A Study for the Moment of Wishart Distribution

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Applied Mathematical Sciences, Vol. 9, 2015, no. 73, 3643-3649 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.52173 A Study for the Moment of Wishart Distribution Changil Kim Department of Mathematics Education Dankook University 126 Jukjeon-dong, Yongin-si, Kyunggi-do 448-701, Korea Chul Kang Department of Applied Mathematics Hankyong National University 67 Seokjeong-dong, Anseong-si, Kyunggi-do 456-749, Korea Corresponding author Copyright c 2015 Changil Kim and Chul Kang. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The skewness of a matrix quadratic form XX is obtained using the expectation of stochastic matrix and applying the properties of commutation matrices, where X N p,n (0, Σ, I n ). Keywords: Commutation matrix, Moment, Wishart distribution 1 Introduction Let p n stochastic matrix X be distributed as N p,n (0, Σ, Φ), where Σ : p p and Φ : n n are positive semidefinite, and E(X) = 0. Then, the variancecovariance matrix of the vectorization of X is Cov (vecx, vecx) = Φ Σ. For Φ = I n, XX is said to have Wishart distribution with scale matrix Σ and degrees of freedom parameter n, where I n is an n n identity matrix. Von Rosen (1988) has obtained moments of arbitrary order of matrix X, and using these has calculated the second order moment of quadratic form XAX, where A : n n is an arbitrary non-stochastic matrix. Neudecker and Wansbeek (1987) have also obtained the second order moment of XAX by calculating

3644 Changil Kim and Chul Kang the expectation of Y AY CY BY, where Y = X + M is the matrix of means, and A, B and C are n n arbitrary non-stochastic matrices. Tracy and Sultan (1993) obtained the third order moment of XAX using the sixth moment of matrix X. Kang and Kim (1996a) derived the vectorization of the general moment of XAX. Also Kang and Kim (1996b) derived the vectorization of the general moment of non-central Wishart distribution, using the vectorization of the general moment of XAX. We obtain the skewness of XX using the third moment of XX. Commutation matrices play a major role here. These and some useful results are presented in Section 2. The skewness of matrix quadratic form XX is obtained in Section 3. 2 Some Useful Results Definition 2.1 Let A = (a ij ) be an m n matrix and B = (b ij ) be a p q matrix. Then the Kronecker product A B of A and B is defined as A B = a 11 B a 12 B a 1n B a 21 B a 22 B a 2n B...... a m1 B a m2 B a mn B. Definition 2.2 Let A be an m n matrix and a i the ith column of A ; then veca is the mn column vector veca = [a 1, a 2,, a n]. Definition 2.3 Commutation matrix I m,n is an mn mn matrix containing mn blocks of order m n such that (ij)th blcok has a 1 in its (ji)th position and zeroes elsewhere. One has n m I m,n = (H i,j H i,j) i=1 j=1 where H i,j is an n m matrix with a 1 in its (ij)th position and zeroes elsewhere, and can be written as H i,j = e i e j, where e i is the ith unit column vector of order n. For an m m identity matrix I m and n n identity matrix I n, I m I n = I mn, (1) I 1 m,n = I m,n = I n,m, (2)

A Study for the moment of Wishart distribution 3645 and (I n I n,n I n ) (I n I n I n,n ) (I n I n,n I n ) = (I n I n I n,n ) (I n I n,n I n ) (I n I n I n,n ). For an m n matrix A, I n,m veca = veca, (A B) = A B. For A, B, C, and D conformable matrices, (A B) (C D) = (A B) + (A D) + (B C) + (B D). For A and B conformable matrices, vec A vecb = tr (AB). For an m n matrix A and a p q matrix B, vec (A B) = (I n I m,q I p ) (veca vecb). For A, B, and C conformable matrices, vec(acb) = (B A) vecc. For A, B, C, and D conformable matrices, (AB) (CD) = (A C) (B D). (3) For an m n matrix A and a p q matrix B, I m,p (A B) = (B A) I n,q. (4) For an m n matrix A and a p q matrix B, and an r s matrix C, A B C = I r,mp (C A B)I qm,s = I pr,s (B C A)I n,sq. (5) For an m n matrix A and a p q matrix B, and an r s matrix C, vec(a B C) = (I m I m,qs I pr ) (I mnq I p,s I r ) (veca vecb vecc). Balestra (1976) and Neudecker and Wansbeek (1983) discussed higher order commutation matrices, for which I ab,c = (I a,c I b ) (I a I b,c ) = (I b,c I a ) (I b I a,c ), (6) I a,bc = (I b I a,c ) (I a,b I c ) = (I c I a,b ) (I a,c I b ), (7)

