# > ! 5qKA F KYA ! ( >X< F EA F < @IE HP ! #*S ] H L'# ! E >X< EN# . A$# !8H*JnF < @BA C>< EA H : K < U # C #?# : ( > H ( H8! #ON 5qK&# : EK&# ( > A H >pP: F # > E < A : E%F=A ! NQADo F P @DEpE H< !%! # !N! S H : ! ABN1# > A : . EK&# ! A ?# H J < U A > E P< @ ! < LE(@$# * FRA-#1@BNA : . < : A : JZH > L < EA H : 5qKA F K=A ! < ()( > H 2QADL < EA U ##@LFEK&# ! < L'#pEK < : EK&# #K2(# > EA : JZH > L < EA H : * A ! ( >X< F EA F < @ < : N ! PC% # F EA U # < : N H C% # F EA U #(N < E < A : JZH > L < EA H : !F <*:wC # # < ! AD@LF FH LA( < > #1N S.C : A : NQA F < E H >
nA ! N&# : #ON?E H > #3(@ < F #3EK&# ! A ?#
n5qK&# : N < E <
tn< > # F # : !H > #1N *
A :3< L H > # . # : # >< @ ! #EEA : . EK < : EK&#G##A CP @B@ < :< @GF ! A ! S ) A :< @B@GF * EK&#C( H+! EN# > A H > F#H LE( P E < EA H :]> #1L < A : ! ! ADLE(@$#*S
.#0 (7 (4I686 6 (4I61G6 X26 0
\# : H E%#FW (t)
EK&#HG##A CP @B@5NQA ! E > A CP EA H : J P: F EA H : S G#sK < U #
P (T ≤ te|B1) =
∫
IR
∫
IR
FW (te|µ, β)π(µ|B1, β)π(β) dµdβ,
= 1 −∫
IR
1
1 + xβe
b1(β)
a
π(β) dβ,
= 1 −∫
IR
(1 − α)π(β) dβ = α.
-'KA ! > # ! P @DE ! K H @N ! # U # : A J ] P (9((π(β))
A ! : H E FH LE( < F EOSJC !N! P L`A : .a > β−1
l
*
E[T |B2] =
∫
IR
∫
IR
µ−1/βΓ(1 + 1/β)π(µ|B2, β)π(β) dµdβ,
=
∫
IR
Γ(1 + 1/β)b1/β2 (β)
Γ(a − 1/β)
Γ(a)π(β) dβ = te.
-'KA ! > # ! P @DE K H @BN ! A J ] P ()((π(β)) ⊂ [βl,∞[
JZH > <*: Fβl > 0
S ) A : < @B@GF * < L H N1# H*J NQA ! E > A CP EA H : M A ! @ HQF < EN#ONA :te 6= 0
! P F KMEK < EqEK&#]N1# > A U < EA U #m′
W (t)C #7?# > H SPG#sK < U #
mW (t) =
∫
IR
aba
(b+ tβ)a+1 βt
β−1π(β) dβ.
GiA\EK*] P ()((π(β))
FH LE( < F E * EK,#]N1# > A U < EA U #nA ! N&# : #ON < : N
m′W (t) = −
∫
IR
abatβ−2
(b+ tβ)a+2
[tβ(aβ + 1) − b(β − 1)
]π(β) dβ.
