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1 c Piecewise Constant Intensity Model Piecewise Constant Intensity Model 1. the oldest-old Carey et al ; sibuyam@takachiho.ac.jp ; nob-hanayama@jcom.home.ne.jp

2 Mean residual life Ma #deaths older than x-axis Ma

3 Kaufmann2001 Kannisto1999 Osmond and Gardener (2.1) 1 (1 + γx/a) 1/γ, γ 0, F (x) =F (x; γ,a)= 1 exp ( x/a), γ =0,. 0 <x<, γ 0, 0 <x<a/( γ), γ < 0,. (2.1) GPrt (γ, a) γ =0 Coles2001 X GPrt(γ, a) E(X) =a/(1 γ),γ < 1, γ 1 h(x) = f(x) 1 F (x) = 1 a + γx, γ R, X x 0 X GPrt(γ, a) P {X x 0 y X x 0} = 1 F (x0 + y) 1 F (x 0) = 1+ γy a + γx 0 1/γ, y 0; γ 0, γ <0 y <a/( γ) x 0 X x 0 X x 0 GPrt(γ, a + γx 0) γ =0 mean residual life mean excess functionm(x 0) y(p; x 0)p x 0 1 m(x 0):=E(X x 0 X x a + 0)= γx0 1 γ, γ < 1, 1 F (x0 + y) y(p; x 0):= =(p γ 1)(a + γx 0)/γ, γ 0, γ, 1 F (x 0)

1) GPrt (γ, a) γ =0 Coles2001 X GPrt(γ, a) E(X) =a/(1 γ),γ < 1, γ 1 h(x) = f(x) 1 F (x) = 1 a + γx, γ R, X x 0 X GPrt(γ, a) P {X x 0 y X x 0} = 1 F

4 {} {} m(x 0) x 0 1 (2.1) {x j; j =1, 2,...,n} (2.1) γ <0 γ =0 ˆm. ˆm(x 0):= γ<0 n j=1 (x j x 0)I[x j x 0] n j=1 I[x j x 0], (I[true]=1, I[false] = 0). (2.2) F (x; η, ω) =1 (1 x/ω) η, 0 x<ω; η = 1/γ, ω = a/( γ), X/ω Be(1,η) E(X) =ω/(1 + η), m(x 0)=E(X x 0 X x 0)= α := = γ ω x0 (ω x0) = 1 γ 1+η γ x 0 + a 1 γ γ γ 1 γ = 1 X 1+η = E x0 X x 0 ω x 0 perseverance parameter h(x) = f(x) 1 F (x) = η º 1 x, ω ω ω< γ<0 h(x) γ<0 force of mortality Gompertz γ =0 Gompertz Aarssen and de Haan1994 (2.1) n (2.3) l = 1+γ γ n j=1 1 n log(1 + τx j) n log a, n j=1 1+γ γ log(1 + τx j)=γ, 1 n n j=1 x j 1+τx j = 1 τ, τ = γ/a,

2) F (x; η, ω) =1 (1 x/ω) η, 0 x<ω; η = 1/γ, ω = a/( γ), X/ω Be(1,η) E(X) =ω/(1 + η), m(x 0)=E(X x 0 X x 0)= α := = γ ω x0 (ω x0) = 1 γ 1+η γ x 0 + a 1 γ γ

5 τ γ <0 Smith /2 <γ Fisher n 2 1/a I(γ,a)=, γ > 1/2 (or η>2), (2γ +1)(γ +1) 1/a (γ +1)/a 2 Var((ˆγ, â)) = γ +1 γ +1 a, γ > 1/2 (orη>2), n a 2a 2 γ<0 ω = a/( γ) Var(ˆω) = a2 (γ +1) 2 (2γ +1) a2 nγ 4 nγ, 1 4 ω SD(ˆω) 1 = η, γ 0, n γ n γ = 1/2 (n log n) 1/2. 1/2 <γ< 1 γ 1 Smith1987 γ (i, j),i = 1,...,I; j =1,...,J i j (i, j),i= 100,...,120; j = 1970,...,2000 j i M ij 45 1 Lexis diagram, Keiding1999M ij 3 1.

1 4 ω SD(ˆω) 1 = η, γ 0, n γ n γ = 1/2 (n log n) 1/2. 1/2 <γ< 1 γ 1 Smith1987 γ 1 3. 3.1 (i, j),i = 1,.

