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Isabel Silva Magalhes Integer-valued time series
Integer-valued time series
Isabel Silva Magalhes
Departamento de Engenharia Civil, Faculdade de Engenharia da Universidade do Porto
Unidade de Investigao Matemtica e Aplicaes (UIMA), Universidade de Aveiro
UTAD - February 2010
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Isabel Silva Magalhes Integer-valued time series
Motivation
Discrete time non-negative integer-valued time series counting series
Motivation UTAD - February 2010 3 / 31
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Isabel Silva Magalhes Integer-valued time series
Motivation
Discrete time non-negative integer-valued time series counting series
Examples:
the daily number of seizures of epileptic patients
the yearly number of plants in a region
the daily number of guest nights in a hotel
the monthly incidence of a disease.
Motivation UTAD - February 2010 3 / 31
I b l Sil M lh I l d i i
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Isabel Silva Magalhes Integer-valued time series
Motivation
Discrete time non-negative integer-valued time series counting series
Examples:
the daily number of seizures of epileptic patients
the yearly number of plants in a region
the daily number of guest nights in a hotel
the monthly incidence of a disease.
Usual linear time series models (ARMA processes) are not suitable.
Motivation UTAD - February 2010 3 / 31
Isabel Sil a Magalhes Integer al ed time series
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Isabel Silva Magalhes Integer-valued time series
Motivation
Discrete time non-negative integer-valued time series counting series
Examples:
the daily number of seizures of epileptic patients
the yearly number of plants in a region
the daily number of guest nights in a hotel
the monthly incidence of a disease.
Usual linear time series models (ARMA processes) are not suitable.
Thinning operation
Multiplication counterpart on the integer-valued context.
Motivation UTAD - February 2010 3 / 31
Isabel Silva Magalhes Integer valued time series
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Isabel Silva Magalhes Integer-valued time series
Thinning operation
Binomial thinning operation [Steutel and Van Harn (1979)]
X : non-negative integer-valued random variable (r.v.), 0
X=X
j=1
Yj
{Yj} N0 (counting series): sequence of independent and identically distributed(i.i.d.) r.v., independent ofX, P(Yj = 1) = 1P(Yj = 0) =
X is the number of successes, with probability , in X trials
Thinning operation UTAD - February 2010 4 / 31
Isabel Silva Magalhes Integer-valued time series
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Isabel Silva Magalhes Integer valued time series
Thinning operation
Binomial thinning operation [Steutel and Van Harn (1979)]
X : non-negative integer-valued random variable (r.v.), 0
X=X
j=1
Yj
{Yj} N0 (counting series): sequence of independent and identically distributed(i.i.d.) r.v., independent ofX, P(Yj = 1) = 1P(Yj = 0) =
X is the number of successes, with probability , in X trials
Generalized thinning operation [Gauthier and Latour (1994); Silva and Oliveira (2004, 2005)]
X=X
j=1
Yj
{Yj}: sequence of i.i.d.r.v., independent ofX, with some discrete distribution such thatE[Yj] = , Var[Yj] =
2, E[Yj3] = , E[Yj
4] =
Thinning operation UTAD - February 2010 4 / 31
Isabel Silva Magalhes Integer-valued time series
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Isabel Silva Magalhes Integer valued time series
INteger-valued AutoRegressive (INAR) processes
INAR(1) processes [McKenzie (1985, 1988); Al-Osh and Alzaid (1987)]
Xt = Xt1 + et, t= 1, . . . ,N
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 5 / 31
Isabel Silva Magalhes Integer-valued time series
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g g
INteger-valued AutoRegressive (INAR) processes
INAR(1) processes [McKenzie (1985, 1988); Al-Osh and Alzaid (1987)]
Xt = Xt1 + et, t= 1, . . . ,N
0 <
1
{et} N0 : sequence of i.i.d. discrete r.v. (arrival process)
E[et] = e, Var[et] = e2, E[et
3] = e, E[et4] = e
counting series, {Yj}, are independent, and independent of{et}, and such thatE[Yj] = , Var[Yj] =
2, E[Yj3] = , E[Yj
4] =
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 5 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Poisson INAR(1) process with binomial thinning operation
Xt = Xt1 + et
{et}
Poisson distributed and
{Yj}
Bernoulli distributed
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 6 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Poisson INAR(1) process with binomial thinning operation
Xt = Xt1 + et
{et}
Poisson distributed and
{Yj}
Bernoulli distributed
Xt1 : survivors of the elements of the process at time t1, each withprobability of survival
et : elements which enter in the system in the interval ]t
1, t]
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 6 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Poisson INAR(1) process with binomial thinning operation
Xt = Xt1 + et
{et}
Poisson distributed and
{Yj}
Bernoulli distributed
Xt1 : survivors of the elements of the process at time t1, each withprobability of survival
et : elements which enter in the system in the interval ]t
1, t]
X1 Po
1
{et} Po()
={Xt} Po
1
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 6 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Likelihood function of the Poisson INAR(1) process
p(Xt|Xt1) : convolution of binomial distribution and Poisson distributionp(Xt|Xt1) = exp()
Mt
i=0
(Xt)i
((Xt) i)!
