地域総合研究第40巻第1号

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1 * abstract This paper attempts to show a method to estimate joint distribution for income and age with copula function. Further, we estimate the joint distribution from National Survey of Family Income and Expenditure in In that real data, there exists structural difference of income distribution between the people over sixty and the others. The difference is explained by changing the functional form of copula between those generations. 1 Singh-Maddala 2 n 2 C : [0, 1] [0, 1] [0, 1] [2] 1. C(u, v) u v キーワード: 所 得 分 布,コピュラ, 統 計 分 析 * 本 学 経 済 学 部 講 師 35

2 地 域 総 合 研 究 第 40 巻 第 1 号 (2012 年 ) 2. C(0, v) = C(u, 0) = 0, C(1, v) = v, C(u, 1) = u 3. C(u 2, v 2 ) C(u 2, v 1 ) C(u 1, v 2 ) + C(u 1, v 1 ) 0 ( u 1, u 2, v 1, v 2 [0, 1] u 1 u 2, v 1 v 2 ) (x, y) (F(x), G(x)) C(F(x), G(x)) I(x) A(x) C(I(x), A(x)) I(x) 2 2 f (x θ) = ax ap 1 b ap B(p, q) {1 + (x/b) a p+q, x > 0 (1) } θ 2 θ = (α, β, p, q) L(θ) = p k (θ) = K N! k=1 K n n p k(θ) k (2) k! k=1 uk l k f (x θ)dx N K (n k, l k, u k ) k A(x) k m k = (l k + u k )/2 f (x) = 1 ( ) Nh x mk n k K (3) h k K( ) h [3] 36

3 コピュラによる 所 得 と 年 齢 の 同 時 分 布 の 推 定 1 2 α β p q (0.0235) (1.532) (0.0233) (0.0311) F(a, b) = C(I(a), A(b)) = Φ 2 (Φ 1 (u), Φ 1 (v) ρ) (4) u = v = a b I(x)dx A(x)dx Φ 2 (x, y ρ) ρ *1 Φ 1 (x) i j (i, j) n i,j, (i, j) p i,j K I, K A n = (n 1,1,..., n 1,KA, n 2,1,..., n 2,KA,..., n KI,1,..., n KI,K A ) p(n) = N! K I i=1 K A j=1 n i,j! K I K A i=1 j=1 n p i,j i,j L(θ) = N! K I i=1 K A j=1 n ij! K I K A i=1 j=1 n p i,j(θ) ij (5) (i, j) p i,j (θ) p i,j (θ) = F(ui I, ua j ) F(uI i 1, ua j ) F(uI i, ua i 1 ) + F(uI i 1, ua i 1 ) (6) ρ ( ) x mk K = 1 exp { (x m k) 2 } h 2π 2h 2 *1 Φ 2 (x, y ρ) [1] 37

4 地 域 総 合 研 究 第 40 巻 第 1 号 (2012 年 ) Density Income 1 2 Density Age 2 38

5 コピュラによる 所 得 と 年 齢 の 同 時 分 布 の 推 定 2 ) ρ ( ) ρ ˆρ = p i,j 4 p i,j p i,j

6 地 域 総 合 研 究 第 40 巻 第 1 号 (2012 年 ) 5 ) ρ 1 ρ ( ) ( ) 6 ρ (ρ 1, ρ 2 ) 60 I U60 ρ = ρ 1 p i,j (θ a < 60, ρ = ρ 1 ) = p i,j(θ ρ = ρ 1 )) i p i,j (θ ρ = ρ 1 ), i I U60 (7) 60 I O60 ρ = ρ 2 p i,j (θ a 60, ρ = ρ 2 ) = p i,j(θ ρ = ρ 2 )) i p i,j (θ ρ = ρ 2 ), i I O60 (8) p(ρ 1, ρ 2 ) = p 1 p i,j (θ a < 60, ρ = ρ 1 ) + p 2 p i,j (θ a 60, ρ = ρ 2 ) (9) p i,j (θ) p 1 60 p 2 60 *2 ( ˆρ 1, ˆρ 2 ) = ( 0.018, 0.373) ( 5 ) 6 p i,j * (p 1, p 2 ) 40

7 コピュラによる 所 得 と 年 齢 の 同 時 分 布 の 推 定 [1] Genz, A. "Numerical Computation of Rectangular Bivariate and Trivariate Normal and t Probabilities", Statistics and Computing, Vol. 14, No. 3, pp , [2] Jondeau, E., Ser-Huang Poon and Michael Rockinger, Financial Modeling under Non-Gaussian Distributions, Springer, [3]

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