90 36 National Life Table Experience Life Table Generalized Linear Models GLM Miller946 Gompertz825 Makeham 860 Makeham Heligman Pollard980 8 Heligman

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1 Vol. 36No. 3 May Population Research * GLM GLM 0 ~ 89 B - GLM B The Application of Generalized Linear Model in the Graduation of Life Table Mortality Rates Zhang Lianzeng Duan Baige AbstractAttempt has been made in this research to apply generalized linear models in graduating China' s life table mortality rates.using demographic data of deaths by age and gender from China Population Statistical Yearbooks and Statistical Yearbooks of China' s Population and Employment the relationships between mortality and age and between mortality and year are explored by fitting death rates at ages from 0 to 89 using Poisson regression and negative binomial regression.upon comparison of the fitting effects of the two modelsthe paper proposes to use B - spline function to smooth the death rates.implications of this study are discussed for constructing China' s empirical life tablesproviding theoretical foundation and practical reference for mortality analysis by China Insurance Regulatory Commissionand achieving market - oriented rates of life insurance and scientific management of the life insurance industry in China. KeywordsMortality RatesGeneralized Linear ModelNegative Binomial RegressionPoisson RegressionB - spline Graduation AuthorsZhang Lianzeng is Professorand Duan Baige is PhD StudentDepartment of Risk Management and InsuranceSchool of EconomicsNankai UniversityTianjin zhlz@ nankai. edu.cn. * NKZXTD0

2 90 36 National Life Table Experience Life Table Generalized Linear Models GLM Miller946 Gompertz825 Makeham 860 Makeham Heligman Pollard980 8 Heligman - Pollard Carriere992 Lee Carter992 Haberman Renshaw996 GLM Moving Weighted Average Graduation MWAG MWAG Copas Haberman983 Kernel Smoothing Locally - weighted Regression LOESS Generalized Additive Models GAM Bayes Wang Müller Capra 998 Wang 2005Debón Montes Sala2006 da Rocha Neves Migon CL ~ 80 CL00-03 CL ~ ~ 200

3 3 9 GLM 0 ~ 89 B - 2 GLM Nelder Wedderburn 972 GLM McCullagh Nelder 989 GLM GLM GLM GLM Over dispersed Poisson distribution Gamma Exponential Dispersion Family EDF GLM GLM GLM de Jong Heller 2008 GLM GLM Ohlsson Johansson 200 GLM GAM de Jong Heller GLM GLM Y = Y Y 2 Y n ' EDF EY= μ X X X j X J X X = 2 X 2 j X 2 J X n X n j X n J β = β 0 β β J ' gμ= η η = Xβ 2 g X i j i = nj = J j X j i β 0 β β J X X 2 X J J J J - J J 2 β j j J - J j J μ = e β 0 j μ = e β 0 + β j = e β 0 e β j J j e β j β j = 0 j J β j > 0 β j < 0 X X 2 X J J Z Z 2 Z K K J + K GLM GLM 2 GLM J - J

4 92 36 { } lβ = ln Π n fy i β = n lnfy i β = n lncy i + y iθ i - aθ i 3 i = i = i = β j j = J 0 lβ η i = n y i - μ i β j i = l = n l θ i β j i = θ i η i 4 X'DY - μ= 0 θ i X η i j = 0 i θ i D i = n 0 η i θ i = η - i η i ( θ ) = η - i μ ( i i μ i θ ) = g'μ i a θ i - = g'μ i Vμ i - 6 i D g'μ i Vμ i - i = n 0 G g'μ i i = n 0 W g'μ i 2 Vμ i - i = n 0 D = WG5 X'WGY - μ= 0 7 gy i gy i gμ i + g'μ i y i - μ i 8 gy gμ+ GY - μ= Xβ + GY - μ X'WgY- X'WXβ 0 0 β β^ X'WX - X'WgY W 2. 4 GLM GLM saturated model ~ n l = lncy i + y i θ珘 i - a θ珘 i i = { } 2 l = y i - μ i θ i = y i - a'θ i = 0 3 θ i θ珘 i a' θ珘 i = y i deviance Δ = 2l ~ - l^ = 2 n i = { } y i θ珘 i - θ^ i- a θ珘 i + aθ^ i l ~ ^ ~ l^ l l Δ 0 Δ χ 2

5 3 93 Δ ~ χ 2 n - p 5 n p n - p Δ GLM LR Wald H 0 Cβ = r H Cβ r C r LR l^β^ 珓 l β 珘 LR = 2( l^ β^ - 珓 l β珘 ) 6 LR q χ 2 LR ~ χ 2 q 7 q C Wald Wald - β^ ~ N{ β X'WX } C β^ - r ~ N{ 0 CX'WX - C' } Wald 8 9 W = C β^ - r' { CX'WX - - C' } C β^ - r 20 W q χ 2 W ~ χ 2 q 2 q C C β j = r C j 0C = Cβ = r C 0 0 C = C J J + J 3 GLM C

6 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 89 Figure Log Death Rates at Ages 0-89 by Gender in China GLM 3. 2 GLM 2 GLM GLM Y i t i t μ i t 2 3 GLM 5

