21世紀の統計科学 <Vol. III>

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1 21 III HP, ( ), 1 tatsuya@e.u-tokyo.ac.jp 63

2 1 (Linear Mixed Model, LMM) (Best Linear Unbiased Predictor, BLUP) C.R. Henderson 50 LMM (Generalized Linear Mixed Model, GLMM) LMM LMM (Empirical Best Linear Unbiased Predictor, EBLUP) LMM LMM LMM 64

3 EBLUP EBLUP LMM 2 LMM BLUP 3 EBLUP 4 LMM LMM GLMM LMM GLMM (1992), McCulloch and Searle (2001), McCulloch (2003), (2007), Searle, Casella and McCulloch (1992), Demidenko (2004), Rao (2003) [1]. Battese, Harter and Fuller (1988) k ( ) k 250h (segment) n i i j y ij, LANDSAT, 0.45h 65

4 (picture element, ), k i j, x 1ij, x 2ij y i (x 1ij, x 2ij ) y ij = x ijβ + u ij, i = 1,..., k, j = 1,..., n i x ij = (1, x 1ij, x 2ij ), β = (β 0, β 1, β 2 ) x ijβ = β 0 + x 1ij β 1 + x 2ij β 2 u ij y ij x ij v i e ij u ij = v i + e ij (2.1) v i v i v i v i v i, e ij v i N (0, σ 2 v), e ij N (0, σ 2 e) y ij = x ijβ + v i + e ij, i = 1,..., k, j = 1,..., n i (2.2) σv, 2 σe 2 σv 2 σ2 e β (2.2) (LMM) y ij (Variance Component Model) (2.2) (Nested Error Regression Model) [2]. x ij, β 3 1 p 1 66

5 y i = y i1., y = y 1., x i = x i1. y ini, X = x 1. y k, β = β 0. x in i x k β p 1, e y e i = (e i1,..., e ini ), e = (e 1,..., e k ), 1 n i 1 j ni block diag( ) Z = block diag(j n1,..., j nk ) v = (v 1,..., v k ) (2.2) N = k i=1 n i y =Xβ + Zv + e, (2.3) v N (0, σ 2 vi k ), e N (0, σ 2 ei N ) y i Cov (y i ) = Σ i (σ 2 e, σ 2 v) = σ 2 ei ni + σ 2 vj ni I ni n i n i, J ni = j ni j n i 1 n i n i y Cov (y) = Σ(σ 2 e, σ 2 v) = block diag(σ 1 (σ 2 e, σ 2 v),..., Σ k (σ 2 e, σ 2 v)) v i Cov (y i ) = σei 2 ni y i σei 2 ni + σvj 2 ni (MCMC) 67

6 [3]. (2.3) y =Xβ + Zv + e, (2.4) v N q (0, G), e N N (0, R) y N 1 X N p Z N q y Cov (y) = Σ = R + ZGZ (2.5) G, R α = (α 1,..., α m ) Σ = Σ(α) 2.2 (BLUP) [1] BLUP. (2.4) β v v G, R v y (BLUP) v β β Henderson (1950) ( ) ( ) X R 1 X Z R 1 X X R 1 Z Z R 1 Z + G 1 ) ( β v = X R 1 y Z R 1 y (2.6) β = (X Σ 1 X) X Σ 1 y, v = GZ Σ 1 (y X β) (2.7) (X Σ 1 X) X X Σ 1 X β β 2 (GLS) a R p, b R q µ = a β + b v BLUP µ = a β + b GZ Σ 1 (y X β) (2.8) 68

