Improving the sample covariance matrix for a complex elliptically contoured distribution

Transcription

1 September 13, 006 Improving the sample covariance matrix for a complex elliptically contoured distribution Yoshihiko Konno Faculty of Science, Japan Women s University -8-1 Mejirodai, Bunkyo-ku, Tokyo , Japan Abstract. In this paper the problem of estimating the scale matrix in a complex elliptically contoured distribution(complex ECD is addressed. An extended Haff-Stein identity for this model is derived. It is shown that the minimax estimators of the covariance matrix obtained under the complex normal model remain robust under the complex ECD model when the Stein loss function is employed. Key words: covariance matrix, shrinkage estimator, complex Wishart distribution, minimax, unbiased risk of estimate, complex normal distribution, eigenvalue distribution in sample covariance matrix. MSC: primary; 6H1; secondary: 6F10 1 Introduction The focus of the present paper is on estimating the scale matrix Σ that appear in a standard complex MANOVA model Y = Cβ + E, (1 in which Y is an N p matrix of response complex variables, C is an N m known matrix of complex entries, β is an m p unknown parameter matrix of complex entries, and E is an Address: konno@fc.jwu.ac.jp Supported in part by the Japan Society for the Promotion of Science through Grants-in-Aid for Scientific Research(C(No

2 N p error matrix of complex entries. The following assumptions on (1 are made to hold; (i the error matrix E follows a complex elliptically contoured distribution with the density function Det(Σ N f(tr(σ 1 e e e C N p, in which Σ is a p p unknown positive definite Hermitian matrix, and f( is a nonnegative function on the nonnegative real line such that f(t dt = 1; (ii N m p +1 > 0. 0 Here C N p is the space of N p matrices of complex entries, stands for the conjugate transpose of a matrix of complex entries, Tr and Det mean the usual trace and the determinant of a squared matrix, respectively. The model (1, along the previous assumptions, includes the complex MANOVA model with the complex normal error, which is extensively explored by Andersen et al. (1995. The practical significance of this model in signal processing and communications is discussed, for example, in DoGondžić and Neborai (003. See Krishnaiah and Lin (1986 and Micheas el al. (005 for construction of an elliptical family of complex distributions via an extension of elliptical family of real distributions. Stein (1977 pointed out that the eigenvalues of the sample covariance matrix obtained from the multivariate real normal sample spread out more than the corresponding eigenvalues of the parameter covariance matrix, and that the problem gets more serious when the parameter covariance matrix is near spherical. Since this fact was pointed out, there has been growing literature on improved estimators of the covariance matrix of the multivariate real normal distributions. We refer to Lin and Perlman (1985, Young and Berger (1994, Kubokawa and Srivastava (1999, Daniels and Kass (1999, 001, and the cited references therein for further information on the improved estimation of the real normal covariance matrix. The sample covariance matrix obtained from independently and identically distributed complex random vectors may be improved by applying shrinkage-expansion techniques to its eigenvalues since the eigenvalues of the sample covariance matrix are biased away from those of the true parameter covariance matrix. Abraham and Dey (1994 demonstrated that the adaptive beamforming and detection performance loss due to the estimation of an unstructured covariance matrix is reduced by modifying the eigenvalues of the sample covariance matrix. The methods of Abraham and Dey (1994 heavily depend on the results of Dey and Srinivasan (1985 for estimating the covariance matrix of a real multivariate normal random vector. It is conjectured in the paper of Abraham and Dey (1994 that the results of Dey and Srinivasan (1985 may be extended to the complex multivariate normal

