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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36 3 202 5 Vol. 36No. 3 May 202 89 Population Research * GLM GLM 0 ~ 89 B - GLM 20 90 B - 30007 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 995-2006 and Statistical Yearbooks of China' s Population and Employment 2007-200 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 30007.Emailzhlz@ nankai. edu.cn. * NKZXTD0

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 Migon2007 94 958 980 200 CL90-93 8 ~ 80 CL00-03 CL90-93 995 ~ 2006 2007 ~ 200

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 Heller2008 2. 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 2. 2 2 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 - 2. 3 GLM GLM 2 GLM J - J

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 9 9 7 X'WgY- X'WXβ 0 0 β β^ X'WX - X'WgY W 2. 4 GLM 2. 4. 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 4 5 4 l ~ ^ ~ l^ l l Δ 0 Δ χ 2

3 93 Δ ~ χ 2 n - p 5 n p n - p Δ 0 2. 4. 2 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 = 0 0 2 Cβ = r C 0 0 C = 0 0 0 0 0 0 0 22 C J J + J 3 GLM C

94 36 3. 995 ~ 2006 2007 ~ 200 994 ~ 2009 90 0 ~ 89 997 996 85 994 995 997 998 4 85 ~ 89 996 996 85 ~ 89 4 85 ~ 89 2 994 ~ 2009 6 0 ~ 89 994 ~ 2009 0 ~ 89 Figure Log Death Rates at Ages 0-89 by Gender in China 994-2009 GLM 3. 2 GLM 2 GLM GLM 3 3. 2. Y i t i t μ i t 2 3 GLM 5

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 = 2 6 994 2009 X i t 89 i t β β = β 0 β β 89 β 04 ' 26 + 89 + 5 = 05 q i t n i t q i t 3. 2. 2 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 = 0 3. 3 3. 3. 2 994 0-3. 674 e = 0. 0254 e - 3. 2639 = 0. 03822005 50 e - 3. 674-0. 3292 -. 3237 = 0. 0049 e - 3. 2639-0. 4396-2. 650 = 0. 0028 e - 3. 729 = 0. 0240 e - 3. 3793 = 0. 0342005 50 e - 3. 729-0. 2750 -. 309 = 0. 0049 e - 3. 3793-0. 4448-2. 0855 = 0. 0027 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

96 36 Table Parameter Estimates of two Regression Models β β 0-3. 674-3. 2639-3. 729-3. 3793-2. 2539-2. 4987-2. 2530-2. 4878 2-2. 6494-2. 990-2. 583-2. 8860 3-2. 9404-3. 3278-2. 8690-3. 2486 4-3. 975-3. 6899-3. 39-3. 5887 5-3. 3758-3. 9256-3. 3028-3. 7767 6-3. 538-4. 627-3. 4538-4. 03 7-3. 5650-4. 3348-3. 5665-4. 2422 8-3. 5758-4. 4296-3. 522-4. 2765 9-3. 7092-4. 576-3. 684-4. 427 0-3. 7085-4. 5400-3. 5906-4. 3470-3. 8055-4. 6305-3. 7204-4. 486 2-3. 8468-4. 5697-3. 7800-4. 3809 3-3. 8467-4. 5653-3. 7606-4. 350 4-3. 7627-4. 505-3. 6287-4. 3292 5-3. 6377-4. 3726-3. 538-4. 285 6-3. 5506-4. 3482-3. 3636-4. 220 7-3. 3944-4. 2568-3. 204-4. 0365 8-3. 229-4. 0538-3. 0205-3. 7748 9-3. 028-4. 050-2. 873-3. 7023 20-2. 950-3. 874-2. 7923-3. 672 2-2. 9745-3. 8605-2. 7405-3. 6680 22-2. 9056-3. 774-2. 7036-3. 5334 23-2. 9050-3. 7437-2. 8264-3. 4620 24-2. 8553-3. 665-2. 7267-3. 3997 25-2. 833-3. 6460-2. 7259-3. 4247 26-2. 8563-3. 6842-2. 728-3. 4999 27-2. 88-3. 6375-2. 7343-3. 4548 28-2. 7946-3. 6360-2. 6435-3. 4364 29-2. 750-3. 5742-2. 5645-3. 347 30-2. 6603-3. 524-2. 53-3. 2867 3-2. 6633-3. 5266-2. 4536-3. 3897 32-2. 5920-3. 4629-2. 4953-3. 297 33-2. 5975-3. 4936-2. 4494-3. 3328 34-2. 509-3. 443-2. 3658-3. 2670 35-2. 4226-3. 3335-2. 298-3. 977 36-2. 469-3. 3759-2. 2755-3. 254 37-2. 3266-3. 2876-2. 398-3. 069 38-2. 2734-3. 255-2. 782-3. 0864 39-2. 205-3. 275-2. 00-2. 8957 40-2. 0806-3. 047 -. 9897-2. 934 4-2. 0865-3. 0466 -. 9677-2. 8222 42 -. 9879-2. 9397 -. 927-2. 8262