B D
GiK&# :a > β−1
l
*F K HH8! A : .b = b3
U A ! A C @LF < @B@ H 5 ! E HH C E < A :m′(te) = 0
S'0 # ! AN1# !?* ∀x > 0* 5`#nK < U #
m′W (t|B3) =
∫
IR
aba3tβ−2
(b3 + tβ)a+2 (aβ + 1)π(β)
tβe − tβ
dβ
5qK H+! # ! A . : A ! EK&# ! < L'# EK <*:te − t
S -'K&# : EK&# P: A F ADE%F H J EK,#sL H N1#nA ! # : ! P> #1NTS
Annexe C K 4
/.10 r4 7 "! 7 6 G6 32 4 $# 4 X2 7&% 7 (' 6r7 4 *)
N,+ J < AD@B@ < : F # ! > +?##@B@$# ! S * S * S * S * S * S * S * S * QS * *S
F # : ! P> # !.- N > H A\E%# S * S * S * =KRS * K0KS * QS * S * S
/;10 2 xS43 P> +2# ! N&# U A$#X
B L H A ! D N1# F#H LE( H8! <*: E ! N P F A > F P ADE ! # FH : N < A > #S
I : E%# > U < @D@$#sN1# FH : <*: F # B * AD Y < @$# P> L5+ONQA < : # 2(# > E E1
B * /D 2(# > E E2
B * /D /10 2 xS 63 L (A : A H : ! N #K2(# > E ! P> @ < N P> +?#sN1# U A$#
XS
Annexe D K
.10 6 4 7 2 4 T2 2 T2
P LN ? D' LN$DO%P%P UN N L N M# : ! #1L C @$#N&# ! FH : NQA\EA H : ! I P A ( > + ! AN1# : E - @ < ( > H N P F EA H : N P: + F K < : EAB@D@ H :N1# N H :: +2# !5u ( < > #K21##LE(@$# RE < AB@D@$# * E%F4(# N1#sN H :: +2# !2*U < @BANQA\E + H P F # : ! P> #*S YH A > ( < > #K21##LE(@$# < NQA$# P>K:78"gDz B D ( H P> (@ P ! N1#1( > + F A ! A H : ! S
Z$(L$(O ) $(O J ? U ? !H A\E(x1, . . . , xn)
P: + F K < : EAB@B@ H : N1# @ H A I P #1@ F#H : I P #S < ! E < EA ! EA I P #nN H > N > #iN1#
@ + F K <*: EAB@D@ H : * : H E +2#x∗i* # ! Eq@ <
i U < @$# P> N1#s@ + F K <*: EAD@B@ H : E > A +1( <*> H > N > # F > H A !%! <*: EqN1# U < @$# P> x∗1 ≤ x∗2 ≤ . . . ≤ x∗n.
YH A > ( <*> #K21##LE(@$# <*P N H A : B /DC( H P>^P: # > # U P #nE > !!F @ < A > #]N1# ! ( > A : F AL( < @$# !!F < >X< F E + > A ! EA I P # ! N1#@ < NQA ! E > A CP EA H : N1# !C! E < EA ! EA I P # ! N H > N > #S
> UVN N N + ) J ) ! P ()( H+!H : !1I P # Xn!H A\E P: # ! P ADEN#pN&# U < > A <*C @-# ! < @ + < E H A > # ! SL : NQADE
I P # XnF#H : U # > . # U # > !
X $ >+ - |9>HV$>#hE>#f9:I P<*: N
n→ ∞ * #E H : + F > A\E Xn p.s.−−→ X! A
P (ω : Xn(ω) → X(ω)) = 1.
I;@ ! < . ADE N1#x@ <[: H EA H : @ < (@ P ! :< E P> #1@D@$#xN&# FH : U # > . # : F #^N P: # ! P ADEN#xN&# U < > A < C @$# < @ + < E H A > # !2* I P A ((# P E! # >X< (9( > HQF K&# > N1#!@ <s: H EA H : N1# FH : U # > . # : F # ! ABLE(@$#xN1# !]! P A\E%# ! N < ()(@BA F < EA H : !B < U # F P: ( H A : E'N1# U P #( > H C< C AD@BA ! EN#DS
%? N$(O%S L @ O P%O$ P: L H N @-# ( <*>< L +E > A I P # M(θ)H
θ ∈ Θ# ! E]ABN1# : EA < C @$# ! A
θ1 6= θ2 ⇔ M(θ1) 6=M(θ2)
u < P E > #1L.# : EqNA\E * # P 2(N1#1( < >X< L E > # ! NQA # + > # : E ! N H :: # : E!N1# ! L H N @-# ! NQA # + > # : E ! S N N @L0B ) O N N %N,+#L <*> F K&#M@ H/. A I P # ((# > L'#EE <*: E`N1# F < @ F P @$# > H PY> + U A ! # > @ < ( > H C< C AB@BA\E +MN P: #
