Research Article On the Marginal Distribution of the Diagonal Blocks in a Blocked Wishart Random Matrix

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1 Hindawi Publishing Corporation nternational Journal of Analysis Volume 6, Article D 59678, 5 pages Research Article On the Marginal Distribution of the Diagonal Blocks in a Blocked Wishart Random Matrix Kjetil B. Halvorsen, Victor Ayala,,3 and Eduardo Fierro 4 NordUniversity,87Nesna,Norway nstituto de Alta nvestigación, Universidad de Tarapacá, Antofagasta N. 5, Arica, Chile 3 Universidad Católica del Norte, Región de Antofagasta, Chile 4 Departamento de Matemáticas, Universidad Católica del Norte, Casilla 8, Antofagasta, Chile Correspondence should be addressed to Victor Ayala; vayala@ucn.cl Received 9 June 6; Accepted September 6 Academic Editor: Shamsul Qamar Copyright 6 Kjetil B. Halvorsen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let A be a (m +m ) (m +m ) blocked Wishart random matrix with diagonal blocks of orders m m and m m.thegoalof the paper is to find the exact marginal distribution of the two diagonal blocks of A. We find an expression for this marginal density involving the matrix-variate generalized hypergeometric function. We became interested in this problem because of an application in spatial interpolation of random fields of positive definite matrices, where this result will be used for parameter estimation, using composite likelihood methods.. ntroduction The goal of this paper is to find an exact and useful form for the marginal distribution of the diagonal blocks of a blocked Wishart random matrix. This problem arises in an applied problem, to estimate the parameters of a Wishart random field, which will be reported elsewhere. Let A be a (m +m ) (m +m ) Wishart random matrix, where the diagonal blocks are of orders m m and m m, respectively. n our intended application m, m will be small integers (and m m,butwechoosetotreatthemore general case). Write A( A A A A ). Denote the number of freedom parameters by n and the scale parameter, which is a matrix blocked in the same way as A, byσ( Σ Σ Σ Σ ). We are mostly interested in the special case Σ( Σ ρσ ρσ Σ ) where the absolute value of ρ is less than one, but the general case is not more difficult. All matrices are real. Notation: we use Tr(A) for the trace ofthesquarematrixa and etr(a) exp(tr(a)). Wewrite P(m) for the convex cone of real m mpositive definite matrices, and we write O(m) for the orthogonal group, that is, the set of m m orthogonal matrices. The Stiefel manifold, that is, the set of m m column orthogonal matrices is written as V m,m (m m ).Weindicatethetransposeofamatrix by superscript. ntheconvexconeofpositivedefinitematrices,weusethe cone order, defined by A<Bmeaning that B Ais positive definite, written as B A >. ntegrals over cones are written as g(a)(da) meaning the integral is taken over the cone <A<. The multivariate gamma function is denoted by Γ m (a) for R(a) > (m )/; see Muirhead [] for proofs and properties. n Section we give some background information, especially about the Jacobians which we need to evaluate the integrals. n Section 3 we state our results and give proofs. n Section 4 we give some comments on the result.. Background The single most important reference for background material for this paper is Muirhead []. Some results therefrom will not be cited directly. When doing change of variables in a multiple integral we need to know the Jacobian. Here we will list the ones we need;