3 97 43 -. 970-2. 892 -. 8666-2. 79 44 -. 8990-2. 8060 -. 7650-2. 642 45 -. 7838-2. 690 -. 7293-2. 5653 46 -. 7377-2. 6256 -. 6339-2. 4708 47 -. 6434-2. 5243 -. 5364-2. 3925 48 -. 5749-2. 4393 -. 4435-2. 2649 49 -. 4737-2. 3080 -. 354-2. 74 50 -. 3237-2. 650 -. 309-2. 0855 5 -. 3262-2. 584 -. 994-2. 0024 52 -. 2064-2. 029 -. 554 -. 8968 53 -. 505 -. 956 -. 334 -. 8260 54 -. 043 -. 832-0. 9806 -. 7430 55-0. 9405 -. 757-0. 8560 -. 6768 56-0. 8928 -. 7076-0. 7946 -. 5887 57-0. 7586 -. 5849-0. 7567 -. 4855 58-0. 6802 -. 4967-0. 5636 -. 32 59-0. 5342 -. 3537-0. 436 -. 2320 60-0. 3659 -. 844-0. 379 -. 0985 6-0. 3506 -. 673-0. 2898 -. 0869 62-0. 232 -. 0379-0. 299-0. 8978 63-0. 790-0. 9673-0. 0487-0. 7955 64-0. 0378-0. 8305 0. 0269-0. 690 65 0. 0853-0. 7220 0. 0939-0. 588 66 0. 27-0. 6899 0. 2285-0. 5007 67 0. 2469-0. 545 0. 2863-0. 43 68 0. 3835-0. 3958 0. 425-0. 2685 69 0. 5273-0. 2459 0. 5356-0. 008 70 0. 6696-0. 027 0. 6753-0. 007 7 0. 7047-0. 0600 0. 78 0. 060 72 0. 8526 0. 07 0. 8507 0. 2032 73 0. 98 0. 599 0. 9208 0. 2695 74 0. 9909 0. 2495 0. 9730 0. 380 75. 0827 0. 3533. 082 0. 562 76. 734 0. 4474. 990 0. 5438 77. 2499 0. 536. 302 0. 7062 78. 380 0. 6727. 4025 0. 7823 79. 5270 0. 8209. 4890 0. 8557 80. 667 0. 9762. 5766. 024 8. 7002. 0323. 7504. 234 82. 8005. 333. 7406. 78 83. 8743. 255. 8736. 3246 84. 950. 3058. 9322. 3967 85 2. 064. 3707. 9998. 4466 86 2. 0656. 450 2. 054. 5493

98 36 87 2. 647. 5422 2. 753. 5808 88 2. 2463. 6397 2. 2605. 6923 89 2. 38. 7363 2. 390. 877 995 0. 0202-0. 033 0. 0202-0. 052 996-0. 027-0. 098-0. 005-0. 0757 997-0. 0426-0. 005-0. 0363-0. 058 998-0. 0747-0. 73-0. 0670-0. 654 999-0. 283-0. 867-0. 97-0. 770 2000-0. 22-0. 790-0. 402-0. 2083 200-0. 864-0. 283-0. 74-0. 2783 2002-0. 638-0. 2528-0. 555-0. 2480 2003-0. 254-0. 3070-0. 2353-0. 2999 2004-0. 2855-0. 3608-0. 2679-0. 3507 2005-0. 3292-0. 4396-0. 2750-0. 4448 2006-0. 4888-0. 5829-0. 463-0. 5724 2007-0. 485-0. 5468-0. 4609-0. 5457 2008-0. 4504-0. 5462-0. 496-0. 55 2009-0. 5639-0. 6550-0. 5399-0. 6566 05 0. % % 5% 0% 3. 3. 2 2 2 2 Table 2 Test Results of two Regression Models Null deviance 0253755 900579. 0 28287. 6 200520. 2 Null deviance 439 439 439 439 Residual deviance 3443 33. 4 662. 4 726. 8 Residual deviance 335 335 335 335 AIC 342 0670 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 Negative Binomial Distribution 3. 3. 3 B - GLM B - 997200020032006 5 085 3 5 3 B - 3 B - 2 3 335 B - 42 3. 3. 4 2005 B - 4 2005 0 ~ 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 Negative Binomial Distribution and B - spline Graduation B - Null deviance 65224. 4 47759 Null deviance 439 439 Residual deviance 675. 2 724 Residual deviance 42 42 AIC 447 0762 Fisher 4 3. 3. 5 50 B - 4 6 4 GLM 994 ~ 2009 0 ~ 89 0 ~ 89 B -

3 0 GLM 4 B - 2005 0 ~ 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 - 994 ~ 2009 50 Table 4 Log Death Rates and Log Smoothed Death Rates at Age 50 by Gender in China 994-2009 Using Negative Binomial Distribution and B - spline Graduation 994 995 996 997 998 999 2000 200 2002 2003 2004 2005 2006 2007 2008 2009-5. 30-5. - 5. 9-5. 36-5. 24-5. 33-5. 2-5. 0-5. 26-5. 33-5. 20-5. 3-6. 0-5. 38-5. 2-5. 6-5. 02-5. 00-5. 02-5. 05-5. 0-5. 3-5. 6-5. 8-5. 20-5. 23-5. 27-5. 33-5. 39-5. 45-5. 50-5. 53-5. 70-5. 42-5. 43-5. 94-5. 6-5. 80-5. 6-5. 70-5. 89-5. 89-5. 78-5. 82-6. 50-5. 82-6. 49-6. 07-5. 44-5. 46-5. 50-5. 55-5. 60-5. 64-5. 65-5. 67-5. 70-5. 75-5. 83-5. 90-5. 95-5. 99-6. 04-6. 09

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