Xt1i
i(1)(Xt1)i,Mt = min{Xt1,Xt}
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 7 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Likelihood function of the Poisson INAR(1) process
p(Xt|Xt1) : convolution of binomial distribution and Poisson distributionp(Xt|Xt1) = exp()
Mt
i=0
(Xt)i
((Xt) i)!
Xt1i
i(1)(Xt1)i,Mt = min{Xt1,Xt}
X = {X0,X1, . . . ,XN}L(X,,) =
[/(1)]X0X0!
exp
1
N
t=1
p(Xt|Xt1)
L(X,,|X0)
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 7 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Likelihood function of the Poisson INAR(1) process
p(Xt|Xt1) : convolution of binomial distribution and Poisson distributionp(Xt|Xt1) = exp()
Mt
i=0
(Xt)i
((Xt) i)!
Xt1i
i(1)(Xt1)i,Mt = min{Xt1,Xt}
X = {X0,X1, . . . ,XN}L(X,,) =
[/(1)]X0X0!
exp
1
N
t=1
p(Xt|Xt1)
L(X,,|X0)Poisson distribution mean = variance problem in practical applications
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 7 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Likelihood function of the Poisson INAR(1) process
p(Xt|Xt1) : convolution of binomial distribution and Poisson distributionp(Xt|Xt1) = exp()
Mt
i=0
(Xt)i
((Xt) i)!
Xt1i
i(1)(Xt1)i,Mt = min{Xt1,Xt}
X = {X0,X1, . . . ,XN}L(X,,) =
[/(1)]X0X0!
exp
1
N
t=1
p(Xt|Xt1)
L(X,,|X0)Poisson distribution mean = variance problem in practical applicationsOverdispersion: variance > mean binomial, negative binomial, geometric or
generalized Poisson
Underdispersion: variance < mean generalized PoissonINteger-valued AutoRegressive (INAR) processes UTAD - February 2010 7 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
INAR(p) processes [Du and Li (1991); Latour (1998)]
Xt = 1 Xt1 + +p Xtp + et, t= 1, . . . ,N,
i 0, i = 1, . . . ,p1, and p > 0, such that pi=1i < 1,{et} N0 : sequence of i.i.d. discrete r.v. (arrival process)
E[et] = e, Var[et] = e2, E[et
3] = e, E[et4] = e,
all counting series, {Yj,i}, of the thinning operationsi Xti =Xtij=0 Yj,i, i = 1, . . . ,p,
are mutually independent, and independent of{et}, and such thatE[Yj,i] = i, Var[Yj,i] = i
2, E[Yj,i3] = i, E[Yj,i
4] = i.
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 8 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
INAR(p) processes [Du and Li (1991); Latour (1998)]
Xt = 1 Xt1 + +p Xtp + et, t= 1, . . . ,N,
i 0, i = 1, . . . ,p1, and p > 0, such that pi=1i < 1,{et} N0 : sequence of i.i.d. discrete r.v. (arrival process)
E[et] = e, Var[et] = e2, E[et
3] = e, E[et4] = e,
all counting series, {Yj,i}, of the thinning operationsi Xti =Xtij=0 Yj,i, i = 1, . . . ,p,
are mutually independent, and independent of{et}, and such thatE[Yj,i] = i, Var[Yj,i] = i
2, E[Yj,i3] = i, E[Yj,i
4] = i.
INAR(p) has the same second-order structure as an AR(p) process
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 8 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Autocovariance function [Gauthier and Latour (1994)]
R(0) = Vp +pi=1iR(i),R(k) =
pi=1iR(i k), k Z\{0},
Spectral density function [Silva and Oliveira (2004, 2005)]
f() = 12
Vp1pk=1keik2 , ,One-step-ahead prediction error [Silva (2005)]
Vp = e2 +Xpi=1i2.
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 9 / 31
Isabel Silva Magalhes Integer-valued time series
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INteger-valued AutoRegressive (INAR) processes
Autocovariance function [Gauthier and Latour (1994)]
R(0) = Vp +pi=1iR(i),R(k) =
pi=1iR(i k), k Z\{0},
Spectral density function [Silva and Oliveira (2004, 2005)]
f() = 12
Vp1pk=1keik2 , ,One-step-ahead prediction error [Silva (2005)]
Vp = e2 +Xpi=1i2.