2 nternational Journal of Analysis most can be found in Muirhead [] or in Mathai []. We are following the notation of Muirhead []. First there is a very brief summary. For any matrix X,let dx denote the matrix of differentials dx ij. For an arbitrary m m matrix X, thesymbol(dx) denotes the exterior product of the mn elements of dx: m m (dx) dx ij. () ji f X is a symmetric m m matrix, the symbol (dx) will denote the exterior product of the m (m + )/ distinct elements of dx: m (dx) dx ij, () i j m with similar definitions for other kinds of structured matrices. The following invariant form in the orthogonal group represents the Haar measure, (H dh) m i m ji+ h j dh i.here H represents an orthogonal matrix. This form normalized to have total mass unity is represented by (dh). We also need to integrate over a Stiefel manifold; then (H dh) represents a similarly defined invariant form; see Muirhead []. Some needed Jacobians are not in Muirhead [], so we cite those Jacobians here, from Díaz-García et al. [3, 4]. Lemma (Jacobianofthesymmetricsquarerootofapositive definite matrix). Let S and R be in P + (m) such that SR and let Δ beadiagonalmatrixwiththeeigenvaluesofr on the diagonal. Then, (ds) m det (Δ) m i j m i<j (Δ i +Δ j ) (dr). (Δ i +Δ j ) (dr) ThisresultcanalsobefoundinMathai[]. We need the generalized polar decomposition of a rectangular matrix. Let C be m m rectangular matrix with m m.thenwealwayshavecuhwhere H is positive semidefinite and positive definite if C has full rank, and U is m m column orthogonal matrix. n that last case, U is unique; see Higham [5]. Lemma (Generalized polar decomposition). Let X be m m matrix with m m and of rank m,withm distinct singular values. Write X U H,withU V m,m and H P(m ).ThenHhas m distinct eigenvalues. Also let Δ be the diagonal matrix with the eigenvalues of H on the diagonal. Then m (dx) det (Δ) m m (Δ i +Δ j ) (dh) (U du). (4) i<j Note that since those results are used for integration, the assumption of distinct singular values is unimportant, since the subset where the singular values are equal has measure zero. (3) 3. Results Let us state our main result. Theorem 3 (The marginal distribution of the diagonal blocks of a blocked Wishart random matrix with blocks of unequal sizes). Let A( A A A A ) be a (m +m ) (m +m ) blocked Wishart random matrix, where the diagonal blocks are of sizes m m and m m,respectively.thewishartdistribution of A has n m +m degrees of freedom and positive definite scale matrix Σ( Σ Σ Σ Σ ) blocked in the same way as A. The marginal distributions of the two diagonal blocks A and A have density function given by c etr { (Σ A +F C FA )} etr { C A } det (A ) (n m )/ det (A ) (n m )/ F where C A / FA F A / G ), 4 (5) Σ Σ Σ Σ, F C Σ Σ,andG. c (m +m )n/ Γ m (n/)γ m (n/)(det Σ) n/. F is the generalized matrix-variate hypergeometric function, as defined in Muirhead []. Note that the definition of the matrix-variate hypergeometric function is by a series expansion, which is convergent in all cases we need; see Muirhead []. The rest of this section consists of a proof of this theorem. ntroduce the following notation: the Schur complements of Σ ( Σ Σ Σ Σ ) is C Σ Σ Σ Σ and C Σ Σ Σ Σ. Then define FC Σ Σ. n the following we will be using some standard results on blocked matrices without quoting them. The Wishart density function of A written as a function of the blocks is c etr ( (Σ A +F C FA F A +C A )) det (A ) γ det (A A A A ) γ, where c (m +m )n/ Γ m +m ((/)n)(det Σ) n/ and γ (n m m )/.nthefollowingwewillworkwiththedensity concentrating on the factors depending on A.Toprovethe theorem we need to integrate out the variable A.Theother variables, which are constant under the integration, will be concentrated in one constant factor. So we repeat formula (6) written as a differential form with the constants left out (6) K etr (FA ) det (A A A A ) γ (da ), (7) where K c etr( (/)(Σ A +F C FA )etr( (/ )C A )) det(a ) γ.now,tofindthemarginaldistribution of the diagonal blocks, we need to integrate over the offdiagonal block A. Under this integration the value of the