7 3 95 q i t i t n i t i t q i t Eq i ( ) t = E Y i t n i t = μ i t n i t 23 2 GLM ( ) g μ i t n i t ( ) = ln μ i t n i t = X i t β 24 Y i t ~ Poissonμ i t i = 0 lnμ i t = lnn i t + X i t β 25 t = X i t 89 i t β β = β 0 β β 89 β 04 ' = 05 q i t n i t q i t Y i t ~ NBμ i t κ i t lnμ i t = lnn i t + X i t β 27 Y i t μ i t μ i t + κ i t μ i t κ i t = e = e = e = e = e = e = e = e = GLM GLM Newton - Raphson Fisher IWLS GLM Wald SAS Wald P R Wald Z P P t μ i t 2 X X i t 05 i t 0

8 96 36 Table Parameter Estimates of two Regression Models β β

10 % % 5% 0% Table 2 Test Results of two Regression Models Null deviance Null deviance Residual deviance Residual deviance AIC Fisher 4 4 AIC AIC AIC 2 Null deviance Residual deviance

3 99 335 2 994 ~ 2009 0 ~ 89 2 994 ~ 2009 0 ~ 89 Figure 2 Fitted Log Death Rates at Ages 0-89 by Gender in China 994-2009 Using

11 ~ ~ ~ ~ 89 Figure 2 Fitted Log Death Rates at Ages 0-89 by Gender in China Using Negative Binomial Distribution B - GLM B B - 3 B B B ~ 89 κ 0

00 36 3 B - 994 ~ 2009 0 ~ 89 Figure 3 Log Death Rates at Ages 0-89 by Gender in China 994-2009 Using Negative Binomial Distribution and B - spline Graduation Table 3 3 B - Model Test Results Using

12 B ~ ~ 89 Figure 3 Log Death Rates at Ages 0-89 by Gender in China Using Negative Binomial Distribution and B - spline Graduation Table 3 3 B - Model Test Results Using Negative Binomial Distribution and B - spline Graduation B - Null deviance Null deviance Residual deviance Residual deviance AIC Fisher B GLM 994 ~ ~ 89 0 ~ 89 B -

13 3 0 GLM 4 B ~ 89 Figure 4 Log Age - Specific Death Rates and Log Smoothed Age - Specific Death Rates by Gender in China 2005 Using Negative Binomial Distribution and B - spline Graduation 4 B ~ Table 4 Log Death Rates and Log Smoothed Death Rates at Age 50 by Gender in China Using Negative Binomial Distribution and B - spline Graduation

14 ~ ~ GLM 90 GLM 90 3 R R GLM R GLM /References Miller N. D Elements of Graduation. Actuarial Society of America and American Institute of Actuaries. New York Gompertz B On the Nature of the Function Expressive of the Law of Human Mortality and on a New Mode of Determining Life Contingencies. The Royal Society of LondonPhilosophical Transactions. Series A Makeham W. M On the Law of Mortality and the Construction of Annuity Tables. Journal of the Institute of Actuaries and the Assurance Magazine Heligman L. Pollard J. H The Age Pattern of Mortality. Journal of the Institute of Actuaries Carriere J. F Parametric Models for Life Tables. Transactions of the Society of Actuaries Lee R. D. Carter L. R Modeling and Forecasting U. S. Mortality. Journal of the American Statistical Association Haberman S. Renshaw A. E Generalized Linear Models and Actuarial Science. Journal of the Royal Statistical Society Copas J. B. Haberman S Non - parametric Graduation Using Kernel Methods. Journal of the Institute of Actuaries

15 Wang J. L. Müller H. G. and Capra W Analysis of Oldest - Old MortalityLife Tables Revisited. The Annals of Statistics Wang J. L Smoothing Hazard Rates. Encyclopedia of Biostatistics2nd Edition. Willey Debón A. Montes F. and Sala R A Comparison of Nonparametric Methods in the Graduation of MortalityApplication to Data from the Valencia Region Spain. International Statistical Review da Rocha Neves C. and Migon H. S Bayesian Graduation of Mortality RatesAn Application to Reserve Evaluation. InsuranceMathematics and Economics Nelder J. A. and Wedderburn R. W. M Generalized Linear Models. Journal of the Royal Statistical SocietySeries A McCullagh J. and Nelder J. A Generalized Linear Models2nd Edition. New YorkChapman and Hall de Jong P. Heller G. Z Generalized Linear Models for Insurance Data. CambridgeCambridge University Press Ohlsson E. and Johansson B Non - life Insurance Pricing with Generalized Linear Models. European Actuarial A- cademy Series - Textbook. Springer - Verlag Berlin Heidelberg

~ 10 2 P Y i t = my i t W Y i t 1000 PY i t Y t i W Y i t t i m Y i t t i 15 ~ 49 1 Y Y Y 15 ~ j j t j t = j P i t i = 15 P n i t n Y

~ 10 2 P Y i t = my i t W Y i t 1000 PY i t Y t i W Y i t t i m Y i t t i 15 ~ 49 1 Y Y Y 15 ~ j j t j t = j P i t i = 15 P n i t n Y * 35 4 2011 7 Vol. 35 No. 4 July 2011 3 Population Research 1950 ~ 1981 The Estimation Method and Its Application of Cohort Age - specific Fertility Rates Wang Gongzhou Hu Yaoling Abstract Based on the