One-sided general linear representation [Silva (2005)]
Xt =
u=0ut
u,
{t
}white noise process
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 9 / 31
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INteger-valued AutoRegressive (INAR) processes
Third-order characterization (in terms of moments)[Silva and Oliveira (2004, 2005) and Silva (2005)]
X(0, 0) = pi=1
pj=1
pk=1ijkX(ij, i k) + 3pi=1pj=1ji2X(ij)
+3X(2e +e
2)pi=1i + 3e
pi=1
pj=1ijX(ij)
+3Xep
i=1i
2 +Xp
i=1(i
3ii2
3
i
) + e
X(0, k) = pi=1iX(0, k i) +eX(0), k> 0
X(k, k) = pi=1
pj=1ij X(k
i, k
j) +
pi=1i
2X(k
i) + 2eX(k)
X(e2e2), k> 0
X(k, m) = pi=1iX(k, m i) +eX(k), m > k> 0
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 10 / 31
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INteger-valued AutoRegressive (INAR) processes
Third-order characterization (in terms of cumulants)[Silva and Oliveira (2004, 2005) and Silva (2005)]
CX(0, 0) = pi=1
pj=1
pk=1ijkX(ij, i k) + 3pi=1pj=1ji2X(ij) + e
+3(eX)pi=1pj=1ijX(ij) + 3X(eX)pi=1i2 + 23X6eX2pi=1i3e(e2 +2e ) +Xpi=1 (i3ii23i )
CX(0, k) = pi=1iCX(0, k i), k> 0
CX(k, k) = pi=1
pj=1ijCX(k i, kj) +pi=1i2CX(k i), k> 0
CX(k, m) = pi=1iCX(k, m i), m > k> 0
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 11 / 31
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INteger-valued AutoRegressive (INAR) processes
Third-order characterization (in terms of cumulants)[Silva and Oliveira (2004, 2005) and Silva (2005)]
CX(0, 0) = pi=1
pj=1
pk=1ijkX(ij, i k) + 3pi=1pj=1ji2X(ij) + e
+3(eX)pi=1pj=1ijX(ij) + 3X(eX)pi=1i2 + 23X6eX2pi=1i3e(e2 +2e ) +Xpi=1 (i3ii23i )
CX(0, k) = pi=1iCX(0, k i), k> 0
CX(k, k) = pi=1
pj=1ijCX(k i, kj) +pi=1i2CX(k i), k> 0
CX(k, m) = pi=1iCX(k, m i), m > k> 0
INAR processes have a non-linear structure
1st and 2nd order moments and cumulants are not sufficient to describe dependence structure
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 11 / 31
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INteger-valued AutoRegressive (INAR) processes
Some extensions of INAR processes and relations with other models
Switching INAR(1) process [Franke and Seligmann (1993)]
Inclusion of explanatory variables [Brnns (1995)]
Panel data [Brnns (1994, 1995)]
INteger-valued Moving Average (INMA) model [Al-Osh and Alzaid (1988); Mckenzie (1988);
Brnns and Hall (2001)]
INteger-valued AutoRegressive-Moving Average (INARMA) model [Mckenzie (1985, 1986);
Al-Osh and Alzaid (1991)]
Multivariate INAR(p) process [Franke and Subba Rao (1995); Latour (1997)]
INAR(p) of Alzaid and Al-Osh (1990) ARMA(p,p1)Poisson INAR(1) process is a M/M/ queueing system observed at regularly spaced interval of
times [Steutel et al. (1983); Mckenzie (1988)]
INAR(p) is a Multitype Branching processes with immigration, BGWI(p) [Dion et al. (1995)]
INteger-valued AutoRegressive (INAR) processes UTAD - February 2010 12 / 31
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Parameter estimation for INAR(p) processes
Time domain
Method of Moments Second-order: Yule-Walker estimation (YW)
Third-order: Estimation using the Cumulant Third-Order Recursion equation
(TOR),
Least Squares (LS) estimation Second-order: Conditional Least Squares (CLS)
Third-order: LS estimation based on high-order statistics (LS_HOS),
Frequency domain
Whittle estimation (WHT),
Parameter estimation for INAR(p) processes UTAD - February 2010 13 / 31
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Parameter estimation for INAR(p) processes
Time domain
Method of Moments Second-order: Yule-Walker estimation (YW)
Third-order: Estimation using the Cumulant Third-Order Recursion equation
(TOR),
Least Squares (LS) estimation Second-order: Conditional Least Squares (CLS)
Third-order: LS estimation based on high-order statistics (LS_HOS),
Frequency domain
Whittle estimation (WHT),
YW, CLS, WHT, TOR, LS_HOS: do not assume the Poisson distribution for the
arrival process
more adaptive and flexible methods
Parameter estimation for INAR(p) processes UTAD - February 2010 13 / 31
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Parameter estimation for INAR(p) processes
Theorem (Asymptotic distribution of the Yule-Walker estimators of an INAR(p) process [Silva and Silva (2006)])
Let {Xt} be an INAR(p) process and the Yule-Walker estimator of
[Al-Osh and Alzaid
(1987); Du and Li (1991)], that is
Rp
1= rp
R(0) R(1) R(p1)R(1) R(0) R(p2)
.