7 (2.6) (2.7) (2.6) 2 Z R 1 X β + (Z R 1 Z + G 1 ) v = Z R 1 y v = (Z R 1 Z + G 1 ) 1 Z R 1 (y Xβ) (2.9) (Z R 1 Z + G 1 ) 1 Z R 1 (Z R 1 Z + G 1 ) 1 Z R 1 =GZ R 1 G { (Z R 1 Z + G 1 ) G 1} (Z R 1 Z + G 1 ) 1 Z R 1 =GZ R 1 GZ R 1 Z(Z R 1 Z + G 1 ) 1 Z R 1 =GZ { R 1 R 1 Z(G 1 + Z R 1 Z) 1 Z R 1} =GZ Σ 1 Σ 1 = (ZGZ + R) 1 = R 1 R 1 Z(G 1 + Z R 1 Z) 1 Z R 1 (2.10) (2.9) (2.7) v v (2.6) 1 X R 1 X β + X R 1 Z v = X R 1 y X R 1 X β + X R 1 ZGZ Σ 1 (y X β) = X R 1 y X R 1 (Σ ZGZ )Σ 1 X β = X R 1 (Σ ZGZ )Σ 1 y Σ = ZGZ + R R 1 (Σ ZGZ ) = I, X Σ 1 X β = X Σ 1 y (2.7) β [2]. (2.6) y v G 1/2 R 1/2 { ( exp 1 2 v y Xβ Zv ) ( G R 1 ) ( v y Xβ Zv )} 69

8 exp{ } ( 2) h(β, v) = v G 1 v + (y Xβ Zv) R 1 (y Xβ Zv) β v β, v h(β, v) β h(β, v) v = 2X R 1 (y Xβ Zv), =2G 1 v 2Z R 1 (y Xβ Zv), h(β, v)/ β = 0, h(β, v)/ v = 0 (2.6) 1 y v (y, v) ( ) Σ ZG Cov (y, v) = GZ (2.11) G y v E[v y] = GZ Σ 1 (y Xβ),, y v ( v y N q GZ Σ 1 (y Xβ), G GZ Σ 1 ZG ) (2.10) y y N N (Xβ, Σ) { Σ 1/2 exp 1 } 2 (y Xβ) Σ 1 (y Xβ) (2.12) β 2 β v GZ Σ 1 (y Xβ) β GZ Σ 1 (y X β) v 70

9 2 (2.11) EM [3] BLUP. 2.1 (2.2), µ i = x iβ + v i BLUP x i = n i j=1 x ij/n i G(σ 2 v) = σ 2 vi k, Σ i (σ 2 e, σ 2 v) = σ 2 ei ni + σ 2 vj ni, Σ(σ 2 e, σ 2 v) = block diag(σ 1 (σ 2 e, σ 2 v),..., Σ k (σ 2 e, σ 2 v)) Σ 1 i = 1 σ 2 e ( I ni ) σv 2 J σe 2 + n i σv 2 ni θ = σ 2 v/σ 2 e µ i BLUP µ i (θ) (2.8) µ i (θ) = x i β(θ) + θn { } i y 1 + θn i x i β(θ) i (2.13) y i = n i j=1 y ij β GLS { k ( β(θ) = xi x i n2 i θ 1 + n i θ x ) } ix 1 i i=1 k i=1 ( xi y i n iθ 1 + n i θ x iy i ) 2.3 (2.4) 71

10 (2.2) µ i = x iβ + v i y i n i 1 5 (2.13) BLUP µ i (θ) y i x i β(θ) y i n i y i x i β(θ) y i BLUP n i θ y i x i β(θ) n i BLUP BLUP [1]. v i β = 0 µ i y i v i (y i, v i ) ( ) σv 2 + σ 2 Cov (y i, v i ) = e/n i σv 2 y i v i E[v i y i ] = θn i (1 + θn i ) 1 (y i x iβ) y i x iβ y i [2]. (2.2) y i E[y i ] = x iβ i β β y 1,..., y k β(θ) y i σ 2 v σ 2 v 72

11 Efron and Morris (1975) 2.4 (EBLUP) [1] (ML) (REML) (2.4) G, R α = (α 1,..., α m ) Σ(α) = R(α) + ZG(α)Z BLUP (2.8) µ(α) = a β(α)+b G(α)Z {Σ(α)} 1 {y X β(α)} α α µ( α) (EBLUP) α (Maximum Likelihood, ML) (Restricted Maximum Likelihood, REML) y (2.12) y N N (Xβ, Σ(α)) ML β GLS β(α) α ML log Σ(α) + (y X β(α)) Σ(α) 1 (y X β(α)) X r K K X = 0 N (N r) K y N N r (0, K Σ(α)K) REML log K Σ(α)K + y K(K Σ(α)K) 1 K y REML P (α) = Σ(α) 1 Σ(α) 1 X { X Σ(α) 1 X } X Σ(α) 1 y P (α)y =(y X β(α)) Σ(α) 1 (y X β(α)), P (α) =K(K Σ(α)K) 1 K ( / α i ) log Σ =tr (Σ 1 Σ/ α i ), P / α i = P ( Σ/ α i )P, ( / α i ) log K ΣK =tr (P Σ/ α i ) 73