K F4( H EK ! #*S ##EE%#^N,+#L < > F K&#`# ! E > + . A$#*( < > @ P EAB@DA ! < EA H : N1# > . @$# !! E > A F EN# ! N1# FH L C A :< A !H : N1# ! ( > H C< C AD@BADE + !2* N&# ! I P ##@B@-# ! N,+ > A U #s@$#sEK + H > L'#]N1#0 < F+# ! S <*: ! @ < ((# > ! (# F EA U # C< F + ! A$# :: # * P: #( > H C<*C AB@DADE +: # ! E1( < ! A : EN# > ( > +E +?# FH LL'#3@-#E( < !N! < . # - @ < @DABL`A\E%#N P: # J > + I P # : F # * L < A ! (@ P E*E F#H L`L'#3@ < E >X< N P F EA H :V:RP L + > A I P #[N P: +#E < ExN1# F#H ::< A !%! < : F # * ( <*> #K21##LE(@$#]@$#[N1# . > +]N1# FH : <*: F # < F#FH > N + - P: #K F4( H EK ! #*S
N$ LP%P ? RUNMS LN !H ADE(X1, . . . , Xn)
P: + F K <*: EAB@D@ H : N1#3@ < @ H A%N1#( > H C < C AB@DADE +Pθ
SxL :?< ()((#1@D@$#A : EN# > U < @D@$# N&# F#H : <*: F #'N1# : A U # <*P B H P ! # P AD@;D
α! P>IP: # JZH : F EA H :
h(θ)P: A : E%# > U < @D@$# < @ + < E H A > #
[T1, T2]*
HT1 ≤ T2
!H : E^N1# P 2 ! E < EA ! EA I P # !?*RJZH : F EA H : N1#n@ + F K <*: EAB@D@ H : * E%##@B@-# !I P #Pθ(h(θ) ∈ [T1, T2]) = 1 − α.
T T5 − ,1m;OD]RA H :> + < @BA ! #
M! ABL P @ < EA H : ! A : N,+3(# : N < : EN# ! N1#q@ < @ H A
Pθ* @$# !U <*> A < C @$# ! < @ + < E H A > # !
(T1, T2)( > # : N > H : E
N1# ! U < @-# P> ! ( <*> EA F P @BA 1> # !(t1, t2)
B NQADEN# !<$8 g38?: f DS # : F < N > ##L'# : E ! # ><3< @ H > !9U >< A H P J < P 2vS IH P>α = 0.05
*(! A H :?> +3( EN# JZH A ! @ < ! + > A$#3N&#M#324( + > A$# : F # ! ( H P> H C E%# : A > 8A : EN# > U < @B@$# !2* H : ((# P E
! < EE%# : N > # -8F # I P #`N # : E > #J# P 2 : # FH : EA-# :: # : E7( < ! @ < U >X< A$# U < @$# P> N1#h(θ)
S UVN9$(O UN + O U$DLP P: # JZH : F EA H : (A U*H E < @-#( H P> @-#A( < >X< L E > #
θ# ! E P: # U <*> A < C @$# < @ + < E H A > # JZH : F EA H :
N1# !]H C ! # > U < EA H : !(X1, . . . , Xn)
#E[N1#θN H : E3@ < @ H A : #pN,+3(# : N ( < ! N1#
θS A : E + > E]L < # P> N P: #
JZH : F EA H : (A UH E < @-#:# ! E!N1#1(# > L'#EE > #sN #K2( > ABL.# >qP: A : E%# > U < @D@$#sN1# FH : < : F # ! P>θS
L?L1 $ !? N JMO ) LN P: ( <*>< L E > #^N1# :P A ! < : F #θ2
< ()( < >X< E N <*: ! P: L H N @$# ! E < EA ! EA I P # M(θ)*
H θ = (θ1, θ2)
F#H L`L'# P: ( < >X< L E > #?N1# U < @$# P> A : F#H ::RP # *`! < : ! > *@$# ( H P> @ # ! EABL < EA H : N P: #JZH : F EA H : N A : E + > ExN,+6((# : N < : E%# P: A I P ##L'# : EnN1#
θ1B NQADE ( <*>< L E > #]N A : E + > EDS <*> #K21#1LA(@-# *! A5@ H :
! A : E + > # !N! # P: A I P ##L'# : E - @ # ! ( + >X< : F #9N P: #@ H A : H > L < @$# * P: + F < > E E%F4((#IA : FH ::RP # ! E P: ( < >X< L E > #IN&#:RP A ! <*: F #S? : ( > H C @ L'#'L < # P> N1#'@ < ! E < EA ! EA I P #`# ! E9N H : F N1# > +1N P A > # UH A > # N1# ! P ()( > ADL'# > @ A : 'P # : F #N1# F # ( < >X< L E > # ! P> @ A : J + > # : F #*S YH A > ( < > #K21#1LA(@-#10 < ! P B =D!( H P> (@ P ! N( > + F A ! A H : ! S
?O@ J P: #!E > A CP BZH Pσ− < @ . 1C> # D A N P: # ! ( < F # JZH : N < L'# : E < @
Ω# ! E P: # : ! #1L C @$# : H : U ABN&# N1# ( < > EA$# !