..
.
.... .
.
..R(p1) R(p2) R(0)
1
2.
..
p
=
R(1)
R(2).
..R(p)
.
Then
N1/2 () is AN(0p, Vyw),where 0n is a vector ofn zeros and Vyw = D
TRrD, for Rr given by
Rr(i,j) = Cov(VRr(i), VRr(j)) and DT =
R(1)Ip R(p)Ip
(Rp11)T
Rp11
Ip2 0p2p
+
0pp2 Rp1
1
, with In the nn identity matrix,
0nm
the n
m matrix of zeros and
the Kronecker.
Parameter estimation for INAR(p) processes UTAD - February 2010 14 / 31
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Parameter estimation for INAR(p) processes
Theorem (Cumulant Third-Order Recursion (TOR) equation of INAR(p) processes [Silva (2005)])
Let {Xt} be a stationary INAR(p) process. Then the third-order cumulants ofXt canbe written in a single cumulant Third-Order Recursion (TOR) equation by
CX(k,m)pi=1iCX(i k, im) = (k)p
i=1i
2CX(im),
where 0 k m, m = 0, and (a) is the Kronecker delta function.
Parameter estimation for INAR(p) processes UTAD - February 2010 15 / 31
Isabel Silva Magalhes Integer-valued time series
( )
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Parameter estimation for INAR(p) processes
Theorem (Cumulant Third-Order Recursion (TOR) equation of INAR(p) processes [Silva (2005)])
Let {Xt} be a stationary INAR(p) process. Then the third-order cumulants ofXt canbe written in a single cumulant Third-Order Recursion (TOR) equation by
CX(k,m)pi=1iCX(i k, im) = (k)p
i=1i
2CX(im),
where 0 k m, m = 0, and (a) is the Kronecker delta function.
CX(k,k) = CX(0, k)
C3,X=
CX(0, 0) CX(1, 1) CX(p1,p1)CX(0, 1) CX(0, 0) CX(p2,p2)
.
.
....
. . ....
CX(0,p1) CX(0,p2) CX(0, 0)
1
2...
p
=
CX(0, 1)
CX(0, 2)
.
.
.
CX(0,p)
= c3,X
Parameter estimation for INAR(p) processes UTAD - February 2010 15 / 31
Isabel Silva Magalhes Integer-valued time series
P i i f INAR( )
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Parameter estimation for INAR(p) processes
Estimation using the Cumulant TOR equation
{X1, . . . ,XN=B M} = {X(1)1 , . . . ,X(1)M ,X(2)1 , . . . ,X(2)M , . . . ,X(B)1 , . . . ,X(B)M },
C(i)
X (k, k) =1
M
Mkj=1
(X(i)
j X(i))(X(i)j+kX(i)
)2, k= 0, . . . ,p1,
C
(i)
X (0, k) =
1
M
Mkj=1 (X
(i)
j X(i)
)
2
(X
(i)
j+kX(i)
), k= 1, . . . ,p,
CX(k, k) =1
B
B
i=1C
(i)X (k, k), k= 0, . . . ,p1,
CX(0, k) =1
B
B
i=1C
(i)X (0, k), k= 1, . . . ,p,
Solve C3,X= c3,X.
Parameter estimation for INAR(p) processes UTAD - February 2010 16 / 31
Isabel Silva Magalhes Integer-valued time series
P t ti ti f INAR( )
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Parameter estimation for INAR(p) processes
Estimation using the Cumulant TOR equation
{X1, . . . ,XN=B M} = {X(1)1 , . . . ,X(1)M ,X(2)1 , . . . ,X(2)M , . . . ,X(B)1 , . . . ,X(B)M },
C(i)
X (k, k) =1
M
Mkj=1
(X(i)
j X(i))(X(i)j+kX(i)
)2, k= 0, . . . ,p1,
C
(i)
X (0, k) =
1
M
Mkj=1 (X
(i)
j X(i)
)
2
(X
(i)
j+kX(i)
), k= 1, . . . ,p,
CX(k, k) =1
B
B
i=1C
(i)X (k, k), k= 0, . . . ,p1,
CX(0, k) =1
B
B
i=1C
(i)X (0, k), k= 1, . . . ,p,
Solve C3,X= c3,X.