12 ML REML ( ) 1 Σ(α) [ML] tr Σ(α) = y P (α) Σ(α) P (α)y (2.14) α i α ( i [REML] tr P (α) Σ(α) ) = y P (α) Σ(α) P (α)y (2.15) α i α i y P (α){ Σ(α)/ α i }P (α)y =(y X β(α)) Σ(α) 1 { Σ(α)/ α i }Σ(α) 1 (y X β(α)) = (y X β(α)) { Σ(α) 1 / α i }(y X β(α)) ML REML McCulloch and Searle (2001) 6.10 REML ML REML 3.1 [2]. ML, REML (2.14), (2.15) (2.2) n 1,..., n k σ 2 e ˆσ 2UB e = S 1 N k p + λ, S 1 = k n i } 2 {(y ij y i ) (x ij x i ) β1 i=1 (2.16) p λ k ni i=1 j=1 (x ij x i )(x ij x i ) (2.2) λ = 1, λ = 0 β 1 k ni i=1 j=1 {(y ij y i ) (x ij x i ) β} 2 β y 1,..., y k ˆσ e 2UB y i N (x iβ, σe/n 2 i + σv), 2 i = 1,..., k, σv 2 III β 0 = (X X) 1 X y S = (y X β 0 ) (y X β 0 ) N = k i=1 n i, N = N tr (X X) 1 k i=1 n2 i x i x i S E[S] = (N p)σ2 e + N σv 2 σv 2 j=1 ˆσ 2UB v = N 1 {S (N p)ˆσ e 2UB } 74

13 σv 2 ˆσ2UB v Kubokawa (2000) σe, 2 σv 2 θ = σv/σ 2 e 2 { 1 { S ˆθ = max N ˆσ e 2UB (N p) }, 1 } k 2/3 (2.17) ˆθ (2.13) EBLUP µ i (ˆθ) EBLUP EBLUP EBLUP EBLUP 3.1 Fay and Herriot (1979) y i = x iβ + v i + e i, i = 1,..., k, (3.1) e i e i N (0, σe/n 2 i ) (2.2) Fay-Herriot σe 2 σe 2 (2.2) y i = n i j=1 y ij/n i, x i = n i j=1 x ij/n i y i, x i 75

14 y = (y 1,..., y k ), X = (x 1,..., x k ), e = (e 1,..., e k ) y =Xβ + v + e, (3.2) v N (0, σ 2 vi k ), e N (0, D) D D = diag (σ 2 e/n 1,..., σ 2 e/n k ) (3.1) σ 2 v θ = σ2 v/σ 2 e (y 1,..., y k ) θ ˆθ = ˆθ(y 1,..., y k ) µ i = x iβ + v i EBLUP (2.13) ˆθ µ i (ˆθ) = y i ˆγ i (ȳ i x i β(ˆθ)), ˆγ i = γ i (ˆθ) = (1 + n iˆθ) 1 (3.3) β GLS ( k β(θ) = i=1 n i x i x ) 1 k i 1 + n i θ i=1 n i x i y i 1 + n i θ (3.4) θ ML, REML (2.14), (2.15) [ML] [REML] σ 2 e σ 2 e k i=1 k i=1 n i k 1 + n i θ = i=1 n i 1 + n i θ σ2 etr = n 2 i {y i x i β(θ)} 2 ( k i=1 (1 + n i θ) 2 n i x i x ) 1 k i n 2 i x i x i 1 + n i θ i=1 (1 + n i θ) 2 k n 2 i {y i x i β(θ)} 2 i=1 (1 + n i θ) 2 0 ˆθ ML, ˆθ REML Fay and Herriot (1979) [FH] σ 2 e(k p) = k n i {y i x i β(θ)} 2 i=1 1 + n i θ 76