N1#Ω! A
*S ∀A ∈ A * A ∈ A uQS IH P> E H P E%# J < L`AD@B@$#nN + : H L C>X< C @$# Ai; i ≥ 1 N +#@ +#L'# : E ! N1# A *
⋃
i≥1
Ai ∈ A.
U P%O N @ < E > A CPwC H > +#@BA$# :: #3N1#χ* : H E + B(χ)
* # ! E!@ < E > A CP # : . # : N > +2#( <*> @ < J < L`AB@D@$# F N1# !!H P U # > E !N1#χ*QF # ! E - NQA > #q@ < (@ P ! (#EADEN#!E > A CP
σ(F)F#H : EN# :< : E F S H > ! I P #
χ ∈ IR∗+ B ( < > #K21##LE(@$#χ+E <*: E
P: # : ! #1L C @$#sN1#nEN#1LE( ! N1#]N,+ J < AB@D@ < : F # D *QH :<
B(χ) = σ (]x1, x2]; 0 < x1 < x2) .
. 6 4 7 2 4 7&% 6 4 # . @ H+!%! < A > # > # . > H P (#8@$# ! E%# > L'# ! N A : . + : A$# > A-#p@$# ! (@ P !sJ > + I P #1LL'# : E ##LE(@ H F + ! N < : !3F #MN HRF P L'# : E1S
IH P>qP: #s@DA ! E%# (@ P !^FH LA(@ EN# *QUH A > ( < > #326#1LE(@-#0 < F K < >3:78"gDz B * ( < . # ! DS
# JHLV@O%P O$ < (QEADE P N&#MN P: ## : EA\E + - < F1FH LA(@DA >sP: # JZH : F EA H :W> # I P A ! # * N < : ! N&# !]F#H : NQADEA H : ! N H :: +?# !N P EAD@BA ! < EA H : #ErN1#VL < A : EN# :< : F # * +P !I P -YF # I P P: +E < Ep@BADL`ADEN# !H ADE < EE%##A : E B : H > L'# [ K *DS
O%L @ O P%O$ < (QEA\E P N1# N P: # # : EA\E + - < F#F#H LE(@BA >pP: # JZH : F EA H :j> # I P A ! # * N < : ! N1# !(FH : NQADEA H : ! N H :: +?# !N P EAD@BA ! < EA H : * (# : N < : E P: EN#1LA( ! N H :: + B : H > L'# ! [ ) K _#E [ ) /DS
. LO N$ N LV@ O P%O%$ < (QEA\E P N1#N P: #Y# : EADE + - E > #(L < A : E%# :RP # H P > +E < C @BA$# *! P>P: A : E%# > U < @B@-#N1#rEN##LE( !N H :: + * N <*: ! N1# !5FH : NQADEA H : ! N P EAB@DA ! < EA H : ##E N < F#F#H LE(@BA !%! ##L'# : E5N1#9@ < L < A : EN# :< : F #9N H :: +?# ! B : H > L'# 
? 1 H T
QS + N$(O < F <*: E ( H P> H C% # F EA J N1# > +ON P A > #8@ < ( > H C<*C AB@DADE +8N1#MN + J < AB@B@ <*: F # H P N&#pN,+ . >< N < EA H : N P: # # : EADE + H P N P: ! # > U A F # u ##@B@-# # ! EN + F @$# : F K +?# ! #1@ H :=P: + F K + <*: F A$# > +E < C @BA - ( < > EA >N P:V: H L C> # ( > +1N,+#EN# > L`A : +]N P: A\E + ! N P ! < . # B L < A : EN# :< : F # 8: #hE8=: - |9> D`##E H P N1# !!F > A\E #> # !( > +1N +EN# > L`A : + !2*&! A . : A F < EA J ! N&#s@ +E < E!N1#]N,+ . >X< N < EA H : N1#s@ # : EA\E + H P N P ! # > U A F # B L < A : E%# :<*: F #c fva;: f f >#ggG> DS