Asymptotic distribution: sixth-order moments and cumulants
Parameter estimation for INAR(p) processes UTAD - February 2010 16 / 31
Isabel Silva Magalhes Integer-valued time series
P t ti ti f INAR( )
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Parameter estimation for INAR(p) processes
Theorem (Asymptotic distribution of the CLS estimators of an INAR(p) process [Latour (1998)])
Let {X1, . . . ,XN} be observations of an INAR(p) process such that E [e3t] < andE [(Y
(k)i )
3] < , for k= 1, . . . ,p and k N. Let = [ 1 p e ]T and let denote the Conditional Least Squares estimator of, given by
QN() = min {QN()} = min Nt=p+1 (Xt1Xt1 . . .pXtpe)2 .Then
N() N(0p+1, V1WV1),
where 0p+1 is a (p + 1)1 vector of zeros and the entries ofV and W are defined by
[V]i,j =
X(ij) i,j = 1, . . . ,p,X i = p + 1 or j = p + 1, i = j,1 i = j = p + 1,
[W]i,j = e2[V]i,j +
p
k=1
k2X(k i, kj).
Parameter estimation for INAR(p) processes UTAD - February 2010 17 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR( ) processes
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Parameter estimation for INAR(p) processes
LS estimation based on high-order statistics (HOS)
{x
1,x
2, . . . ,x
n}: realization of a non-negative integer-valued stationary stochastic
process with third-order moments (0, k), k> 0
Parameter estimation for INAR(p) processes UTAD - February 2010 18 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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Parameter estimation for INAR(p) processes
LS estimation based on high-order statistics (HOS)
{x
1,x
2, . . . ,x
n}: realization of a non-negative integer-valued stationary stochastic
process with third-order moments (0, k), k> 0
Approximating model: INAR(p) with parameters 1, ,p,e,2e andthird-order moments X(0, k), k> 0, which can be represented in the following
matrix form: 3,X = M3,X+eX(0)1p
X(0, 1)
X(0, 2)
..
.
X(0,p)
=
X(0, 0) X(1, 1) . . . X(p1,p1)X(0, 1) X(0, 0) . . . X(p
2,p
2)
..
....
. . ....
X(0,p1) X(0,p2) . . . X(0, 0)
1
2
..
.
p
+eX(0)
1
1
..
.
1
X(0) =p
i=1
iX(i) +eX + Vp, with Vp = e2 +X
p
i=1
i2
Parameter estimation for INAR(p) processes UTAD - February 2010 18 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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Parameter estimation for INAR(p) processes
Defining H = [M3,X X(0)1p] and = [ 1 p e ]T
3,X = H
Parameter estimation for INAR(p) processes UTAD - February 2010 19 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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Parameter estimation for INAR(p) processes
Defining H = [M3,X X(0)1p] and = [ 1 p e ]T
3,X = H
may be estimated by least squares, ie, minimizing the squared error between 3,Xand the third-order moments of the data:
3 = [ (0, 1) (0,p) ]T
Parameter estimation for INAR(p) processes UTAD - February 2010 19 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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Parameter estimation for INAR(p) processes
Defining H = [M3,X X(0)1p] and = [ 1 p e ]T
3,X = H
may be estimated by least squares, ie, minimizing the squared error between 3,Xand the third-order moments of the data:
3 = [ (0, 1) (0,p) ]T
Least Squares estimator of using HOS (LS_HOS)
= min{L()} = min
{(3H)T(3H)}
Parameter estimation for INAR(p) processes UTAD - February 2010 19 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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Parameter estimation for INAR(p) processes
Defining H = [M3,X X(0)1p] and = [ 1 p e ]T
3,X = H
may be estimated by least squares, ie, minimizing the squared error between 3,Xand the third-order moments of the data:
3 = [ (0, 1) (0,p) ]T
Least Squares estimator of using HOS (LS_HOS)
= min{L()} = min
{(3H)T(3H)}
In practice: = min{ L()} = min
{(3 H)T(3 H)}
Parameter estimation for INAR(p) processes UTAD - February 2010 19 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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Parameter estimation for INAR(p) processes
Spectral density function easy to obtain
{Xt} is an INAR process[Silva (2005)]
{Xt} is a Non-Gaussian Mixing process: {Xt} is strictly stationary, E[|Xt|k] < , t Z, k N,
s1=
. . .
sk1=
|CX(s1, . . . , sk1)|< , k 2, (mixing condition)
Parameter estimation for INAR(p) processes UTAD - February 2010 20 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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Parameter estimation for INAR(p) processes
Spectral density function easy to obtain
{Xt} is an INAR process[Silva (2005)]
{Xt} is a Non-Gaussian Mixing process: {Xt} is strictly stationary, E[|Xt|k] < , t Z, k N,
s1=
. . .