3 nternational Journal of Analysis 3 diagonal blocks A and A will remain fixed, and the region of integration will be a subset of R m m consisting of the matrices A such that the block matrix A ( A A A A ) is positive definite. This seems like a complicated set, but we can give a simple description of it using the polar decomposition of a matrix. Note that this is one of the key observations for the proof, and this author has not seen any use of this observation earlier. Now we need to assume that m m.fortheopposite inequality a parallel development can be given, using the other factorization det A det(a ) det(a A A A ). From, for instance, Theorem. in Zhang [6] it follows that the region of integration is the set {A R m m :<A A A <A }. (8) ntroduce E A / A A / where we use the usual symmetric square root. Then in terms of the new variable E the region of integration becomes {E R m m :<EE <} (9) andwiththegeneralizedpolardecompositionintheform E U P with P P + (m ), U V m,m, EE P so the region of integration can be written as {P P + (m ), U V m,m :<P <} () which is a Cartesian product of a cone interval with a Stiefel manifold. The Jacobian of the transformation from A to E is (de) (de ) det(a ) m/ det(a ) m/ (da ).The Jacobian of the polar decomposition E U P is (de) (de )(det Δ) m m m i<j (Δ i +Δ j )(dp)(u du),whereδ is adiagonalmatrixwiththeeigenvaluesofp on the diagonal; see Lemma. A last transformation will be useful. Define P X. The Jacobian of this transformation is (dx) m det Δ m i<j (Δ i +Δ j )(dp); Δ is as above. See Lemma. Applying this transformation the integral of (7) can be written as K etr (X / A / FA/ U) det ( X)γ V m,m () det (X) (m m )/ (dx) (U du), where the constant K m c etr ( ((Σ +F C F) A +C A )) (det A ) γ+m / (det A ) γ+m /. () We are ready to perform the integration over the Stiefel manifold. For this purpose we need a generalization of Theorem 7.4. from Muirhead [], which we cite here. Let X be m nreal matrix with m nand H[H : H ]n n orthogonal matrix, where H is n m.then etr (XH ) (dh) F O(n) 4 XX ). (3) But we have an integral over the Stiefel manifold, not the orthogonal group, so we need now to generalize the result (3) to an integral over the Stiefel manifold. What we need is the following. Let V m,m bethemanifoldofm m column orthogonal matrices with m m, and let f be a function defined on the Stiefel manifold. We can extend this function to a function defined on O(m ) in the following way. Let U be m m orthogonal matrix, and write it in block form as [U : U ] such that U V m,m. How can we characterize the set of U which is complementing U to form an orthogonal matrix? First, let U be a fixed but arbitrary matrix complementing U. Then clearly any other m (m m ) column orthogonal matrix with the same column space also works. The common column space is the orthogonal complement of the column space of U. The set of such matrices can be described as {V V m m,m : V U Q for Q O(m m )}. For this set we write V H m m,m. As a set we can identify this with O(m m ). Specifically, we can identify U with the very special column orthogonal matrix ( m m m ), where Q Q O(m m ) which clearly forms a proper submanifold of the Stiefel manifold V m m,m.thefunctionf can now be extended to the orthogonal group by defining f(u) f([u : U ]) f(u ) andfortheintegralwefindthat f(u )(U du) O(m ) V m,m V H m m,m f([u :U ]) (U du) V m,m f(u )(U du ) V H m m,m (Q dq) Vol O (m m ) V m,m f(h )(H dh ). (4) Returning to our integral, the integral over the Stiefel manifold occurring in () can now be written as V m,m etr (X / A / FA/ U)(U du) m )) etr (X / A / FA/ U )(U du), O(m ) where U consists of the m first columns of U )) m )) etr (X / A / FA/ U ) (du) O(m ) (5)