. RLN O ) 1 .?4? N HL ? LT$DO%UVN ( > HQF # !N! P ! (K4F ! A I P # *"F KABLA I P # H P3<*P E > # < F < : E# : E >X<: + P: #'N + J < AB@B@ <*: F #B : H > L'# [ ) K /DS
. U ? 3? ,? LO%P P%LN <# #M#E ( < > @-# I P ##@ P: # N,+ J < AD@B@ <*: F #]# ! E H C ! # > U +?# B : H > L.# [ ) K /DS 21##LE(@$# ! !N! P> # * >P (QE P> #S $ y? UVN ;$(O%UVN N 1 N$ 1# : ! ##L C @-#sN1# ! ( > H ( > A +E + !*I P ATN,+ F > A U # : E @ < NQA ! ( H : A C AB@DADE +sN P: #H# : EA\E +J#E
@-# !^J < F EN# P> ! I P A%@ < F#H : NA\EA H :: # : E <*C AD@BADE + * L < A : EN# : < C AB@DADE + #E @ H/. A ! EA I P #[N&#[L < A : EN# : < : F # B : H > L'#I]1L /DS : ) ><*: F # *F #EC# : ! #1L C @$#sA : F @ P E < P !N! A @ < ! + F P> ADE +S
LJ t? ? LO P%P%LN N + ! A . : #s@ < ( > H ( H > EA H : N # : EA\E + ! I P A H : E ! P> U + F P - P: A : ! E < : E <*>C ADE >X< ADE > #t#E
: # !H : E (@ P ! # : U A$# - @ A : ! E < : Et+1* ( > H ( H > EA H :[>X< L'# : +2# - @ P: ADE +^N1#'E%##LE( !7B : H > L.# [ ) 0K DS
1 +)!? UN9$DO%UVNMN 1 N$ @ < N P> +2#8N1# JZH : F EA H : #$## F EA U #`N P: #.# : EADE + *! F : H : FRL'#3N&# 3 EN#1LE( ! N1#! # > U A F # *(! #K2( > ADL'#[@$#(@ P !:!H P U # : ET# : EN#1LA( ! F P L P @ +N K,# P> # ! N1# JZH : F EA H :: #1L.# : E H P # :?: H L C> #F P L P @ +]N&# !H @B@BA F A\E < EA H : ! S
Annexe E
0 2 2 ' 2 2 B 4DS9- H ! P> U A U # H J E HnJ < AB@ "EK < E A ! EK&# I P # ! EA H : *'xh`z :<8=: :;z$* * ()(TS S
0xF 2 T 2 2 2 B DS 3 ! A B xA . < L`L < D ) P: F EA H : S! * !K0`# >: H P @B@DA < : N P @$# > IH @LF : H L`A < @ ! < : NrEK&# P @$# > E < F @ <*P> A : ) H > L P @ < !"SI : 28"fva*b 1d _ 8=: "9>h 8?: c 8"g9 |Qf c : f1( ;: "!9 $h3|QgG8 /" $8 "' / 8"fva 8?: ")>#h 8?: c 8 gw 8*b#gG>"P* EK ( > A : EA : . S [ #35$# H > @ H"U # > S
&% ' 2 2() *5 + 2 2 B K/DS " = a"f >+ _ 9;;:3w9>-, "Rf - |9>"/. E < > F10 @ 0 @4@ 0 > [ 0 5# H > @ S
2 2 2J3 54687& 2 B K0K/DSC .90 : 0 >X< @ F @ <9:;: H*J FH<0 o F A 0 : E : H*J NQA U90 > .=0 : F>0]H JtH : 0 NA : E > A CP EA H :J > H L < : H EK 0 > . Qz@? *z : 8?:;z ,OzA. . ()(5S S
x 2 6 3 CBED 6 3 6 D12 2F 54 2 B DS] 0 <*> F KA : .wJZH > FH :: 0 ::P: A :C > H P ( N 01F A : A H : L <=@ A : . SxC 0 < F 0 : A <*: ( 0 >;: ( 0OF EA U=09.C? 8 $ :xa > $>-, "9>"$E, ")>>. : A U=0 >-: ADE +N 0 ] < >< .*H :;: 0 S