sk1=
|CX(s1, . . . , sk1)|< , k 2, (mixing condition)
IN(j) =1
2N
N
t=1
Xteijt
2
f(j)2
22 ,
Parameter estimation for INAR(p) processes UTAD - February 2010 20 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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Parameter estimation for INAR(p) processes
Spectral density function easy to obtain
{Xt} is an INAR process[Silva (2005)]
{Xt} is a Non-Gaussian Mixing process: {Xt} is strictly stationary, E[|Xt|k] < , t Z, k N,
s1=
. . .
sk1=
|CX(s1, . . . , sk1)|< , k 2, (mixing condition)
IN(j) =1
2N
N
t=1
Xteijt
2
f(j)2
22 ,
Whittle estimator [Silva and Oliveira (2004, 2005)]
= min{L(X)} = min
[N/2]
j=1
log(f(j)) +
IN(j)
f(j)
Parameter estimation for INAR(p) processes UTAD - February 2010 20 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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(p) p
Spectral density function easy to obtain
{Xt} is an INAR process[Silva (2005)]
{Xt} is a Non-Gaussian Mixing process: {Xt} is strictly stationary, E[|Xt|k] < , t Z, k N,
s1=
. . .
sk1=
|CX(s1, . . . , sk1)|< , k 2, (mixing condition)
IN(j) =1
2N
N
t=1
Xteijt
2
f(j)2
22 ,
Whittle estimator [Silva and Oliveira (2004, 2005)]
= min{L(X)} = min
[N/2]
j=1
log(f(j)) +
IN(j)
f(j)
Asymptotic variance: Fourth-order cumulant spectral density function
Parameter estimation for INAR(p) processes UTAD - February 2010 20 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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(p) p
e
= X1p
i=1
i , 2e = VpXp
i=12
i,
where Vp = R(0)pi=1 i R(i) and 2i is an estimator of Var[Yj,i], i = 1, . . . ,p. Forinstance, 2i = i(1 i), for the binomial thinning operation.
Parameter estimation for INAR(p) processes UTAD - February 2010 21 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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(p) p
e
= X1p
i=1
i , 2e = VpXp
i=12
i,
where Vp = R(0)pi=1 i R(i) and 2i is an estimator of Var[Yj,i], i = 1, . . . ,p. Forinstance, 2i = i(1 i), for the binomial thinning operation.
Simulation results
Non-admissible estimates constrained estimation (CLS, WHT, LS_HOS)
Parameter estimation for INAR(p) processes UTAD - February 2010 21 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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( )
e = X1p
i=1i , 2e = VpX
p
i=12
i,
where Vp = R(0)pi=1 i R(i) and 2i is an estimator of Var[Yj,i], i = 1, . . . ,p. Forinstance, 2i = i(1 i), for the binomial thinning operation.
Simulation results
Non-admissible estimates constrained estimation (CLS, WHT, LS_HOS)Sample bias, variance, mean square error and univariate skewness decrease as
the sample size increases consistency and symmetry
Parameter estimation for INAR(p) processes UTAD - February 2010 21 / 31
Isabel Silva Magalhes Integer-valued time series
Parameter estimation for INAR(p) processes
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e = X1p
i=1i , 2e = VpX
p
i=12
i
,
where Vp = R(0)pi=1 i R(i) and 2i is an estimator of Var[Yj,i], i = 1, . . . ,p. Forinstance, 2i = i(1 i), for the binomial thinning operation.
Simulation results
Non-admissible estimates constrained estimation (CLS, WHT, LS_HOS)Sample bias, variance, mean square error and univariate skewness decrease as
the sample size increases consistency and symmetryLS_HOS, WHT and CLS provides good results
in terms of smaller sample bias, variance and mean square error
Parameter estimation for INAR(p) processes UTAD - February 2010 21 / 31
Isabel Silva Magalhes Integer-valued time series
Application to real data
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Poliomyelitis incidence in the United States
Monthly number of U.S. cases of
poliomyelitis, from 1970 to 1983
[Zeger (1988): Parameter-driven
model], and sample
autocorrelation and partialautocorrelation functions
1970 1972 1974 1976 1978 1980 1982 19840
2
4
6
8
10
12
14
year
monthlycounts
0 5 10 15 20 25 300.2
0
0.2
0.4
0.6
0.8
1
1.2
k
(k)
0 5 10 15 20 25 300.2
0.1
0
0.1
0.2
0.3
k
(k)
Application to real data UTAD - February 2010 22 / 31
Isabel Silva Magalhes Integer-valued time series
Application to real data
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Poliomyelitis incidence in the United States
X= 1.33 and S2 = 3.48
INAR(1) with binomial thinning operation and discrete arrival process
Xt = Xt1 + et, t= 2, . . . , 168, E[et] = e, Var[et] = e2,
Application to real data UTAD - February 2010 23 / 31
Isabel Silva Magalhes Integer-valued time series
Application to real data
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Poliomyelitis incidence in the United States
X= 1.33 and S2 = 3.48
INAR(1) with binomial thinning operation and discrete arrival process
Xt = Xt1 + et, t= 2, . . . , 168, E[et] = e, Var[et] = e2,
Method e 2e
YW 0.2948 0.9403 2.9041
CLS 0.3063 0.9414 2.8862
WHT 0.2799 0.9601 2.9279
LS_HOS 0.2083 0.9277 3.0504
LS_HOS_C 0.2344 0.7477 3.0040
TOR_1B 0.1475 1.1367 3.1650
TOR_2B 0.1431 1.1425 3.1737
Table: Parameter estimates of the INAR(1) model fitted to the polio data.