15 ˆθ F H β 2 = ( k j=1 n jx j x j) 1 k j=1 n jx j y j S 2 = k i=1 n i(y i x i β 2 ) 2 (2.16) [TR] { 1 {S ˆθT R 2 = max (k p) } 1 }, n σe 2 k 2/3 n = N tr ( k n i x i x i) 1 i=1 k n 2 i x i x i n 1 = = n k = n θ ML n 1 max{n k i=1 {y i x i β(θ)} 2 /(kσe) 2 1, 0}, REML n 1 max{n k i=1 {y i x i β(θ)} 2 /((k p)σe) 2 1, 0} REML FH REML ˆθ T R i=1 3.2 EBLUP EBLUP EBLUP µ i = x iβ + v i M i (θ, µ i (ˆθ)) = E [{ µ i (ˆθ) µ i } 2 ] /σ 2 e (3.5) (Mean Squared Error, MSE) MSE EBLUP y i N (x iβ, σ 2 e/n i + σ 2 v) Stein EBLUP MSE (2007) Datta, Kubokawa, Rao and Molina (2011) MSE MSE n i k k ˆθ 77

16 Bias θ (ˆθ) = E[ˆθ θ], V ar θ (ˆθ) = E[(ˆθ E[ˆθ]) 2 ] Bias θ (ˆθ) = O p (k 1 ) ˆθ ML, REML, FH, TR g 1i (θ) =n 1 i n 1 i γ i (θ), { k g 2i (θ) = {γ i (θ)} 2 x i j=1 g 3i (θ) =n i {γ i (θ)} 3 V ar θ (ˆθ), γ j n j x j x j} 1xi, EBLUP MSE k M i (θ, µ i (ˆθ)) = g 1i (θ) + g 2i (θ) + g 3i (θ) + O(k 3/2 ) MSE MSE { 2 M i U (ˆθ) = g 1i (ˆθ) + g 2i (ˆθ) + 2g 3i (ˆθ) Biasˆθ(ˆθ) γ i (ˆθ)} (3.6) E[ M U i (ˆθ)] = M i (θ, µ i (ˆθ)) + O(k 3/2 ) θ ˆθ ML, ˆθREML, ˆθF H, ˆθT R V ar θ (ˆθ ML ) = V ar θ (ˆθ REML ) = 2/ k i=1 (n iγ i ) 2 + O(k 3/2 ), V ar θ (ˆθ F H ) = 2k/( k i=1 n iγ i ) 2 + O(k 3/2 ), V ar θ (ˆθ T R ) = 2 k i=1 γ 2 i /N 2 + O(k 3/2 ) Bias θ (ˆθ REML ) = Bias θ (ˆθ T R ) = O(k 3/2 ), Bias θ (ˆθ ML ) = tr ( i n iγ i x i x i) 1 i (n iγ i ) 2 x i x i i (n iγ i ) 2 + O(k 3/2 ), Bias θ (ˆθ F H ) =2 k i (n iγ i ) 2 ( i n iγ i ) 2 ( i n iγ i ) 3 + O(k 3/2 ) MSE EBLUP MSE M i (θ, µ i (ˆθ)) ˆθ g 3i (θ) V ar θ (ˆθ) ˆθ MSE V ar θ (ˆθ ML ) = V ar θ (ˆθ REML ) V ar θ (ˆθ F H ) FH ML, REML MSE (3.6) M U i (ˆθ) Bias(ˆθ) = 0 θ REML 78

17 Datta et al. (2011), Rao (2003), Datta, Rao and Smith (2005), Kubokawa (2011b) Butar and Lahiri (2003), Hall and Maiti (2006a,b), Kubokawa and Nagashima (2011) 3.3 EBLUP EBLUP i µ i = x β +v i σ 2 e µ i µ i y i y i µ i N (µ i, σ 2 e/n i ) I i : y i ± z α/2 σ 2 e /n i (3.7) z α/2 α/2 1 α n i y i (3.1) µ i µ i N (0, σ 2 vi k ) µ i γ i = (1 + n i θ) 1 µ B i (β, θ) = x iβ + (1 γ i )(y i x iβ) y i µ i µ i y i N ( µ B i (β, θ), (σ 2 e/n i )(1 γ i ) ) µ i 1 α I B i (β, θ) : µ B i (β, θ) ± z α/2 (σ 2 e /n i )(1 γ i ) β, θ ˆθ y 1,..., y k θ β (3.4) 2 β(ˆθ) µ i µ EB i (ˆθ) = x i β(ˆθ) + (1 ˆγ i ) ( ) y i x 1 i β(ˆθ), ˆγ i = (1 + n iˆθ) 79

