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1 Recent Advances in Statistical Inference - in Honor of Professor Masafumi Akahira Shrinkage estimators for covariance matrices in multivariate complex normal distributions December 12, 2008

2 (Large Covariance matrix) (1) Wishart ; (2) ; (3) ; (4) (SURE ); (5) (SURE) ; (6) 1

3 X (1) X =ReX + 1ImX, [X] = ReX, ImX X. ( ReX ImX ) ; X CN(0, 1) ( ) ( ) ReX 0 [X] = N ImX 2 (, ( ) ). X (w.r.t. Lebesgue measure on C ) f X (x) = 1 π exp( xx), x C, x x complex conjugate. 2

4 (2) Z CN(0, 1), θ C, σ R+ X := θ + σz CN(θ, σ 2 ). X C p c C p θ C p Σ Herm(p, C) + c X CN(c θ, c Σc) X CN p (θ, Σ). c c transpose complex conjugate X CN p (θ, Σ) (w.r.t. Lebesgue measure on C p ) f X (x) = 1 π p(det Σ) 1/2 exp{ (x θ) Σ 1 (x θ)}. 3

5 (3) Z CN p (θ, Σ) [Z] := ( Re Z Im Z ) ( Re θ N 2p ( Im θ ), ( Re Σ Im Σ Im Σ Re Σ ) ) Re Σ, Im Σ symmetric skew-symmetric. 4

6 Wishart (1) p Z 1, Z 2,, Z n CN p (0, Σ) n W := Z i Z i Σ, p,n Wishart CW p (Σ, n) i=1 n p P(W ) =1 W (w.r.t. Lebesgue measure on Herm + (C, p) ) f W (w) = Det (w) n p exp( Tr (wσ 1 )) Det (Σ) n π p(p 1)/2 Π p j=1 Γ(n +1 j), w Herm+ (C, p) Γ( ) Euler s gamma function. 5

7 (1) Z 1, Z 2,...,Z n CN p (0, Σ) Z i (i =1, 2,..., n) p ( ) p Σ p p n ( -1) p Wishart (p p ) W := n k=1 Z iz i transpose complex conjugate Σ L( Σ, Σ) =Tr( ΣΣ 1 I p ) 2 Σ Σ I p p p Tr W L R( Σ, Σ) :=E[L( Σ, Σ)] Σ 6

8 (2) 0 ; Wishart W n p; Wishart W (n ) n p ; Σ A ΣA ; Σ AΣA (A p p ) L( Σ, Σ) =Tr( ΣΣ 1 I p ) 2 ; L S ( Σ, Σ) =Tr( ΣΣ 1 ) log Det( ΣΣ 1 ) p. Det n <p L S n 1 W L S ) 7

9 (1) n 1 W E[n 1 W ]=Σ n 1 W Σ (Marchenko-Pastur law). n<p Σ n 1 W 8

10 (2) n p L S n 1 W Shrinkage-expansion method. Svensson (2004), Konno (2007a, 2007b), Konno(2009). SURE eigenvaluecaluculus n<p L Konno (2009)(Haff (1980) Wishart ) n<p 9

11 (3) n<p Wishart S W p (Σ, n) Σ ; Ledoit and Wolf (2004): Tr( Σ Σ) 2 n 1 S I p (n/p ) ; Wu and Pourahmadi (2003), Bickel and Levina (2008): banding approach. ; Furrer and Bengtsson (2007): tapering ; AOS (2009) 10

12 Z 1, Z 2,...,Z n CN p (0, Σ) Z i (i =1, 2,..., n) p ( ) p Σ p p n ( -1) p ; Wishart (p p ) W := n k=1 Z iz i Σ L( Σ, Σ) =Tr( ΣΣ 1 I p ) 2 Σ Σ ; W L R( Σ, Σ) :=E[L( Σ, Σ)] Σ 11

13 W = n i=1 Z iz i l 1 l n W W = U 1 LU 1, L = Diag(l 1,...,l n ); U 1 p n s.t. U 1U 1 = I n. Σ = U 1 Ψ(L)U 1, (1) Ψ := Ψ(L) = Diag(ψ 1,ψ 2,...,ψ n ) ψ k := ψ k (L)(k =1, 2,...,n) R n R. Σ E[Tr ( ΣΣ 1 I p ) 2 ] 12