4 4 nternational Journal of Analysis )) m )) F ( m 4 A/ F A / XA/ FA/ ), (6) 4. Some Final Comments To help interpret our main result, we calculated the conditional distribution of the matrix A given the matrix A.We will not give the full details of the calculation here but only give the result. The density of A given that A a has the density given by wherewediduse(3).herevol(o(m)) m π m / /Γ m (m/) isthevolumeoftheorthogonalgroup;seemuirhead[].the differential form (du) denotes Haar measure normalized to total mass unity. Now write GA / FA F A / ; then we can write () as K )) m )) (det X) (m m )/ det ( X) γ F ( m GX ) (dx) 4 (7) and to evaluate this integral we need Theorem 7.. from Muirhead []; we do not state it here. Usingthiswefindaresultweneedfortheintegralof a hypergeometric function, by using the series expansion definition of the hypergeometric function and integrating term by term. Theorem 4. f Y is a symmetric m mmatrix one has that det (X) a (m+)/ det ( X) b (m+)/ p F q ( a,...,a p b,...,b q Γ m (a) Γ m (b) Γ m (a+b) p+ F q+ ( XY ) (dx) a,...,a p,a b,...,b q,a+b Y ) (8) so both degrees of the hypergeometric function are raised by one. The proof is a simple calculation that we leave out. Nowusing(8)tocalculate(7)weget,finally,theresult K )) Γ m (m/) Γ m ((n m )/) m )) Γ m (n/) F ( m m, n 4 G ) (9) butnotethatonepairofupperandlowerargumentsto the hypergeometric function are equal with those arguments canceled. With a little algebra we complete the proof of our main theorem. mn/ Γ m (n/) det (C ) n/ etr ( C A ) det (A ) (n m )/ etr ( Ω) F 4 ΩC A ), () where we have given the conditional density only for the special case Σ ( Σ ρσ ρσ Σ ).Forthiscasewehave,withthe notation from the main theorem, C C ( ρ )Σ, F (ρ/( ρ ))Σ,andF C F (ρ /( ρ ))Σ. We have defined Ω ρ C a,whichcanbeseenas a noncentrality parameter. The density above is equal to the noncentral Wishart distribution given in Theorem.3. in Muirhead []. We see that the conditional distribution is a kind of noncentral Wishart distribution, where the noncentrality parameter Ω depends on the conditioning matrix A.nthisway,theeffectoftheconditioningis to change the distribution of A,whichinthemarginal case is central Wishart, to a noncentral Wishart distribution, with noncentrality parameter depending on the conditioning matrix. As said in ntroduction, this result will be used for modelling of a spatial random field of tensors, where we will estimate the parameters using composite likelihood. This application will be reported elsewhere. For that application we will need to calculate values of matrix-variate hypergeometric functions numerically. A paper giving an efficient method for summing the defining series is Koev and Edelman [7], with associated Matlab implementation. Butler and Wood [8] give a Laplace approximation for the case we need, the F function. Competing nterests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments Kjetil B. Halvorsen was partially supported by Proyecto Mecesup UCN7 and Convenio de Cooperación Minera Escondida, Universidad Católica del Norte. Victor Ayala partially was supported by Proyecto Fondecyt no. 375 and no. 59. Eduardo Fierro was partially supported by Proyecto Mecesup UCN7.

5 nternational Journal of Analysis 5 References [] R. J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, 98. [] A.M.Mathai,Jacobians of Matrix Transformations and Functions of Matrix Arguments, World Scientific, 997. [3] J. A. Díaz-García and G. González-Farías, Singular random matrix decompositions: Jacobians, Journal of Multivariate Analysis, vol. 93, no., pp. 96 3, 5. [4] J. A. Díaz-García, R. Gutierrez Jaimez, and K. V. Mardia, Wishart and pseudo-wishart distributions and some applications to shape theory, Journal of Multivariate Analysis, vol.63, no., pp , 997. [5] N. J. Higham, Functions of Matrices, Theory and Computation, SAM, 8. [6] F.Zhang,Ed.,The Schur Complement and ts Applications, vol. 4,Springer,NewYork,NY,USA,5. [7] P. Koev and A. Edelman, The efficient evaluation of the hypergeometric function of a matrix argument, Mathematics of Computation,vol.75,no.54,pp ,6. [8] R. W. Butler and A. T. Wood, Laplace approximation for Bessel functions of matrix argument, Journal of Computational and Applied Mathematics,vol.55,no.,pp ,3.