3 x 6 D 2 2 2GH %I 3 2 B 0K/DS 0 < F 0 : A < :(> H CP: E : 0 :;: L H N 0 @BA : . P: A : . > 06. P @ < > @LF U < > FA : .NQA : E > A CP EA H :: . 8 >"+ 8"fDxf 8 g 8+-.J. (9(TS =K S
3 6 GK 2G ' 0 6 GK 2 2 B DS H : E > H @B@ 0 NTE E JZH > H (QEADL < @ :< LE(@BA : ..8? 80 $ :!: >-, "Qf1 - |9>LNM9M=O L # / u 8" >%$3a"| u (? P. : A U90 >;: ADE + < > A : < P (KA : 0 S
3 6 K 2 Q6 2 8R 6 x S<T 2UD123 V 6 D12 2 B DS60 < F 0 : A <*: H LE( P E < EA H : < @C()( > H < F K 0 : E H E H N 0 @ ] 0 @ 01F EA H : .+W f gf >8$ 8"f aXW f " 8"|+ 8"f z&*f 8 g+Y $ ,>++f'&9. ]QLADEK . l S S .# H P: .. S xS < : N!G < @ @ N 0 : S1C[S B N : D . < L C> AN .=0 : A U=0 >-: ADE%F > 0 :;: S
3 6 D 2 R 2 B DS > 0 N 0 : F10 < : N l H CP: E : 0 :E: 0 0 K < U A H > . >K:$ f.V. ()(5S S 3 2 B K DS w "9> " 8 h3h 8!_X|QfZ,K: f. H @\E l A : 0 K < > EGiA :: E H : . [ 0 5$# H > @ S S[K 2U\ 2 * *S^] 2 2_a`, -76 3 T 2UD12 B DSC : < @LF : A : H*J ( > 0 :E:P> A 0 N 5 < E 0 >r> 0 < F E H >
( > ABL < > F FHRH @ < : Eq@ 0 <=@ 0#U90 : E : F <*P: 0 C FpEK 0 > L < @ J < EA . P 0=. <? Y $ ,>>af'& Sb< Q* G 2 B K DSdc I : J + > 0 : F>0 : E < EA : EA I P 0 ( H P> N 0 : L H N @ 0 : N 0 N P> + 0 : N 0sU A 0^0 E < ()(@BA F < EA H :: .;w " e"3>
a >a ,K: $8=:f. : A U90 >;: ADE +sN 0 l H P 0 : Sb< Q* G 2+ Kg6 N6 ChS ` 2 B K DS0 l E I] P:V< @ .*H > ADEKL 0 N 0 : EADL < EA H :C< F + : A 0 :: 0 ( H P> L H N @ 0 :- N H :: + 0 : A : FH LE(@ E 0 : .Y $ ,>>Xaf&[a >"iikj +l |V$f >+[a >. :<8=: : - |9>>.m P + C 0OF Sb< Q* ( 2 &Kg6 N6 ChS ` 2 ( V 6 2 \k33 7& 2( ] 6 2 B DS :h 8?: f a >h a9e#gG>+
a >a"|V$>"]a/>. > _ $ : >h >#f9:n,>#f |V$>"-. F > H @B@ 0 : S
33 68h6 − b< 3 \ 2P 3 6 N* P 2 B DS :<8=: : ,8"gPxf 8"g 8+ __? >gz 8*b%g;: 8"fva _K> w >+ :f'& a/>g /.E <*> F>0 @ 0 @4@ 0 > Sb< 2 B =DS1L : EK 00 @BADL`A :< EA H : H*J :RP A :<*: F10 ( <*>< L 0 E 0 >;: . V. . ()(TS K0KSb< 2 6 3 2 2b<3 6 6 6 B DS 0 < F 0 : A < :Y< : < @LF : A : H J!F#H LE( 0 EA : . > A :@8: 5qADEK ( < > EA < @B@GF
L <9:@ 0 N F < P: 0nH J J < AB@ P> 09. 0vg >Xa :<8=: : ,+/. C. (9(TS Sb< 7 2^ 2D 2 3 b 6 6 D 2 H2 B /DS U < @ P 0 : JZH > F#H LE( H : ADE 0 :RP @B@!L H N 0 @ : . ^. . ()(TS
QSb 6 6 D12H2 B D :<8=: : , 8 g A>-,"+ f w "9> $ 8"f a 8 >"+ 8"fpxf 8"g 8+ B : N 0 NQADEA H : D . ]4( > A : .=0 >
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