3 setting. The underlying techniques in the paper of Dey and Srinivasan (1985 are twofold; one is integration-by-parts for the real Wishart distribution, which is derived by Stein (1977 and Haff (1979 independently; the other is an algebraic derivation of the eigenstructure of a positive definite matrix of real entries, which is developed by Stein (1977 and Haff (1991. Later, Maiwald and Kraus (1994 used well-known isomorphism between real and complex random matrices in order to give the real representation of integration-by-parts for the complex Wishart distribution. More recently, Svensson and Lundberg (004 obtained an unbiased estimate of the risks for unitary invariant estimators of the scale matrix of the complex Wishart distribution and used the techniques of Haff (1991 in order to derive the variational form of the Bayes estimator. In this paper we consider the problem of estimating the Hermitian scale matrix Σ for the complex ECD model (1 in a decision theoretic manner. We employ the Stein loss function in order to evaluate the performance of every estimator of Σ. We first generalize the techniques of Dey and Srinivasan (1985 to the complex elliptically contoured distributions. Namely, we develop integration-by-parts for the complex ECD model (1 and some algebraic derivations which are applied to the eigenstructure of a positive definite Hermitian matrix. Using these results we show that the dominance results for the real ECD model given by Kubokawa and Srivastava (1999 are generalized to the complex setting. In other words we obtain dominance results in the complex case similar to those in the real case given by James and Stein (1961 and Dey and Srinivasan (1985. The present paper is organized in the following way. In Section, we give integration-byformula for the complex elliptically contoured distribution via an extension of a formula on the real ECD models obtained by Kubokawa and Srivastava (1999. We also give calculus lemmas on the eigenstructure of Hermitian matrices. These lemmas are similar to algebraic lemmas on the eigenstructure of the symmetric matrices given by Haff (1991. These lemmas appear in a unpublished manuscript of Svensson (004. Utilizing the Haff-Stein identify for the complex ECD models we show that minimax estimators which are obtained in a similar way to those under the real normal distributions remain robust under the complex ECD models (1. In Section 3 the proofs of the calculus lemmas are presented since our proofs of the lemmas are more elementary. 3

4 Main Result To describe a canonical form of our estimation problem, let P be an N N unitary matrix such that PC =((C C (1/, 0 m (N m and PP = P P = I N. Also set µ =(C C (1/ β. Let n = N m and let X and Z be m p and n p matrices of complex entries such that (X, Z = PY. Then the joint density function of X and Z has the form Det(Σ N f(tr(σ 1 (x µ (x µ+σ 1 z z, x C m p, z C n p. ( Set S = Z Z. Based on S, we consider the problem of estimating Σ under the Stein loss function L ( Σ, Σ =Tr( ΣΣ 1 Det ( ΣΣ 1 p, (3 in which Σ is an estimator of a p p Hermitian matrix Σ. The loss function given by (3 has been suggested by James and Stein (1961 for the problem of estimating the real normal covariance matrix. Certainly there are many other possible loss functions which we may employ. However the above loss function has the attractive feature that it is relatively easy to work with [see Muirhead (198]..1 Integration by parts formula Let C denote the field of complex numbers. The imaginary unit is denoted by 1. For every c in C we denote by Re c and Im c the real and the imaginary parts of c, respectively. The conjugation of a complex number c is given by c =Rec 1Im c. The absolute value of c is given by c =(c c 1/. We denote by C n p the set of all n p matrices of complex entries. To describe integration-by-parts for S = Z Z, we need the following notation: For a nonnegative x and f( in (, set F (x = x E F [h(x, Z] = E Here, h(, is a real-valued integrable function and f(t dt and define [ h(x, Z F (R f(r R =Tr(Σ 1 (X µ (X µ+σ 1 Z Z. ]. (4 The notation in (4 was introduced by Kubokawa and Srivastava (1999 in order to describe integration-by-parts for the real elliptically contoured distributions. Let G(S beap p 4

5 matrix of complex entries, the (i, j-element g ij (S of which is a function of S = (s ij. For a p p Hermitian matrix S =(s jk, let D S =(d jk S beap p operator matrix, the (j, k-element of which is given by d jk S = 1 { (1 + δ jk (Re s jk + } 1, j, k =1,,...,p. (5 (Im s jk Here, δ jk is the Kronecker delta ( = 1 if j = k and = 0 if j k. Thus the (j, k-element of D s G(S is {D S G(S} jk = p l=1 d jl S g lk(s = 1 (1 + δ jl p l=1 } (Re s jl (S+ g lk 1 (Im s jl (S. { glk Lemma.1 Assume that each entry of G(S is a partially differentiable function with respect to Re s jk and Im s jk, j, k =1,,...,p. Also assume that, for any real a, (i (ii E[ Tr(G(SΣ 1 ] is finite; lim z jk G(SS 1 F ( z jk + a =0, z jk for j =1,,...,nand k =1,,...,p. Then we have E[Tr(G(SΣ 1 ] = E F [(n ptr(g(ss 1 +Tr(D S G(S]. (6 Proof. The proof is given in Section 3. Remark.1 When f(x =π np exp( x, then the model (1 becomes the multivariate linear complex normal model and S follows the complex Wishart distribution CW p (n, Σ, the density function of which is Det (s n p exp( Tr (sσ 1 Det(Σ n π p(p 1/ p s Cp p j=1 Γ(n +1 j, Here Γ( indicates the usual gamma function and C p p S stands for the space of p p semipositive definite Hermitian matrices. See Goodman (1963, Khatri (1965, and Andersen et al. (1995 for further information of the complex Wishart distribution. Under the assumption of the complex normal error, E F in the right hand side of (6 becomes the usual expectation taken with respect to the above density function. Therefore the identity (6 becomes a complex extension of the Haff-Stein identity for the real Wishart distribution given by Haff (1979. This complex analogue of the Haff-Stein identity for the complex Wishart distribution is given by Svensson and Lundberg ( S.

