The main aim is to estimate the noncentrality matrix of a noncentral Wishart distribution. The method used is Leung's but generalized to a matrix loss function. Parallelly Leung's scalar noncentral Wishart identity is generalized to become a matrix identity. The concept of L¨owner partial ordering of symmetric matrices is used

Neudecker, Heinz

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Statistics & Operations Research TransactionsSORT 28 (2) July-December 2004, 191-200Statistics &Operations ResearchTransactionsEstimation of the noncentrality matrix of a noncentralWishart distribution with unit scale matrix.A matrix generalization of leung’s domination resultHeinz Neudecker∗CesaroAbstractThe main aim is to estimate the noncentrality matrix of a noncentral Wishart distribution. The methodused is Leung’s but generalized to a matrix loss function. Parallelly Leung’s scalar noncentral Wishartidentity is generalized to become a matrix identity. The concept of Lo¨wner partial ordering of symmetricmatrices is used.MSC: 62H12, 15A24, 15A45Keywords: Noncentral Wishart matrix identity, noncentrality matrix, decision-theoretic estimation,matrix loss function, Lo¨wner matrix ordering, Haffians1 IntroductionWe consider S ∼ Wm(n, Im,M′M). Following Leung (1994) we recall that the habitualunbiased estimator of M′M is T := S − n Im. Under certain conditions Tα := T +α(tr S )−1 Im dominates T for a suitable choice of α, as was shown by Leung, who usedthe loss functionλ[(M′M)−1,R]:= tr{(M′M)−1 R − Im}2.He extended work by Perlman & Rasmussen (1975), Saxena & Alam (1982), Chow(1987) and Leung & Muirhead (1987).∗ Address for correspondence: Oosterstraat, 13. 1741 GH Schagen. The Netherlands. Postal address: NL 1741 GH13. E-mail: heinz@fee.uva.nlReceived: May 2001Accepted: October 2003192 Estimation of the noncentrality matrix of a noncentral Wishart distribution with unit scale matrix...In this article we propose to use a matrix loss function, viz L[(M′M)−1,R] :={(M′M)−1 R − Im}′ {(M′M)−1 R − Im} and apply the concept of Lo¨wner partial orderingof symmetric matrices. We shall show that Leung’s result still holds approximately, theerror term being of order o(n−1). For accomplishing this we need a matrix version ofLeung’s Identity for the noncentral Wishart distribution. This will be presented first.A matrix version of an ancillary lemma by Leung, viz his Lemma 3.1 will next beestablished. The generalized domination result will then follow straightforwardly.We shall employ an approximation of E(tr S )−1S , where E is the expectationoperator. A lemma on the matrix Haffian ∇ϕF, where ϕ and F are scalar and matrixfunctions of S , will be proved in Appendix 1. In Appendix 2 we shall prove a lemmaon the scalar Haffian tr∇F2AF1, when F1 and F2 are matrix functions of S and A is aconstant matrix.2 A matrix version of Leung’s identity for the noncentral Wishart2 distributionWe quote Leung’s Theorem 2.1, where without loss of generality we take h = 1, h beinga scalar function of S in Leung’s work:E trΣ−1F = 2E tr∇F + (n − m − 1)E tr S −1F + E1 trΣ−1M′MS −1F, (1)where S ∼ Wm(n,Σ,Σ−1M′M), E denotes the expectation with