18 I B i (β, θ) Ii EB (ˆθ) : µ EB i (ˆθ) ± z α/2 (σ 2 e /n i )(1 ˆγ i ) µ EB i (ˆθ) y i 1 α k = 50, p = 3, 1 α = 0.95 β, n i, x i θ 1 Ii EB (ˆθ) 95% 0.99 I i * I i EB I i AEB 0.98 I i AEB I i * I i EB θ 1: I i, IEB i, Ii AEB 3.2 k 1 α, n 1 = = n k σe 2 Basu, Ghosh and Mukerjee (2003) n 1,..., n k σe 2 - (2005), Kubokawa (2010) Basu et al. (2003), z α/2 z α/2 {1 + (2k) 1 h(ˆθ)} I AEB i : µ EB i (ˆθ) ± z α/2 [1 + (2k) 1 h(ˆθ) ] (σ 2 e/n i )(1 ˆγ i ) (3.8) 80

19 h(ˆθ) h(ˆθ) = kn iˆγ i 2 Biasˆθ(ˆθ) + (1 + z 2 kn 2 i ˆγ i 4 1 ˆγ α/2) i 4(1 ˆγ i ) V arˆθ(ˆθ) 2 + kn iˆγ i 2 { x i 1 ˆγ i ( k j=1 n j x j x ) (ˆθ)} 1xi j + 2n iˆγ i V arˆθ 1 + n j ˆθ (3.9) Bias θ (ˆθ) = O p (k 1 ), ˆθ/ y i = O p (k 1 ) P [µ i I AEB i ] = 1 α + o(k 1 ), (k ) (3.1) ˆθ T R Bias θ (ˆθ T R ) = o(k 1 ), V ar θ (ˆθ T R ) = 2 k i=1 (1 + n iθ) 2 /N 2 + o(k 1 ) (3.9) n 1 = = n k = n h(ˆθ) = 1 + z2 α/2 2n 2ˆθ nˆθ { kx i ( k j=1 1xi } x j x j) Ii AEB 95%, θ > % 2 Ii AEB Ii Ii AEB LMM Kubokawa (2011b) Chatterjee, Lahiri and Li (2008), Hall and Maiti (2006b), Kubokawa and Nagashima (2011) 3.4 EBLUP EBLUP m 2 81

20 I i * I i * I i EB I i AEB I i AEB I i EB θ 2: I i, IEB i, Ii AEB i n i k = 48 n i y ij (2.2) y ij = β 0 + x 1i β 1 + x 2ij β 2 + x 3ij β 3 + v i + e ij x 1i i x 2ij (i, j) (i), x 3ij (i, j) x ij = (1, x 1i, x 2ij, x 3ij ), x i = (1, x 1i, x 2i, x 3i ), k ni i=1 j=1 (x ij x i )(x ij x i ) 2 (2.16) ˆσ e 2 λ λ = 2 ( ) µ i = β 0 + x 1i β 1 + x 2i β 2 + x 3i β 3 + v i ˆθ 3.1 ˆθ T R ˆθ T R, β(ˆθ T R ), ˆσ 2 e ˆθ T R = , β(ˆθ T R ) = (12.927, , , ), ˆσ 2 e = β 1 82

21 1: 1m 2 (EBLUP i (4.6) ) No. n i ˆv i y i EBLUP i β(ˆθt R ) 1/n i M U i EBLUP i