6 The next two lemmas were given by Svensson (004. Since our proofs are more elementary and self-contained, we record the proofs of the lemmas in Section 3. Lemma. Decompose S = ULU in which U =(u jk is a p p unitary matrix and L = Diag(l 1,l,...,l p with l 1 l l p > 0. Then we have, for j, k, s, t = 1,,...,p, d st S l k = u sk u tk, d st S u jk = a k u ja u ta u sk l k l a, d st S u jk = u ja u sa u tk, l k l a a k in which d st S is given by (5. Proof. The proof is given in Section 3. Let R p = {(x 1,x,...,x p R p : x 1 x x p > 0}. The next lemma is a complex analogue of Theorem 5.1 given by Haff (1991. Lemma.3 Let ϕ 1 (L, ϕ (L,...,ϕ p (L be differentiable real-valued functions on R p with ϕ k (L 0,,,...,p, and let Φ(L =Diag(ϕ 1 (L, ϕ (L,..., ϕ p (L. Then, D S [UΦ(LU ]=UΦ (1 (LU, in which Φ (1 (L =Diag(ϕ (1 1 (L, ϕ (1 (L,...,ϕ (1 p (L and ϕ (1 k (L = ϕ k (L ϕ a (L + ϕ k(l, k =1,,...,p. l k l a l k a k Proof. The proof is given in Section 3. Lemma.4 Let ϕ 1 (L, ϕ (L,...,ϕ p (L be differentiable real-valued functions on R p with ϕ k (L 0,,,...,p, and let Φ(L =Diag(ϕ 1 (L, ϕ (L,..., ϕ p (L. Then, under suitable conditions corresponding to those of Lemma.1, we have [ p { E[Tr(UΦ(LU Σ 1 ] = E F ϕk (L+ ϕ k (L ϕ a (L +(n p ϕ }] k(l. l k l k l a l k a k Proof. The proof is obtained immediately from Lemmas.1 and.3. Remark. As in Remark.1, set f(x =π np exp( x. Then the identity in Lemma.4 reduces to that appearing in Theorem 1 of Svensson and Lundberg (004. 6

7 . Alternative estimators The next proposition is a complex analogue of Proposition 1 of Kubokawa and Srivastava (1999. Proposition.1 Decompose S = TT in which T is a p p lower triangular matrix. Also set V = Diag(v 1,...,v p with the v k s, k =1,..., p, being positive constants. Then we have E [ Tr (SΣ 1 +Tr(TVT Σ 1 ] p = E [np F + (n + p +1 kv k ], where E F is defined by (4. Proof. The above equality follows from two equalities, i.e., E [ Tr (SΣ 1 ] = E F [np] and E [ Tr (TVT Σ 1 ] = E [ F p (n + p +1 kv k]. The first equality is derived from Lemma.4 with ϕ k (L =l k,k =1,,..., p. By an invariance argument it suffices to prove that the second inequality holds in the case that Σ = I p. From an application of Lemma.1 and the fact that Tr (D S S=p, we have E[Tr (S] = E F [p(n p+tr(d S S] = np E F [1]. Furthermore, since the distribution of S is invariant under the permutation of the coordinates, we see that E[s kk ]=n E F [1] (7 for k =1,,...,p. Also, note that the Jacobian of the transformation from S to T depends only on the diagonal elements of T. Thus, it suffices to show that E[ t 1 ]=E F [1] in the case that p = in order to prove that, for k l, E[ t kl ]=E F [1]. (8 To prove that E[ t 1 ]=E F [1], let Z =(z 1, z with an n 1 vector z. Set v = Qz such that Qz 1 =(t 11, 0. We decompose v =(v 1, v with a scalar v 1. Recall that S = Z Z to see that s 1 = z z 1 = u Q Qz 1 = v 1 t 11. Furthermore the distribution of (X, z 1,v 1, v is Det (Σ m f(tr {Σ 1 (x θ (x θ} +Tr(u 1 u 1+ v 1 + v v. Proceed in a manner similar to that used in the derivation of (17 in Section 3 to see that E[ t 1 ]=E[ v 1 ]=E F [1]. By using (7 and (8 some algebraic computations complete the proof of the second equality. 7