respect to this distribu-tion, E1 denotes the expectation with respect to the distribution Wm (n + m + 1,Σ,Σ−1 M′M), F = F(S ) and n > m + 1. The matrices S ,Σ, F and ∇ are square ofdimension m, whereas M has dimension n × m. It is assumed that M has full columnrank. Further ∇F is the matrix Haffian as denoted by Neudecker (2000b). Inspired byHaff (1981), who did it for the central Wishart distribution, we shall establish a matrixversion of (1).Theorem 1EF1Σ−1F2 = 2EF1∇F2 + 2(EF′2∇F′1)′++ (n − m − 1)EF1S −1F2 + E1F1Σ−1M′MS −1F2,(2)for F1 and F2 satisfying the conditions of Lemma 5.Proof. Take F = F2e je′i F1, with unit vectors ei and e j. We then use the identity:tr∇F2AF1 = tr (∇F2) AF1 + tr (∇F′1) A′F′2,with constant A. For a proof see Lemma 5.Heinz Neudecker 193Taking A = e je′i we getE trΣ−1F2e je′i F1 = 2E tr (∇F2) e je′i F1 + 2E tr(∇F′1)eie′jF′2++ (n − m − 1)tr ES −1F2e je′i F1 + E1 trΣ−1M′MS −1F2e je′i F1or equivalently(EF1Σ−1F2)i j = 2 (EF1∇F2)i j + 2(EF′2∇F′1)ji ++ (n − m − 1)(EF1S −1F2)i j +(E1F1Σ−1M′MS −1F2)i j .¤Note: It was assumed that (1) holds for all F = F2e je′i F1, which puts strongerconditions on the input matrix than was necessary for (1). By choosing F1 = Im andtaking traces we derive (1) from (2).For discussion of the central Wishart case we refer to Haff (1981).3 A matrix version of Leung’s lemma 3.1Lemma 2E (tr S )−1 (M′M)−1 S (M′M)−1 < nE (trS )−1 (M′M)−2 −−2(n − 4)E (tr S )−2 (M′M)−2 + E1 (tr S )−1 (M′M)−1 −−2E1 (tr S )−2 (M′M)−1 ,where S ∼ Wm(n, Im,M′M) and M′M is assumed to be nonsingular. The inequalityA < B, for symmetric A and B, stands for the Lo¨wner ordering meaning that B − A ispositive definite.Proof. Take F1 = (tr S )−1 (M′M)−1 and F2 = S (M′M)−1. By Theorem 1 (withΣ = Im):E (tr S )−1 (M′M)−1 S (M′M)−1 = 2E (tr S )−1 (M′M)−1 ∇S (M′M)−1 ++2{E (M′M)−1 S∇ (tr S )−1 (M′M)−1}′++(n − m − 1)E (tr S )−1 (M′M)−2 + E1 (tr S )−1 (M′M)−1 == (m + 1)E (tr S )−1 (M′M)−2 − 2E (tr S )−2 (M′M)−1 S (M′M)−1 ++(n − m − 1)E (tr S )−1 (M′M)−2 + E1 (tr S )−1 (M′M)−1 = (i)194 Estimation of the noncentrality matrix of a noncentral Wishart distribution with unit scale matrix...= nE (tr S )−1 (M′M)−2 − 2E (tr S )−2 (M′M)−1 S (M′M)−1 ++E1 (tr S )−1 (M′M)−1 (i)FurtherE (tr S )−2 (M′M)−1 S (M′M)−1 = 2E (tr S )−2 (M′M)−1 ∇S (M′M)−1 ++2{E (M′M)−1 S∇ (tr S )−2 (M′M)−1}′+ (n − m − 1)E (tr S )−2 (M′M)−2 ++E1 (tr S )−2 (M′M)−1 ,where we applied Theorem 1 (with Σ = Im) using F1 = (tr S )−2 (M′M)−1 and F2 =S (M′M)−1. Proceeding as before we getE (tr S )−2 (M′M)−1 S (M′M)−1 = (m + 1)E (tr S )−2 (M′M)−2 −−4E (tr S )−3 (M′M)−1 S (M′M)−1 + (n − m − 1)E (tr S )−2 (M′M)−2 ++E1 (tr S )−2 (M′M)−1 = nE (tr S )−2 (M′M)−2 −−4E (tr S )−3 (M′M)−1 S (M′M)−1 + E1 (tr S )−2 (M′M)−1 .We use the Lo¨wner ordering S < (tr S )Im, which yields (M′M)−1 S (M′M)−1 <(tr S ) (M′M)−2. Hence we get(n − 4)E (tr S )−2 (M′M)−2 + E1 (tr S )−2 (M′M)−1 < E (tr S )−2 (M′M)−1 S (M′M)−1 .Insertion in (i) finally yieldsE (tr S )−1 (M′M)−1 S (M′M)−1 < nE (tr S )−1 (M′M)−2 − 2(n − 4)E (tr S )−2 (M′M)−2−2E1 (tr S )−2 (M′M)−1 + E1 (tr S )−1 (M′M)−1 .