6 V ij (3.1) M ij M i+1,j+1 =: D ij, k = j i M ij [i, i +1)j D ij D ij k j i i i +1 M ij =0 j j i i +1 W ij N ij N ij 4 N ij 2 q(k, i)

7 123 (N ij + N i,j+1)/ j i k = j i M ij P (V ij) (3.2) q(k, i) :=F k (i +1) F k (i) =F k (i) F k (i +1), k = j i, F k (t) GPrt (γ k,a k ) F k (t) k γ <0 (2.2) q(i) =F (i) F (i +1), F (t) =F (t; ω, η) := k l = k i=1 l ω = m i q(i) ω i=1 ηt F (t) = 1 t η 1, ω ω 2 ω m i log q(i), m i = M ij M i+1,j+1, (F (i) F (i +1)), l η = k i=1 η F (t) =F (t)log 1 t ω 1, t < 0, (1 t/ω) η, 0 t<ω, 0, ω t. m i (F (i) F (i +1)), q(i) η, 0 <t<ω, ω k j k + v, j+ w, v, w, 0,1 y (3.3) y = j + w (k + v) =(j k)+(w v) =j k + u, u = w v ( 1, 1) j k j k 1 j i V ij j i j i +1 i.i.d. piece

8 Piecewise Constant Intensity ModelPCI Keiding1990 PCI V rs 2 N 2 = {(i, j) :i, j N} M rs = ɛ 1,ɛ 2 {0,1,2} V rs = (r + ɛ 1, s+ ɛ 2), s = k + r; r, s =0, 3, 6,... (i,j) M rs W ij, j = k + i; r, s =0, 3, 6,... k(= r s =0, 3,...) k +1k 1, k +2 V rs λ rs. k h(x; k) =1/(a k + γ k x) V rs (3.4) λ rs = h(r +1, s r), r,s =0, 3, 6,... V rs x rs+ person-years M ij L r,s r,s λ drs rs exp( x rs+λ rs) λ drs rs exp( λ rs M ij), d rs = (i,j) M rs (i,j) M rs N ij M ij V ij (3.1) Keiding1975 (3.3) W ij N ij u y = j i + u u>0 k = j i u <0 k 1 (3.3) ky [y] =i k + v + y j j k + v + y<j+1 i y + v<i+1.

k h(x; k) =1/(a k + γ k x) V rs (3.4) λ rs = h(r +1, s r), r,s =0, 3, 6,.

9 125 [y] =i y (3.5) = j i y<i+1 v, ρ 0(k, i) := 1 0 F k (i +1 v)dv F k (i), F k (t) (3.2) ρ 0(k, i) (3.6) = j +1 i +1 v y<i+1, ρ 1(k, i) :=F k (i +1) 1 0 F k (i +1 v)dv, ρ 1(k, i) k 1 i j k k 1 ρ 1(k 1,i). P (V ij) (3.2) q(k, i) P (V ij) =R k ρ 1(k, i)+r k 1 ρ 0(k 1,i+1), R k 1,R k k 1 k (3.5), (3.6) : (γ =0) 1 1 F (t v)dv =1 exp( (t v)/a)dv =1 a exp 0 (γ <0) (3.7) F (t v)dv =1 ω 1+η 1+η ω +1 t ω = F (t)+o(ω 2 ), ω. 1 a 1 exp t, a 1+η ω t ω k 1,k 2 3 (N ij,n i,j+1,n i,j+2), k = j i, i = 0, 1,..., 3 (γ k 1,a k 1 ), (γ k,a k ), (γ k+1,a k+1 ) 2 i 4 i (N i,k+i log(r k ρ 0(k, i)) + N i,k+i+1 (log(r k ρ 1(k +1,i)) + log(r k+1 ρ 0(k +1,i)) +N i,k+i+2 log(r k+2 ρ 1(k +2,i))), R k,r k+1,r k+2 k, k +1,k+2 ρ 0(k, i) = 1 (i +1+η k ω k ) 1 i ηk (i +1 ω k ) 1 i +1 ηk, 1+η k ω k ω k ρ 1(k, i) = ω k i 1 i ηk ω k i + η k 1 i +1 ηk, 1+η k ω k 1+η k ω k

6) : (γ =0) 1 1 F (t v)dv =1 exp( (t v)/a)dv =1 a exp 0 (γ <0) (3.7) 1 0 0 F (t v)dv =1 ω 1+η 1+η ω +1 t ω = F (t)+o(ω 2 ), ω.

10 *** , , , , (3.2) D ij; j = k + i, N ij, N i+1,j, N i,j+1, N i+1,j D ij; j = k + i, i = 100, 101,...,N ij, N i+1,j, N i,j+1, N i+1,j * D ij N ij N i+1,j+1 compare death numbers, 1890 * + N[i,j] * %% + * % + - * + * + %- - % * + - % - % dm[i,j] 5. +: N(ij), : N(i +1,j), : N(i, j +1), : N(i +1,j+ 1)

2) D ij; j = k + i, N ij, N i+1,j, N i,j+1, N i+1,j+1 5 1880 D ij; j = k + i, i = 100, 101,.

12 Mean residual life, Male, (3.2) PCI γ γ =0 a/γ γ γ =

13 129 4.

14 Piecewise Constant Intensity Model 6. GPrt V rs; r =95, 98,...,113,s= 1980, 1983,...,1998 PCI (3.4) γ γ =0 ω = a/γ PCI γ >0

GPrt 50 1880 1892 1878 1896 5 V rs; r =95, 98,.