14 SURE ) E[Tr ( ΣΣ 1 I p ) 2 ] R( Σ) ( ϕ 1,..., ϕ n l 1,...,l n W ) E[Tr ( ΣΣ 1 I p ) 2 ]=E[ R( Σ)] E[Tr (n 1 W Σ 1 I p ) 2 ] R( Σ) E[Tr (n 1 SΣ 1 I p ) 2 ] SURE E[Tr ( ΣΣ 1 I p ) 2 E[Tr (n 1 W Σ 1 I p ) 2 ] (1) R( Σ) 13

15 SURE (1) (z ij ) i=1,..., n; j=1,..., p := [Z 1, Z 2,...,Z n ] CN n p (0, I n Σ); n p Z ( ) ( 1 Z = = 2 (Re z ij ) z ij i=1, 2,..., n j=1, 2,..., p 1 2 (Im z ij ) ) ; i=1, 2,..., n j=1, 2,..., p Z A (i, j) ( Z A) ij = p k=1 a kj z ik for i =1, 2,...,n; j =1, 2,..., p. 14

16 SURE (2) [Z 1, Z 2,...,Z n ] CN n p (0, I n Σ) W = n i=1 Z iz i p p G = G(W ) E [Σ 1 WG]=E [ng +(Z Z ) G]. [Tr (Σ 1 WG)] = E [ntr (G)+Tr(Z E Z G )]. 15

17 SURE (3) 1 G = U 1 Diag(l 1 1 ψ 1,...,l 1 n ψ n )U 1 2 E [Σ 1 U 1 ΨU 1]=E [ ] U 1 Ψ (1c) U 1 +Tr(L 1 Ψ)(I p U 1 U 1). Ψ (1c) = Diag(ψ (1c) 1,ψ (1c) 2,..., ψ n (1c) ) ψ (1c) k = n b k ψ k ψ b l k l b + ψ k l k (k =1, 2,...,n). complex analog of Kubokawa and Srivastava (2008) s identity E [Tr {Σ 1 U 1 ΨU 1}] =E n k=1 (p n)ψ k l k + ψ k l k + n b k ψ k ψ b l k l b. 16

18 SURE (4) 3 Σ = U 1 Ψ(L)U 1 E[Tr {Σ 1 U 1 ΨU 1Σ 1 U 1 ΨU 1}] =E[Tr {Σ 1 U 1 Ψ(1) U 1 }]. Ψ (1) = Diag( ψ (1) 1, ψ (1) 2 (1),..., ψ n ) ψ (1) k =(p n) ψ2 k l k +2ψ k ψ k l k +2ψ k n b k ψ k ψ b l k l b, k =1, 2,...,n. 17

19 SURE (5) 4 Σ = U 1 Ψ(L)U 1 [ R( Σ, n Σ) = {(p n) ( ψ(1) k 2 ψ (1) k) ( ψ E + k l k l k l k k=1 n (1) (1) } ( ψ k 2ψ k ) ( ψ b 2ψ b ) + l k l b b k 2 ψ k l k ) ] + p. (1) ψ k =(p n)ψk 2/l k +2ψ k ( ψ k / l k )+2ψ n k b k (ψ k ψ b )/(l k l b ) (k =1, 2,...,n). 18

20 (1) n <p Σ t = 1 ( t W + p + n Tr W +U 1U ) 1. U 1 p n, W SW + S Moore-Penrose t Σ t (SURE) 0 <t<2(n 1)(p n +1)/{(p n + 1)(p n +2)} Σ R( Σ t, Σ) R(n 1 W, Σ) 19

21 (2) Σ t ( 1 p+n W + t U Tr W + 1 U ) 1 : Σ t = 1 { t W + p + n Tr W I } p. Σ t SURE 20

+ 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm.....

+ 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm..... + http://krishnathphysaitama-uacjp/joe/matrix/matrixpdf 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm (1) n m () (n, m) ( ) n m B = ( ) 3 2 4 1 (2) 2 2 ( ) (2, 2) ( ) C = ( 46