Application to real data UTAD - February 2010 23 / 31
Isabel Silva Magalhes Integer-valued time series
Application to real data
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Number of plants within the industrial sector
The number of Swedish
mechanical paper and
pulp mills, from 1921 to
1981 [Brnns (1995)
and Brnns and
Hellstrm (2001):
Explanatory variables]
1920 1930 1940 1950 1960 1970 19815
10
15
20
25
30
35
40
45
50
Numberofplants
Application to real data UTAD - February 2010 24 / 31
Isabel Silva Magalhes Integer-valued time series
Application to real data
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Simple INAR(1)
It is not assumed the Poisson distribution for the arrival process:
X= 20.40 and S2 = 155.16
Application to real data UTAD - February 2010 25 / 31
Isabel Silva Magalhes Integer-valued time series
Application to real data
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Simple INAR(1)
It is not assumed the Poisson distribution for the arrival process:
X= 20.40 and S2 = 155.16
Method e 2e x
2x MSE
CLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997
TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224
Application to real data UTAD - February 2010 25 / 31
Isabel Silva Magalhes Integer-valued time series
Application to real data
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Simple INAR(1)
It is not assumed the Poisson distribution for the arrival process:
X= 20.40 and S2 = 155.16
Method e 2e x
2x MSE
CLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997
TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224
Mean and variance of the estimated models: x = e
1 and 2
x =(1 )(e+ 2e )
(1 )2(1 + )
Application to real data UTAD - February 2010 25 / 31
Isabel Silva Magalhes Integer-valued time series
Application to real data
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Simple INAR(1)
It is not assumed the Poisson distribution for the arrival process:
X= 20.40 and S2 = 155.16
Method e 2e x
2x MSE
CLS 0.9591 0.2017 15.2268 4.9279 192.2764 9.3254LS_HOS 0.9269 1.3635 19.2253 18.6516 145.4513 9.2997
TOR_1B 0.9631 0.7518 14.7219 20.374 213.5073 8.9224
Mean and variance of the estimated models: x = e
1 and 2
x =(1 )(e+ 2e )
(1 )2(1 + )
MSE between the observations and the fitted models based on TOR_1B, LS_HOS and
CLS estimates
Application to real data UTAD - February 2010 25 / 31
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The number of plants and
the fitted values
considering the LS_HOS
and CLS estimates
1920 1930 1940 1950 1960 1970 19815
10
15
20
25
30
35
40
45
50
Numberofplants
Real data
CLS
LS_HOS
Application to real data UTAD - February 2010 26 / 31
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Application to real data
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Sunspot groups
0 20 40 60 80 1000
10
20
30
weeks
numberofsunspotgrou
ps
0 20 40 60 80 1000
10
20
30
weeks
numberofsunspotgrou
ps
0 5 10 15 20
0.2
0
0.2
0.4
0.6
0.8
1
k
(
k)
0 5 10 15 20
0.2
0
0.2
0.4
0.6
0.8
1
k
(
k)
0 5 10 15 200.4
0.2
0
0.2
0.4
k
(
k)
0 5 10 15 20
0.2
0
0.2
0.4
k
(
k)
Boulder Palehua
Application to real data UTAD - February 2010 27 / 31
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Application to real data
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Sunspot groups
Poisson RINAR(1) with binomial thinning operation
Xk,t = Xk,t1 + ek,t, t= 2, . . . , 104, k= 1, 2,
Application to real data UTAD - February 2010 28 / 31
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Sunspot groups
Poisson RINAR(1) with binomial thinning operation
Xk,t = Xk,t1 + ek,t, t= 2, . . . , 104, k= 1, 2,
Method
YW 0.3328 10.8637
CLS 0.3992 9.7947
IWCLS 0.3839 10.0450
WHT 0.3666 10.3133
LS_HOS 0.3451 10.8644
Table: Parameter estimates of the Poisson RINAR(1) model for the total number of sunspot groups.
