Heligman-Pollard Graduation: Adjusting for Local Variability in Parameter Estimation Anna Maria Altavilla, Angelo Mazza, Antonio Punzo Università di Catania XLVII Riunione Scientifica della Società Italiana di Economia, Demografia e Statistica Un mondo in movimento: approccio multidisciplinare ai fenomeni migratori Milano, 27-28 e 29 maggio 2010
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Heligman-Pollard Graduation: Adjusting for Local Variability in Parameter Estimation Anna Maria Altavilla, Angelo Mazza, Antonio Punzo Università di Catania.
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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
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.