15 131 γ =0 5 N ij 3 j i j i Coles2001 gpd.fit p p γ ω Aarssen and de Haan , 80,..., , 91,...,94 1 γ =0 ω , α Kaufmann γ =0 γ<0,γ =0 Kaufmann 2001Aarssen and de Haan competing risks

Aarssen and de Haan1994 1877 1881 1971 1 1 1 1881, 80,...,77 1971 1 1 90, 91,.

16 E((1 + τx) r )=(τ/γ) 0 (1 + τx) r 1 1/γ I[1+τx 0]dx =1/(1 γr), r < 1/γ, τ = γ/a. E(1/(1 + τx)) = 1 E(log(1 + τx))/(γ +1)=1/(γ +1), γ > 1, E((X/(1 + τx)) 2 )=E((1 1/(1 + τx)) 2 /τ 2 )=2γ 2 /(τ 2 (2γ +1)(γ +1)). (2.3) n =1 l γ = 1 γ log 1+ γ 2 a x 1+γ aγ 2 l γ 2 = 2 γ 3 log 1+ γ a x l γ a = 1 a 2 2 l 2(1 + γ) = a2 a 3 x 1+γx/a 1+γ a 3 x 1+γx/a x 1+γx/a, l a = 1+γ x a 2 1+γx/a 1 a, x aγ 2 1+γx/a + 1+γ 2 x, a 2 γ 1+γx/a 2 x, 1+γx/a 2 x γx/a a, 2 + γ(1 + γ) a 4 x X (2.1) (γ,a) parametrization 2 6 γ tail indexextreme index η γ =0 γ γ a γ <

17 ω ω η α τ 7 3 γ, η, α 1 2 γ γ (γ,a) 8 ω τ 4 γ, a, ω, τ (ω,τ) 5 Aarssen, K. and de Haan, L On the maximal life span of humans, Mathematical Population Studies, 4, Carey,R.J.,Liedo,P.Orozco,D.andVaupel,W.J Slowing of mortality rates at older ages in large medfly cohorts, Science, 258, Coles, S An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London. Kannisto, V Trends in the mortality of the oldest-old, Statistics, Registries and Science: Experiences from Finland ed. J. Alho, , Statistics Finland. Kaufmann, E About the longevity of humans, Section 14.2, Statistical Analysis of Extreme ValueswrittenbyR.D.ReissandM.Thomas, 2nd ed., Birkhaueser, Basel. Keiding, N Maximum likelihood estimation in the birth-and-death process, Annals of Statistics, 3, Keiding, N Statistical inference in the Lexis diagram, Philosophical Transactions of the Royal Society of London, Series A, 332, Keiding, N Lexis diagram, Encyclopedia of Statistical Scienceseds.S.Kotz,C.B.Read andd.l.banks, Update Volume 3, , Wiley, New York. Osmond, C. and Gardener, M. J Age, period and cohort models applied to cancer mortality, Statistics in Medicine, 1, Smith, R. L Maximum likelihood estimation in a class of nonregular cases, Biometrika, 72, Smith, R. L Estimating tails of probability distributions, Annals of Statistics, 15,

An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London. Kannisto, V.1999.

18 134 Proceedings of the Institute of Statistical Mathematics Vol. 52, No. 1, (2004) Estimation of Human Longevity Distribution Based on Tabulated Statistics Masaaki Sibuya (Faculty of Business Management, Takachiho University) Nobutane Hanayama (Faculty of Art and Information, Shobi-gakuen University) (Age, period)-specific data for the oldest-old survivors and deaths are analyzed using the extreme value theory and the limit of longevity dstribution is discussed. The data were obtained from National Oldest-old Survivors List and Population Movement Statistics by the Ministry of Health and Labor in Japan. In applying the theory of extreme value statistics of continuous variables to the analysis of (age, period)-specific tabulated data, we propose a procedure for applying the multinomial distribution model based on probabilities calculated from the generalized Pareto distribution, and another procedure using a continuous pseudo random sample generated by adding random numbers. Furthermore, the piecewise constant intensity model on the Lexis diagram, widely used for such data, is also applied. ML estimates show the finite upper limit of longevity distribution. Key words: Cohort analysis, maximum likelihood estimation, mean residual life function, national oldest-old survivors list, Piecewise Constant Intensity Model, population movement statistics.

Information, Shobi-gakuen University) (Age, period)-specific data for the oldest-old survivors and deaths are analyzed using the extreme value theory and the limit of longevity dstribution is

1 Tokyo Daily Rainfall (mm) Days (mm)

1 Tokyo Daily Rainfall (mm) Days (mm) ( ) r-taka@maritime.kobe-u.ac.jp 1 Tokyo Daily Rainfall (mm) 0 100 200 300 0 10000 20000 30000 40000 50000 Days (mm) 1876 1 1 2013 12 31 Tokyo, 1876 Daily Rainfall (mm) 0 50 100 150 0 100 200 300 Tokyo,