22 MSE Sample Mean EBLUP n i : y i MSE EBLUP i MSE M U i (No.1 No.48 ) 1 1m 2 No.1 No n i y i EBLUP i (3.3) β(ˆθ T R ) 1 EBLUP i y i β(ˆθ T R ) n i n i 1 (3.5) (MSE) 1/n i v i y i MSE M i U EBLUP i MSE (3.6) 3 y i MSE M i U n i No.1 No.48 EBLUP i y i n i n i 1 MSE MSE 0 84

23 1 ˆv i 1.42, 2.31, , ˆv i 4 Ii AEB y i Ii No.1 No.48 I AEB i (3.8) Ii n i I AEB i I AEB i 5 n i Ii n i Ii AEB, n i Ii AEB I i * (upper) I i AEB (upper) I i * (lower) I i AEB (lower) : I AEB i I i (No.1 No.48 ) 85

24 I i * I i AEB n i : I AEB i I i n i (n i No.1 No.48 ) 4 (LMM) 2.3 LMM, (2007) 4.1 Laird and Ware (1982), Tsimikas and Ledolter (1997), Das, Jiang and Rao (2004) Diggle, Liang and Zeger 86

25 (1994), Verbeke and Molenberghs (2000), McCullock and Searle (2001), Demidenko (2004), Fitzmaurice, Laird and Ware (2004), Molenberghs and Verbeke (2006) Hsiao (2003) (3.1) T (repeated measures data, longitudinal data) t = 1,..., T y i1,..., y it x i1,..., x it y i = (y i1,..., y it ), X i = (x i1,..., x it ) y i y i = X iβ + j T v i + e i (4.1) e i v i e i N T (0, (σ 2 e/n i )Q), v i N (0, σ 2 v) e i = (e i1,..., e it ) y it = x itβ + v i + e it, i = 1,..., k, t = 1,..., T, e is e it, s t, Q AR(1) T = 4 ρ < 1 Q 1 = Q 2 = 1 1 ρ 2 1 ρ ρ ρ ρ 1 ρ ρ ρ ρ 1 ρ ρ ρ ρ 1 = (1 ρ)i 4 + ρj 4, 1 ρ ρ 2 ρ 3 ρ 1 ρ ρ 2 ρ 2 ρ 1 ρ ρ 3 ρ 2 ρ 1 = (1 ρ2 ) ( 1 ρ i j ) 87

26 k y = y 1., X = X 1., v = v 1., e = e 1., y k X k v k e k Z = block diag(j T,..., j T ) y = Xβ + Zv + e (4.2) p q A = (a ij ), r s B A B = (a ij B) Z Z = I k j T v Cov (v) = G D = diag (σ 2 e/n 1,..., σ 2 e/n k ) y Σ = ZGZ + D Q ZGZ = (I k j T )(G 1)(I k j T ) = G J T Σ = G J T + D Q G = Cov (v) G = σ 2 v {(1 ρ v )I k + ρ v J k } 2 G (2.2), (3.1) Cov (v) = σ 2 vi k β v Σ = Cov (y) Σ =σ 2 vi k J T + D Q = diag (Σ 1,..., Σ k ), Σ i =σ 2 vj T + (σ 2 e/n i )Q, i = 1,..., k Σ 1 = diag ( ) Σ 1 1,..., Σ 1 k θ = σ 2 v /σe 2 Σ 1 i = n i {Q 1 n iθq 1 j T j T Q 1 } σe n i θj T Q 1 j T 88

27 v = (ˆv 1,..., ˆv k ) = GZ Σ 1 (y X β) ( ) ˆv i =σvj 2 T Σ 1 i y i X i β n i θ ( ) = 1 + n i θj T Q 1 j T Q 1 y i X j i β T β GLS ( k β = X i Σ 1 i X i i=1 ) 1 k i=1 X i Σ 1 i y i T µ i = j T X iβ/t + v i µ i = j T X i β/t + ˆv i Q Q 1 = (1 ρ)i T + ρj T Q 1 1 = 1 { ρ } I T 1 ρ 1 + (T 1)ρ J T j T Q 1 1 = {1 + (T 1)ρ} 1 j T ˆv i = n i θ 1 + (T 1)ρ + n i θt T t=1 ( ) y it x it β Q Q 2 = (1 ρ 2 ) 1 (ρ i j ) T = 4 1 ρ 0 0 Q 1 ρ 1 + ρ 2 ρ 0 2 = 0 ρ 1 + ρ 2 ρ 0 0 ρ 1 j T Q 1 2 = (1 ρ)(1, 1 ρ,..., 1 ρ, 1) = (1 ρ) 2{ j T + ρ(1 ρ) 1 (1, 0,..., 0, 1) }, j T Q 1 2 j T = (1 ρ) 2{ T + 2ρ/(1 ρ) } n i θ ˆv i = (1 ρ) 2 + n i θ{t + 2ρ/(1 ρ)} { T ( ) y it x it β + ρ ( y 1 ρ i1 x i1 β + y it x β )} it t=1 n i ρ 0 T t=1( yit x it β ) /T 89