8 Remark.3 If S follows the complex Wishart distribution with the degrees of freedom n and a scale matrix Σ, i.e., CW p (n, Σ, then we have F = f in (4 from which it follows that E F [1] = 1. Thus, Proposition.1 results in the equations E[Tr (TVT Σ 1 ] = p (n + p +1 kv k and E[Tr (SΣ 1 ] = np. Consider estimators of the form Σ (JS = TDiag(ṽ 1,..., ṽ p T, (9 in which T is defined in Proposition.1 and ṽ k =(n + p +1 k 1,,,...,p. Proposition. Using the loss function L ( Σ, Σ given by (3 the James-Stein type estimator Σ (JS given by (9 is better than the usual estimator n 1 S uniformly for every function f( in (. Proof. The proof is obtained immediately from an application of Proposition.1. Next we consider a class of unitary invariant estimators of the form Σ (Φ = UDiag(ϕ 1 (L,...,ϕ p (LU, (10 in which U is defined in Lemma. and the ϕ k (L s, k =1,,...,p, are differentiable functions from R p to R. This class of estimators has been suggested by Abraham and Dey (1994 via an extension of orthogonally invariant estimators for the real normal covariance matrix. Proposition.3 Using the loss function L ( Σ, Σ given by (3 the estimators Σ (Φ given by (10 are better than the James-Stein type estimator Σ (JS given by (9 uniformly for every function f( in (, if the following inequalities hold almost surely; { p ϕ k p (L+ } ϕ k (L ϕ j (L +(n p ϕ k(l 0, (11 l k l k l j l k j k p { log(n + p +1 k + log ϕ } k(l 0. (1 l k Proof. For simplicity, we omit the argument L in ϕ k (L asϕ k for k =1,,...,p. Using Lemma.4 and Proposition.1 the risk difference between the estimators Σ (JS and Σ (Φ is [ p ] L ( Σ (JS, Σ L ( Σ (Φ, Σ = E F [p] E { log(n + p +1 k + log l k } [ p { E F ϕ k + l k j k [ p ] +E log ϕ k. 8 ϕ k ϕ j l k l j +(n p ϕ k l k }]

9 Rearranging the terms inside each of two different types of the expectations separately we obtain the desired result. Theorem.1 Let ṽ k =(n + p +1 k 1,,,...,p, and decompose S = ULU in which U =(u jk is a p p unitary matrix and L = Diag(l 1,l,...,l p with l 1 l l p > 0. Using the loss function L ( Σ, Σ given by (3 the Dey-Srinivasan type estimator Σ (DS = UDiag(ṽ 1 l 1,..., ṽ p l p U is better than the James-Stein type estimator Σ (JS given by (9 uniformly for every function f( in (. Proof. The proof follows immediately from the verification of the inequalities (11 and (1 with ϕ k =ṽ k l k,,,...,p. Remark.4 Sheena and Takemura (199 called (10 order-preserving if it satisfies the condition ϕ 1 (L ϕ (L ϕ p (L. The estimator Σ (DS is further improved upon by a corresponding order-preserving estimators. This result follows from the fact that Lemma 1 of Sheena and Takemura holds for nonincreasing functions f( in (. It can be also seen that an estimator similar to that given by Perron (199 for the real normal covariance matrix improves upon the James-Stein type estimator Σ (JS uniformly for every function f(. 3 Proofs of Lemmas Before we get into proving the lemmas presented in the previous section, we state three approaches to prove the Wishart identity for the real case. Firstly, Stein (1977 used integration-by-parts for the multivariate real normal distribution in which the mean vector is zero and the covariance matrix is identity. Then Stein (1977 used an algebraic lemma on the eigenstructure of a symmetric matrix in order to obtain the result for the real case similar to that presented in Lemma.4. This approach was taken by Kubokawa and Srivastava (1999 in order to extend the Wishart identity to real elliptically contoured distributions. This approach is also taken in the present paper. Secondly, Haff used integration-by-parts for the real Wishart distribution and applied the same algebraic lemma as in Stein (1977. This approach was taken by Svensson (004 in order to obtain the Haff-Stein identity for the complex Wishart distribution. Thirdly, Sheena (1995 used integration-by-parts for the distribution of the eigenvalues of the real Wishart distribution. The third approach gives a direct proof of the result for the real case similar to that presented in Lemma.4. Also, this approach was taken by Konno (005 9