¤Notes:1. Nonsingularity of M′M is not trivial. A case of singularity is M′ = µl′, where then means are proportional.2. Leung assumes n > 4. There is no need for it.3. Taking traces in Lemma 2 yields Leung’s Lemma 3.1.Heinz Neudecker 1954 A matrix version of Leung’s domination resultWe shall now prove the main result of this paper.Theorem 3EL[ (M′M)−1,T]> EL[ (M′M)−1,Tα]for 0 < α 6 4 (n − 4), whereL[ (M′M)−1,R]:={(M′M)−1 R − Im}′ {(M′M)−1 R − Im} ,T := S − nIm and Tα := T + α (tr S )−1 Im.Proof.L[ (M′M)−1 ,T ] − L[ (M′M)−1 ,Tα] = {(M′M)−1 T − Im}′ {(M′M)−1 T − Im}−−{(M′M)−1 Tα − Im}′ {(M′M)−1 Tα − Im} = 2nα (tr S )−1 (M′M)−2 −−α2 (tr S )−2 (M′M)−2 − α (tr S )−1 S (M′M)−2 − α (tr S )−1 (M′M)−2 S++2α (tr S )−1 (M′M)−1 .Its expected value is2nαE (tr S )−1 (M′M)−2 − α2E (tr S )−2 (M′M)−2 − αE (tr S )−1 S (M′M)−2 −−αE (tr S )−1 (M′M)−2 S + 2αE (tr S )−1 (M′M)−1 >> 2αE (tr S )−1 (M′M)−1 S (M′M)−1 ++4α(n − 4)E (tr S )−2 (M′M)−2 − 2αE1 (tr S )−1 (M′M)−1 + 4αE1 (tr S )−2 (M′M)−1 −−α2E (tr S )−2 (M′M)−2 − αE (tr S )−1 S (M′M)−2 − αE (tr S )−1 (M′M)−2 S++2αE (tr S )−1 (M′M)−1= 2αE (tr S )−1 (M′M)−1 S (M′M)−1 + α [4(n − 4) − α] E (tr S )−2 (M′M)−2 ++2α{E (tr S )−1 (M′M)−1 − E1 (tr S )−1 (M′M)−1}++4αE1 (tr S )−2 (M′M)−1 − αE (tr S )−1 S (M′M)−2 − αE (tr S )−1 (M′M)−2 S ,by Lemma 2.We approximate E (tr S )−1 S byµ (n Im + M′M) − 2µ2 (n Im + 2M′M)++2µ3 (mn + 2 tr M′M) (n Im + M′M) ,with µ−1 := tr (n Im + M′M), the remainder being of order o(n−1).196 Estimation of the noncentrality matrix of a noncentral Wishart distribution with unit scale matrix...Insertion yields2αE (tr S )−1 (M′M)−1 S (M′M)−1 − αE (tr S )−1 S (M′M)−2 −−αE (tr S )−1 (M′M)−2 S = O + o(n−1).Hence to the order of approximationEL[ (M′M)−1 ,T ] − EL[ (M′M)−1 ,Tα] > α [4(n − 4) − α] E (tr S )−2 (M′M)−2 ++2α[E (tr S )−1 (M′M)−1 − E1 (tr S )−1 (M′M)−1]+ 4αE1 (tr S )−2 (M′M)−1 > O,as E (tr S )−1 > E1 (tr S )−1.For the auxiliary inequality see Leung (1994, p. 112). ¤Appendix 1: a lemma on the matrix Haffian ∇ϕFLemma 4∇ϕF = ϕ∇F +∂ϕ∂XF,where ϕ is a scalar function of the symmetric matrix variable X and F is a matrixfunction thereof. Further∂ϕ∂X:= 12∑i j∂ϕ∂xi j(Ei j + E ji), where Ei j := ei e′jProof.(∇ϕF)ik =∑jdi j (ϕF) jk =∑jdi jϕ f jk = 12∑j(1 + δi j) ∂ϕ f jk∂xi j==∂ϕ fik∂xii+12∑j,i∂ϕ f jk∂xi j= ϕ∂ fik∂xii +12∑j,i∂ f jk∂xi j +(∂ϕ∂xii)fik++12∑j,i∂ϕ∂xi jf jk = ϕ (∇F)ik +(∂ϕ∂X)i.F.k, hence∇ϕF = ϕ∇F +∂ϕ∂XF.Here f jk and (F) jk are the jkth element of F, Fi. is the ith row of F and F. j is the kthcolumn of F.For more details see Neudecker (2000b). ¤Heinz Neudecker 197Appendix 2: a lemma on the scalar Haffian tr∇F2AF1Lemma 5tr∇F2AF1 = tr (∇F2) AF1 + tr (∇F′1) A′F′2,where F2 and F1 are functions of the symmetric matrix variable X and A is a constantmatrix.Each F satisfies F =∑kϕk Ck or dF =∑lPl(dX)Q′l with constant Ck, Pl and Ql.We consider three cases. The first comprises F1 = ϕC and dF2 = P(dX)Q′, thesecond comprises F2 = ϕC and dF1 = P(dX)Q′, the third comprises dF1 = P(dX)Q′and dF2 = R(dX)T ′. The fourth case with F1 = ϕ1C1 and F2 = ϕ2C2 follows easily.Without loss of generality the summation signs were dropped.Proof.Case 1. We have dF1 = (dϕ)C, hence by Lemma 4 ∇F′1 = ∂ϕ∂X C′. Furtherd (F2AF1) = (dF2) AF1 + F2AdF1= P(dX)Q′AF1 + (dϕ)F2ACwhich implies∇F2AF1 = 