28 4.2 v i t = 1,..., T v i T T t=1 (y it x itβ) T (4.1) j T v i y i = X iβ + v i + e i (4.3) v i = (v i1,..., v it ) N T (0, σ 2 vi T ) y i Σ i = σ 2 vi T + (σ 2 e/n i )Q, i = 1,..., k T µ it = x it β + v it E[v it y i ] = σv(0, 2..., 0, 1)Σ 1 i (y i x iβ) µ it = x β it + σv(0, 2..., 0, 1)Σ 1 i (y i X i β) Σ i = (σe/n 2 i )(n i θi T + Q) ( ) ( A 11 a 12 n i θi T + Q = A =, A 1 A 11 a 12 = a 21 a 22 a 21 a 22 a 22 a 22.1 = a 22 a 21 A 1 11 a 12 a 21 = a 21 A 1 11 /a 22.1, a 22 = 1/a 22.1 µ it µ it = x β it + n { iθ (y a it x β) } it (a 21 A 1 11, 0)(y i X i β) (4.4) 22.1 y it x it β T 1 Q Q 1 = (1 ρ)i T + ρj T a 22.1 = (n i θ + 1 ρ)(n i θ (T 1)ρ)/(n i θ (T 2)ρ), a 21 A 1 11 = {ρ/(n i θ ) 90

29 (T 2)ρ)}j T 1, µ it =x β n i θ(n i θ (T 2)ρ) { it + (y (n i θ + 1 ρ)(n i θ (T 1)ρ) it x β) it ρ T 1 } (y n i θ (T 2)ρ it x it β) t=1 (4.5) n i µ it y it, ρ 0 µ it x it β + {n i θ/(1 + n i θ)}(y it x it β) Q Q 2 = (1 ρ 2 ) 1 (ρ i j ) (4.4) T = 3 µ it =x β it + n iθ (n i θ + 1) 2 ρ 2 (n i θ) 2 n i θ + 1 (n i θ + 1) 2 ρ 2 n i θ(n i θ 1) { (y it x it β) ρ (n i θ + 1) 2 ρ 2 (n i θ) 2 [ ni θρ(y i,t 2 x i,t 2 β) + (n i θ + 1)(y i,t 1 x i,t 1 β) ]} (4.6) (4.5) n i θρ < n i θ + 1 (4.6) n i µ it y it, ρ 0 µ it x it β + {n i θ/(1 + n i θ)}(y it x it β) θ, ρ ML REML t β β t = ( k j=1 n jx jt x jt) 1 k j=1 n jx jt y jt ê it = y it x it β t θ (2.16), S 2 = T k t=1 i=1 n iê 2 it, n = T k k k { n i tr ( n 2 i x it x it)( n i x it x it) 1 } t=1 i=1 i=1 i=1 ˆθ T R = max{ 1 n (S 2 σ 2 e T (k p) ), 1 } k 2/3 91

30 ρ, ˆρ = k i=1 n i ˆρ i /N ˆρ i Q 1, e i = T t=1 êit/t, T 1 ˆρ i = 2 t=1 s=t+1 T (ê is e i )(ê it e i ) /{(T 1) T (ê it e i ) 2 } Q 2, ˆρ i = T t=2 (ê it e i )(ê i,t 1 e i )/ T t=1 (ê it e i ) 2 ˆρ < 1, ML, REML Q 2 (4.4) (4.6) (T = 5) ˆθ T R = , ˆρ = EBLUP i 2001 EBLUP i 5, (4.4) 6 ˆv it ˆv it Nos.1, 3, 4, 13, 14, 33 ˆv it ˆv it (No.1, 3, 4) No.13, 14, 33 t=1 4.3 (GLMM) k i n i y i1,..., y ini 92