Application to real data UTAD - February 2010 28 / 31
Isabel Silva Magalhes Integer-valued time series
Recent developments
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Overdispersion new thinning operations and/or different distributions for thearrival processes
Extreme value theory
INAR with periodic structureOutliers
Forecasting
Heteroskedasticity
Random-coefficient INAR
Recent developments UTAD - February 2010 29 / 31
Isabel Silva Magalhes Integer-valued time series
References I
-
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Al-Osh, M.A. and Alzaid, A.A., 1987.
First-order integer-valued autoregressive (INAR(1))
process.
Journal of Time Series Analysis, vol. 8, pp. 261275.
Al-Osh, M.A. and Alzaid, A.A., 1988.
Integer-valued moving average (INMA) process.
Statistical Papers, vol. 29, pp. 281300.
Al-Osh, M.A. and Alzaid, A.A., 1991.
Binomial autoregressive moving average models.
Communications in Statistics: Stochastic Models, vol. 7,pp. 261282.
Alzaid, A.A. and Al-Osh, M.A., 1990.
An integer-valued pth-order autoregressive structure
(INAR(p)) process.
Journal of Applied Probability, vol. 27, pp. 314324.
Brnns, K., 1994.
Estimation and testing in integer-valued AR(1) models.
Technical Report, Ume University, Sweden, 335.
Brnns, K., 1995.
Explanatory variables in the AR(1) count data model.
Technical Report, Ume University, Sweden, 381.
Brnns, K. and Hall, A., 2001.
Estimation in integer-valued moving average models.
Applied Stochastic Models in Business and Industry, vol.
17, pp. 277291.
Dion, J-P. and Gauthier, G. and Latour, A., 1995.
Branching processes with immigration and integer-valued
time series.
Serdica Mathematical Journal, vol. 21, pp. 123136.
Du, Jin-Guan and Li, Yuan, 1991.The integer-valued autoregressive (INAR(p)) model.
Journal of Time Series Analysis, vol. 12, pp. 129142.
Franke, J. and Seligmann, T., 1993.
Conditional maximum likelihood estimates for INAR(1)
processes and their application to modelling epileptic
seizure counts.
In Developments in Time Series Analysis: in honour of
Maurice B. Priestley, Chapman & Hall, pp. 310330.
Franke, J. and Subba Rao, T., 1995.
Multivariate first order integer valued autoregressions.
Technical report, Math. Dep., UMIST.
References UTAD - February 2010 30 / 31
Isabel Silva Magalhes Integer-valued time series
References II
-
8/8/2019 Presentation Seminario UTAD
62/62
Gauthier, G. and Latour, A., 1994.
Convergence forte des estimateurs des paramtres dtun
processus GENAR(p).
Annales des Sciences Mathmatiques du Qubec, vol. 18,
pp. 4971
Latour, A., 1997.
The multivariate GINAR(p) process.
Advances in Applied Probability, vol. 29, pp. 228248.
Latour, A., 1998.
Existence and stochastic structure of a non-negative
integer-valued autoregressive process.
Journal of Time Series Analysis, vol. 19, pp. 439455.
McKenzie, E., 1985.
Some simple models for discrete variate time series.
Water Resources Bulletin, vol. 21, pp. 645650.
McKenzie, E., 1986.
Autoregressive moving-average processes with
negative-binomial and geometric marginal distributions.Advances in Applied Probability, vol. 18, pp. 679705.
McKenzie, E., 1988.
Some ARMA models for dependent sequences of Poisson
counts.
Advances in Applied Probability, vol. 20, pp. 822835.
Silva, M.E. and Oliveira, V.L., 2004.
Difference equations for the higher-order moments and
cumulants of the INAR(1) model.
Journal of Time Series Analysis, vol. 25, pp. 317333.
Silva, M.E. and Oliveira, V.L., 2005.
Difference equations for the higher-order moments and
cumulants of the INAR(p) model.
Journal of Time Series Analysis, vol. 26, pp. 1736.
Silva, I. e Silva, M.E., 2006.
Asymptotic distribution of the Yule-Walker estimator for
INAR(p) processes.
Statistics & Probability Letters, vol. 76, pp. 1655-1663.
Steutel, F.W. and Van Harn, K., 1979.
Discrete analogues of self-decomposability and stability.
The Annals of Probability, vol. 7, pp. 893899.
Steutel, F.W. and Vervaat, W. and Wolfe, S.J., 1983.
Integer valued branching processes with immigration.Advances in Applied Probability, vol. 15, pp. 713725.
Zeger, S.L., 1988.
A regression model for time series of counts.
Biometrika, vol. 75, pp. 621629.
References UTAD - February 2010 31 / 31