10 in order to obtain integration-by-parts for the Wishart distribution on irreducible symmetric cones. The next lemma describes the chain rule for the differentiation which is useful for the proof of Lemma.1. Lemma 3.1 Let Y =(y jk be an n p matrix of complex entries and set W = Y Y. Furthermore, let y jk = (Re y jk + 1 (Im y jk, w kl = (Re w kl + 1 (Im w kl, for k, l =1,,...,pand j =1,,..., n. Then we have p 1+δ kl = y jk. y jl w kl Proof. The real and the imaginary parts of the (c 1,c -element of the matrix W are, respectively, Re w c1 c = c 3 { Re yc3 c 1 Re y c3 c +Imy c3 c 1 Im y c3 c }, Im w c1 c = c 3 { Re yc3 c 1 Im y c3 c Im y c3 c 1 Re y c3 c }. From these two equations the real part of /y jl is = { (Re wc1 c (Re y jl (Re y c 1 c jl (Re w c1 c + (Im w } c 1 c (Re y jl (Im w c1 c [ = (δc3jδ c1lre y lc + δ c3jδ clre y c3c1 (Re w c 1 c,c c1 c 3 + ( ] δ c3 jδ c1 lim y c3 c δ c3 jδ c lim y c3 c 1, (Im w c1 c while the imaginary part is = { (Re wc1 c (Im y jl (Im y c 1 c jl (Re w c1 c + (Im w } c 1 c (Im y jl (Im w c1 c [ = (δc3jδ c1lim y c3c + δ c3jδ clim y c3c1 (Re w c 1 c,c c1 c 3 + ( ] δ c3 jδ c lre y c3 c 1 δ c3 jδ c1 lre y c3 c c. (Im w c1 c 10

11 Therefore we have (Re y jl + 1 (Im y jl = [ 1+δ c1c (δc1lre y jc + δ clre y jc1 (Re w c 1 c c1 c + ( δ c1 lim y jc δ c lim y jc1 (Im w c1 c + 1 ( δ c1 lim y jc + δ c lim y jc1 (Re w c1 c + 1 ( ] δ c lre y jc1 δ c1 lim y jc (Im w c1 c = [ 1+δ jc 1 Re y jc (Re w c lc +Imy 1 jc (Im w lc + ( ] 1 1 Im y jc (Re w lc Re y 1 jc (Im w lc + [ 1+δ jc1 1 Re y jc1 (Re w c c1 1 l Im y 1 jc 1 (Im w c1 l + ( ] 1 1 Im y jc1 (Re w c1 l +Rey 1 jc 1. (Im w c1 l Since the real part of W is symmetric and the imaginary part of W is skew-symmetric, this completes the proof. 3.1 Proof of Lemma.1 To reduce the equation (9 to the case when Σ = I p, we first observe that E [Tr (D S GS] = E F [Tr (SD S G+pTr G], (13 since the (i, i-element of D S GS can be written as (D S GS ii = = = p { p ( 1+δ ij (D S G ik s ki + g jk (Re s ij + 1 (Im s ij (Re s ki + } 1Im s ki j=1 p { p ( s ki (D S G ik + g jk δij δ ki + δ kj (1 δ ij } p s ki (D S G ik + j=1 j=1 p g jj. 11