31 2.0 No. 1 No.13 No. 3 No. 14 No. 4 No No No No.4 No No.3 No : ˆv it ˆv it v i y ij f(y ij v i ) = exp {[y ij θ ij b(θ ij )]/τ ij + c(y ij, τ ij )}, j = 1,..., n i ; i = 1,..., k, θ ij τ ij (> 0) τ ij y ij E[y ij v i ] = µ ij µ ij g( ) x ij g(µ ij ) = x ijβ + v i v i N (0, σ 2 v) GLMM GLMM McCullagh and Nelder (1989, 14.5 ) Fahrmeir and Tutz (2001), McCulloch (2003) McCulloch and Searle (2000) Lawson, Browne and Vidal Rodeiro (2003), Lawson (2006) GLMM 93

32 (Standardized Mortality Rate, SMR), ( )/( ) SMR GLMM (1988) 4.4 (2.2) y ij = µ ij + e ij, µ ij = x ijβ + v i y ij (µ ij, σ 2 e) N (µ ij, σ 2 e) µ ij µ ij N (x ijβ, σ 2 v) β, σ 2 v, σ 2 e β, σ 2 v, σ 2 e (2.2) β, σ 2 v, σ 2 e (i, j)- µ ij µ (µ, σ 2 e) (β, σ 2 v) π 1 (µ, σ 2 e β, σ 2 v) (β, σ 2 v) π 2 (β, σ 2 v) π 1 (µ, σ 2 e β, σ 2 v) µ ij (β, σ 2 v) N (x iβ, σ 2 v), σe 2 σe 2 dσe 2 σe 2 dσe 2 π 2 (β, σv) 2 σv 2 dσv 2 β (1) dβ, (2) β σ2 v N (β 0, σva), 2 σ 2 v 94

33 (3) β (σ 2 v, λ) N (β 0, λσ 2 va), λ π 3 (λ), β 0, A Kubokawa and Strawderman (2007) Banerjee, Carlin and Gelfand (2004) 5 C. Stein (MSE) n 1 = = n k = n (3.1) µ i = x iβ + v i µ = (µ 1,..., µ k ) µ S = X β { + max 1 (k p } 2)σ2 e n y X β, 0 (y X β) 2 β = (X X) 1 Xy β OLS Stein k p 3 µ S MSE y MSE Stein (1991) (2004) 20 n 1 = = n k = n (3.1) µ EBLUP θ REML (3.3) µ S (k p 2) (k p) Henderson BLUP 1950 Stein EBLUP Henderson Stein 95

34 2.4 LMM LMM, n 1 = = n k = n (2.2) (2.16) S 1 S 2 S 1 /σ 2 1 χ 2 m 1, S 2 /σ 2 2 χ 2 m 2 m 1, m 2 σ 2 1 = σ 2 e, σ 2 2 = σ 2 e + nσ 2 v σ2 1, σ 2 2 σ2 1 < σ 2 2 Srivastava and Kubokawa (1999), Kubokawa and Tsai (2006) LMM Jiang, Rao, Gu and Nguyen (2008) Fence Kubokawa and Srivastava (2010) LMM (AIC) Vaida and Blanchard (2005) AIC Kubokawa (2011a), Kubokawa and Nagashima (2011) LMM Carleton John N.K. Rao LMM Rao (Statistica Canada) PhD PhD Rao ,

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untitled 17 5 13 1 2 1.1... 2 1.2... 2 1.3... 3 2 3 2.1... 3 2.2... 5 3 6 3.1... 6 3.2... 7 3.3 t... 7 3.4 BC a... 9 3.5... 10 4 11 1 1 θ n ˆθ. ˆθ, ˆθ, ˆθ.,, ˆθ.,.,,,. 1.1 ˆθ σ 2 = E(ˆθ E ˆθ) 2 b = E(ˆθ θ). Y 1,,Y