3646 Changil Kim and Chul Kang I n,n 2I n,n 2 = I n 2,n and I n 2,nI n 2,n = I n,n 2. (8) When A = I n = Φ, E(XX ) = nσ where E(XX ) is the first moment of XX according to Magnus and Neudecker (1979). Von Resen (1988) obtained the second moment of XX, E ( 2 (XX ) ) = n(vecσvec Σ) + n 2 ( 2 Σ ) + ni p,p ( 2 Σ ). 3 Skewness of Wishart Distribution Notation P (k, l; m) = I p k,p l I p m, P (k; l, m) = I p k I p l,p m, V = (vecσvec Σ) Σ, and S X = XX. Tracy and Sultan(1993) gave the third moment of matrix quadratic form XX, in square matrix form, E ( 3 S X ) = n 3 ( 3 Σ) + n 2 V + n 2 Q ( 3 Σ) + nqp V P + np Q ( 3 Σ) + n 2 QP V P Q +nqp V +n 2 QP ( 3 Σ)+nV P +nv P Q+n 2 P ( 3 Σ)+nP V +np V P Q +n 2 P V P + n 2 QP Q ( 3 Σ). In fact, XX is distributed as Wishart distribution. Lemma 3.1 Let P = P (1; 1, 1), Q = P (1, 1; 1). Then (a) P 2 = I p 3 Q 2 = I p 3. (b) P 1 = P, Q 1 = Q. (c) P QP Q = QP, QP QP = P Q. P roof. (a) Using (1) and (2), we get Q 1 = Q. P 2 = (I p I p,p )(I p I p,p ) = I p I p 2 = I p 3. Similarly, we get Q 2 = I p 3. (b) They are trivial because of (a) in lemma 3.1. (c) Using (6), (7), and (8), we can derive and P QP Q = (I p I p,p )(I p,p I p )(I p I p,p )(I p,p I p ) = I p,p 2I p,p 2 = I p 2,p = QP QP QP = (I p,p I p )(I p I p,p )(I p,p I p )(I p I p,p ) = I p 2,pI p 2,p = I p,p 2 = P Q.

A Study for the moment of Wishart distribution 3647 Lemma 3.2 QP Q ( 3 Σ) = P ( 3 Σ) QP P roof. Let L = QP Q ( 3 Σ). Then P LP Q = P (QP Q ( 3 Σ)) P Q = QP ( 3 Σ) P Q = 3 Σ by (5) and (c) in lemma 3.1. Using P = P 1 and QP = (P Q) 1, L = P ( 3 Σ) QP. Lemma 3.3 P Q ( 3 Σ) QP = QP ( 3 Σ) P Q = 3 Σ P roof. They are trivial using (4) and (5). Lemma 3.4 QP Q ( 3 Σ) P Q = P ( 3 Σ). P roof. Let N = QP Q ( 3 Σ) P Q. Then P N = P QP Q ( 3 Σ) P Q = QP ( 3 Σ) P Q = 3 Σ by (5) and (c) in lemma 3.1. Using P = P 1, N = P 1 ( 3 Σ) = P ( 3 Σ). Theorem 3.5 Let a p n random matrix X be distributed as N p,n (0, Σ, Φ). Then E (S X S X ES X ) = n 2 V + n 3 ( 3 Σ) + n 2 Q ( 3 Σ). P roof. Since ES X = nσ and E ( 2 S X ) = n(vecσvec Σ) + n 2 ( 2 Σ) + ni p,p ( 2 Σ), E (S X S X ES X ) = E ( 2 S X ) ES X = [nvecσvec Σ + n 2 ( 2 Σ) + ni p,p ( 2 Σ)] nσ = n 2 [(vecσvec Σ) Σ] + n 3 ( 3 Σ) + n 2 [I p,p ( 2 Σ) I p Σ] = n 2 V + n 3 ( 3 Σ) + n 2 Q ( 3 Σ) using (4). Theorem 3.6 Let a p n random matrix X be distributed as N p,n (0, Σ, Φ). Then E (S X ES X S X ) = n 2 P QV QP + n 3 ( 3 Σ) + n 2 P ( 3 Σ) QP. P roof. Using (5), lemma 3.2, and lemma 3.3, E (S X ES X S X ) = E [I p,p 2 (S X S X ES X ) I p 2,p] = I p,p 2E (S X S X ES X ) I p 2,p = P QE (S X S X ES X ) QP = P Q [n 2 V + n 3 ( 3 Σ) + n 2 Q ( 3 Σ)] QP = n 2 P QV QP + n 3 P Q ( 3 Σ) QP + n 2 P ( 3 Σ) QP = n 2 P QV QP + n 3 ( 3 Σ) + n 2 P ( 3 Σ) QP. Theorem 3.7 Let a p n random matrix X be distributed as N p,n (0, Σ, Φ). Then E (ES X S X S X ) = n 2 QP V P Q + n 3 ( 3 Σ) + n 2 P ( 3 Σ).