12 P′Q′AF1 + 12 (tr P)Q′AF1 +∂ϕ∂XF2AC,tr∇F2AF1 = 12 tr P′Q′AF1 + 12 (tr P) tr Q′AF1 + tr∂ϕ∂XF2AC;(∇F2) AF1 = 12 P′Q′AF1 + 12 (tr P)Q′AF1,tr (∇F2) AF1 = 12 tr P′Q′AF1 + 12 (tr P) tr Q′AF1;(∇F′1)A′F′2 =∂ϕ∂XC′A′F′2,tr(∇F′1)A′F′2 = tr∂ϕ∂XC′A′F′2 = tr∂ϕ∂XF2AC.This yields the result.Case 2. We replace F1 by F′2, A by A′ and F2 by F′1 in the first result. This leads totr∇F′1A′F′2 = tr(∇F′1)A′F′2 + tr (∇F2) AF1.198 Estimation of the noncentrality matrix of a noncentral Wishart distribution with unit scale matrix...Using tr∇F′ = tr∇F we gettr∇F2AF1 = tr(∇F′1)A′F′2 + tr (∇F2) AF1.Case 3. Now dF1 = P(dX)Q′ and dF2 = R(dX)T ′. Then2∇F1 = P′Q′ + (tr P) Q′2∇F2 = R′T ′ + (tr R) T ′2∇F′1 = Q′P′ + (tr Q) P′by the Theorem in Neudecker (2000b).FurtherdF2AF1 = (dF2)AF1 + F2AdF1,= R(dX)T ′AF1 + F2AP(dX)Q′,which implies2∇F2AF1 = R′T ′AF1 + P′A′F′2Q′++ (tr R) T ′AF1 + (tr F2AP) Q′= 2 (∇F2) AF1 + P′A′F′2Q′ + (tr F2AP) Q′and hence2 tr∇F2AF1 = 2 tr (∇F2) AF1 + tr [Q′P′ + (tr Q) P′] A′F′2= 2 tr (∇F2) AF1 + 2 tr(∇F′1)A′F′2.¤Note: For an introduction to the scalar Haffian see Neudecker (2000a).5 ReferencesChow, M. S. (1987). A complete class theorem for estimating a noncentrality parameter. Ann. Statist., 15,800-4.Haff, L. R. (1981). Further identities for the Wishart distribution with applications in regression. CanadianJ. Statist., 215-24.Leung, P. L. (1994). An identity for the noncentral Wishart distribution with application. J. MultivariateAnal., 48, 107-14.Leung, P. L. & Muirhead, R. J. (1987). Estimation of parameter matrices and eigenvalues in MANOVAand canonical correlation analysis. Ann. Statist., 15, 1651-66.Neudecker, H. (2000a). A note on the scalar Haffian. Qu¨estiio´, 24(2), 243-9.Heinz Neudecker 199Neudecker, H. (2000b). A note on the matrix Haffian. Qu¨estiio´, 24(3), 419-24.Perlman, M. D. & Rasmussen, U. A. (1975). Some remarks on estimating a noncentrality parameter.Comm. Statist., 4, 455-68.Saxena, K. M. L. & Alam, K. (1982). Estimation of the noncentrality parameter of a chi-squareddistribution. Ann. Statist., 10, 1012-6.Hola

Estimation of the noncentrality matrix of a noncentral Wishart distribution with unit scale matrix. A matrix generalitzation of Lenng's domination result

The main aim is to estimate the noncentrality matrix of a noncentral Wishart distribution. The method used is Leung’s but generalized to a matrix loss function. Parallelly Leung’s scalar noncentral Wishart

identity is generalized to become a matrix identity. The concept of L¨owner partial ordering of symmetric matrices is used.Peer Reviewe

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The main aim is to estimate the noncentrality matrix of a noncentral Wishart distribution. The method used is Leung’s but generalized to a matrix loss function. Parallelly Leung’s scalar noncentral Wishart
identity is generalized to become a matrix identity. The concept of L¨owner partial ordering of symmetric matrices is used.Peer Reviewe

Estimation of the noncentrality matrix of a noncentral Wishart distribution with unit scale matrix. A matrix generalitzation of Lenng's domination result

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