12 The second last equality in the above equations follows from the fact that the real part of S is symmetric and the imaginary part is skew-symmetric. Replacing G with GS in (, we get E [Tr (GSΣ 1 ] = E F [(n ptr(g+tr(d S GS]. Substitute (13 into the above equation to get that E [Tr (GSΣ 1 ] = E F [n Tr (G+Tr(SD S G]. Hence it suffices to prove that, for a complex-valued function h(s, E [h(ssσ 1 ]=E F [nh(si p + SD S h(s]. (14 To this end, write Σ = AA and S = AWA in which W =(w ij isap p symmetric matrix and A =(a ij isap p nonsingular matrix with its inverse A 1 =(a ij. Then 1+δ ij = 1+δ {( ij s ij (Re s c 1 c ij + 1 (Im s ij 4 ( a c 1c Re s c c 3 + } 1Im s c c 3 a c 4c 3 w c,c c1 c 4 3 = δ ic3 δ jc a c1c a c4c3 w c 1 c 4,c,c c1 c 4 3 = a c1j a c4i = {(A 1 D W w A 1 } ij, c 1 c c1 c 4 4 which is the (i, j-element of (A 1 D W A 1. Here D W D W. Using this equation, we can reduce (14 to denotes the transpose of the matrix E [Wh(S] = E F [nh(si p + WD W h(s]. (15 Furthermore, let Z = AY with Y =(y ij. Since W = Y Y, it follows from Lemma 3.1 that c 3 w ic3 1+δ jc3 h (S = 1+δ jc3 y ki y kc3 w jc3 k, c 3 h w jc3 (S = 1 Furthermore, note that h {y ki h(s} =δ ij + y ki (S. y kj y kj Thus, it suffices to prove that E[w ij h(s] = k y ki h y kj (S. (16 n E[y ki y kj h(s] = E [δ F ij + 1 ] y h ki (S, (17 y kj 1

13 in which the expectation is taken with respect to a density function Det (Σ m f( R with R =Tr(Σ 1 x x+y jk y jk + But, from integration-by-parts, we note that y ki y kj h(sf( R dx dy kj = 1 (c 1,c (j, k (c 1,c (j, k dy c1 c y kj {y ki h(s}f ( R dx dy kj y c1 c. (c 1,c (j, k dy c1 c and that {y ki h(s} =δ ij + y ki {h(s}. y kj y kj Here, dx and dy c1 c indicate m j=1 p d (Re x jkd (Im x jk and d (Re y c1 c d (Im y c1 c, respectively. These two equations give the equation (17, which completes the proof of the lemma. 3. Proof of Lemma. Since the real and the imaginary parts of S are symmetric and skew-symmetric matrices respectively, we have ds ab (ds st = 1+δ st ( d(re sab + 1 d(im s ab ( (Re s st + 1 (Im s st = 1+δ { ( ( } st d(re s ab d(im s ab (Re s st (Im s st = 1+δ { } st δas δ bt + δ at δ bs (δ as δ bt δ at δ bs (1 δ st 1+δ st = δ at δ bs. (18 Now taking the differentials of S = ULU and multiplying on the left by U and on the right by U, we get U dsu =(U dul + L(U du + dl. (19 But, from the differentials of U U = I p, we can see that U du is skew-symmetric. Expressing the diagonal and the off-diagonal elements of (19 we get dl k = {U dsu} kk, (0 1 {U l du} ak = k l a {U dsu} ak (a k, (1 0 (a = k. 13

14 Using (18, (0, and (1, we get d st S l k = a, b u ak ds ab (d st S u bk = a, b ds st (u jk = u ja (U du ak (ds st = a k = u ak δ at δ bs u bk = u tk u sk, 1 u ja u c1 aδ c1 tδ c su c k = l k l a a k, c 1,c a k 1 u ja u c1 a ds c1 c l k l (ds st u c k a a k, c 1,c u ja u ta u sk l k l a, which completes the proof of the first and the second equalities in this lemma. The third equality in this lemma is obtained by taking the complex conjugate of (19 and proceeding via a similar calculation to that presented in the derivation of the second equality in this lemma. 3.3 Proof of Lemma.3 Use Lemma. to see that the (i, j-element of D S [UΦ(LU ]is d ik1 S u k 1 k ϕ k u jk = [ ϕ k u jk d ik 1 S u k 1 k + ϕ k u k1 k d ik 1 S u jk + u k1 k u jk k 1,k k 1,k k 3 = [ u k1au k1au ik ϕ k u jk + ϕ k u k1 k l k l a l k l a k 1,k a k ] ϕ k +u k1 k u jk u ik3 u k1 k l 3 k3 k 3 u jk, = k u ik ( ϕk l k + a k ϕ k ϕ a l k l a a k u jau iau k1k ] ϕ k d ik 1 S l l k 3 k3 which completes the proof. Acknowledgements I am grateful to the anonymous referee and the advisory editor, Professor Michael D. Perlman, for their detailed and helpful comments and suggestions which enabled me to improve the presentation of the paper. I would like to thank Dr. Lennart Svensson for providing me with his unpublished manuscript. References Abraham, D.A., Dey, D.K., The application of improved covariance estimation to 14