3648 Changil Kim and Chul Kang P roof. Using (5), lemma 3.3, and lemma 3.4, E (S X ES X S X ) = E [I p 2,p (S X S X ES X ) I p,p 2] = QP E (S X S X ES X ) P Q = QP [n 2 V + n 3 ( 3 Σ) + n 2 Q ( 3 Σ)] P Q = n 2 QP V P Q + n 3 QP ( 3 Σ) P Q + n 2 QP Q ( 3 Σ) P Q = n 2 QP V P Q + n 3 ( 3 Σ) + n 2 P ( 3 Σ). Here is the key result of this paper. Theorem 3.8 E [ 3 (S X ES X )] = n [P V + V P + QP V + V P Q + P V P Q + QP V P + P Q ( 3 Σ)] +n 2 [P V P P QV QP + QP ( 3 Σ)] 2n 3 ( 3 Σ). P roof. E [ 3 (S X ES X )] = E [( 3 S X ) + (ES X S X ES X ) + (ES X ES X S X ) + (S X ES X ES X )] E [(ES X S X S X ) + (S X S X ES X ) + (S X ES X S X ) + ( 3 ES X )] = E ( 3 S X ) E(S X S X ES X ) E(S X ES X S X ) E(ES X S X S X ) + ( 3 ES X ). Using theorem 3.6, theorem 3.7, and theorem 3.8, E [ 3 (S X ES X )] = n [P V + V P + QP V + V P Q + P V P Q + QP V P + P Q ( 3 Σ)] +n 2 [P V P P QV QP + QP ( 3 Σ)] 2n 3 ( 3 Σ). References [1] P. Balestra, La Derivation Matricielle Collection del IME. Sirey, Paris, 1976. [2] H. Neudecker, and T. Wansbeek, Some results on communication matrices, with statistical applications, Cadad. J. Statist., 11 (1983), 221-231. http://dx.doi.org/10.2307/3314625 [3] H. Neudecker, and T. Wansbeek, Fourth-order properties of normally distributed random matrices, Linear Alg. Appl., 97 (1987), no. 3, 13-21. http://dx.doi.org/10.1016/0024-3795(87)90135-2 [4] Von Rosen, Moments for matrix normal variables, Statistics, 19 (1988), 575-583. http://dx.doi.org/10.1080/02331888808802132 [5] D. S. Tracy and S. A. Sultan, Third moment of matrix quadratic form, Stat. Prob. Letters, 16 (1993), no. 1, 71-76.

A Study for the moment of Wishart distribution 3649 [6] C. Kang and B. Kim, The N-th moment of matrix quadratic form, Stat. Prob. Letters, 16 (1996a), no. 18, 291-297. [7] C. Kang and B. Kim, The general moment of noncentral Wishart distribution, J. Korean Statist. Soc., 25 (1996b), 393-406. Received: March 11, 2015; Publsihed: May 2, 2015

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