Heligman-Pollard Graduation: Adjusting for Local Variability in Parameter Estimation

Anna Maria Altavilla, Angelo Mazza, Antonio Punzo

Università di Catania

XLVII Riunione Scientifica dellaSocietà Italiana di Economia, Demografia e Statistica

Un mondo in movimento:approccio multidisciplinare ai fenomeni migratori

Milano, 27-28 e 29 maggio 2010

Observed mortality patterns as instances of a stochastic process

True but unknown mortality pattern qx

Observed mortality patterns as instances of a stochastic process

Observed mortality pattern (Population: 300.000)

Observed mortality patterns as instances of a stochastic process

Observed mortality pattern (Population: 3.000.000)

Observed mortality patterns as instances of a stochastic process

Observed mortality pattern (Population: 30.000.000)

Variability of conditional distributions

true but unknown mortality rate ex with ex and Var ex(1-) crude mortality rate , with and Var (1-)/ exVC() = variation coefficientStandard deviation Variation coefficient

Graduation

The relation between the crude rates and the true but unknown mortality rates may be summarized as follows:

In order to capture the underlying mortality pattern from the crude rates, a graduation function)is used. In other words, it aims at smoothing out irregularities in crude mortality rates due to random variation and age misstatement.In analogy with the usual statistical modeling, the ) function can be specified parametrically or nonparametrically.

Parametric graduation: the Heligman-Pollard model

The Heligman-Pollard model:parameters estimation The classical estimation method consists in minimizing the quantity: where Ω is the set of observed ages. Our proposal is to consider the following weighted index: where

𝑆2=∑𝑥∈Ω

(�̂�𝑥

�̇�𝑥

−1)2

𝑆𝑤2 =∑

𝑥∈Ω

𝑤𝑥( �̂�𝑥

�̇�𝑥

−1)2

𝑤𝑥=VC( �̇� 𝑥 )−1

∑𝑥∈Ω

VC( �̇�𝑥 )− 1

Which estimation method works better?We have tested the proposed estimation method with the following procedure.1. Choose both a model mortality pattern defined by the couples and a population distribution by age .2. For e, draw a value of from and compute= /.3. Estimate the parameters for the Heligman-Pollard model using both estimation procedures based on and and 4. Compute the goodness-of-fit index 5. Repeat steps 2-4 B times.

Results of the simulation

Notes: Number of replications B=1.000 Age structure of exis either USA 2007 male or USA 2007 female.

M(S2) M(S2) Gain

Female 46 700.000 3,65 46,40% 3,57 53,60% 2,03%Male 46 700.000 1,31 47,60% 1,31 52,40% 0,07%Female 60 700.000 4,93 44,40% 4,78 55,60% 3,13%Male 60 700.000 2,50 37,40% 2,38 62,60% 4,92%Female 70 700.000 4,83 39,20% 4,65 60,80% 3,80%Male 70 700.000 2,66 36,60% 2,50 63,40% 5,86%Female 46 3.000.000 0,31 33,80% 0,27 66,20% 11,66%Male 46 3.000.000 0,14 40,20% 0,14 59,80% 3,90%Female 60 3.000.000 0,67 42,80% 0,65 57,20% 3,85%Male 60 3.000.000 0,24 39,20% 0,23 60,80% 5,54%Female 70 3.000.000 0,70 47,20% 0,68 52,80% 2,27%Male 70 3.000.000 0,29 38,90% 0,27 61,10% 6,05%

Classical EstimationH.P. model for Australia:

Population size

Alternative EstimationΨ ܵ�௪ଶظ ܵ�ଶΨ ܵ�ଶظ ܵ�௪ଶ

Conclusions

The variability of the distribution of crude depends on either ex and qx; consequently, it changes across the age range.

Considering this variation while graduating provides a better estimate of the true but unknown rates .

In evaluating different estimation methods, it makes sense to use as benchmark instead of