15 adaptive beamforming and detection. Signals, Systems and Computers, Conference Record of the Twenty-Eighth Asilomar Conference Volume 1, Andersen, H.H., Højbjerre, M., Sørensen, D., Eriksen, P.S., models, Springer-Verlag, New York. Linear and Graphical Daniels, M.J., Kass, R.E., Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models. J. Amer. Statist. Assoc. 94, Daniels, M.J., Kass, R.E., 001. Shrinkage estimators for covariance matrices. Biometrics 57, Dey, D.K., Srinivasan, C., Estimation of a covariance matrix under Stein s loss. Ann. Statist. 13, DoGondžić, A., Neborai, A., 003. Generalized multivariate analysis of variance. IEEE Signal Processing Magazin, Goodman, N.R., Statistical analysis based on a certain multivariate complex Gaussian distribution (An introduction. Ann. Math. Statist. 34, Haff, L.R., Estimation of the inverse covariance matrix: random mixtures of the inverse Wishart matrix and the identity. Ann. Statist. 7, Haff, L.R., The variational form of certain Bayes estimators. Ann. Statist. 19, James, W., Stein, C., Estimation with quadratic loss. In: Proc. Fourth Berkeley Symp. Math. Statist. Prob. 1, , Univ. California Press. Khatri, C.G., Classical statistical analysis based on a certain multivariate complex Gaussian distribution. Ann. Math. Statist. 36, Konno, Y., 005. Estimation of a normal covariance matrix parametrized by irreducible symmetric cones under Stein s loss, accepted for publication in Journal of Multivariate Analysis. Krishnaiah, P.R., Lin, J., Complex elliptically symmetric distributions. Comm. Statist. A Theory Methods 15, Kubokawa, T., Srivastava, M.S., Robust improvement in estimation of a covariance matrix in an elliptically contoured distribution. Ann. Statist. 7, Lin, S.P., Perlman, M.D., A Monte Carlo comparison of four estimators for a covariance 15

16 matrix. In Multivariate Analysis VI (P.R. Krishnaiah, ed., North-Holland, Amsterdam, , Maiward, D., Kraus, D., Calculation of moments of complex Wishart and complex inverse Wishart distributed matrices. Radar, Sonar and Navigation, IEEE Proceedings 147, Micheas, A.C, Dey, D.K., Mardia, K.V., 005. Complex elliptical distributions with application to shape analysis. Journal of Statistical Planning and Inference, in press, corrected proof, available online 7 April 005. Muirhead, R.J., 198. Aspects of multivariate statistical analysis. John Wiley & Sons, Inc. Perron, F., 199. Minimax estimators of a covariance matrix. J. Multivariate Anal. 43, 6-8 Sheena, Y., Unbiased estimator of risk for an orthogonally invariant estimator of a covariance matrix. J. Japan Statist. Soc. 5, Sheena, Y., Takemura, A., 199. Inadmissibility of non-order-preserving orthogonally invariant estimators of the covariance matrix in the case of Stein s loss. J. Multivariate Anal. 41, Stein, C., Lectures on the theory of estimation of many parameters. In Studies in the Statistical Theory of Estimation I ( I.A. Ibragimov and M.S. Nikulin, eds., Proceedings of Scientific Seminars of the Steklov Institute, Leningrad Division 74, 4-65 ( In Russia. English translation of this article is available in J. Sov. Math. 34 ( Svensson, L., 004. A useful identity for complex wishart forms, technical reports, Department of Signals and Systems, Chalmers University of Technology, December 004. Svensson, L., Lundberg, M., 004. Estimating complex covariance matrix. Signals, Systems and Computers, Conference Record of the Thirty-Eighth Asilomar Conference on Volume, 7-10, Young, R., Berger, J.O., Estimation of a covariance matrix using the reference prior. Ann. Statist.,