A recursion formula for expected negative and positive powers of the central Wishart distribution (p. 201-206)

Statistics & Operations Research Transactions SORT 31 (2) July-December 2007, 201-206 Statistics & Operations Research Transactions A recursion formula for expected negative and positive powers of the central Wishart distribution c Institut d’Estad´ ıstica de Catalunya [email protected] ISSN: 1696-2281 www.idescat.net/sort Heinz Neudecker Amsterdam School of Economics Abstract We use Haff’s fundamental identity to express the expectation of S p in lower-order terms, where S follows the central Wishart distribution. MSC: primary 62H10, secondary 15A69 Keywords: Haff’s Fundamental Identity, matrix differentiation, central Wishart distribution. 1 The recursion formula (positive powers) Let S∼Wm(Ω,n). Hence Sfollows the Central Wishart distribution with scale matrix Ω>0andndegrees of freedom. We use Haff’s Fundamental Identity (FI): εF1Ω−1F2=2εF1∇F2+2εF 2∇F 1+(n−m−1) εF1S−1F2, with F1=F1(S)adifferentiable matrix function of Sand n>m+1. Further ∇is a matrix of differential operators with typical element dij =1 21+δij∂ ∂sij (i,j=1,...,m), where δij is a Kronecker delta: δii =1andδij =0(ij). Address for correspondence: Oosterstraat, 13. 1741 GH Schagen. The Netherlands. E-mail: [email protected] Received: May 2004 Accepted: December 2006 202 A recursion formula for expected negative and positive powers of the central Wishart distribution All matrices are square of dimension m. A very useful property is: when dF = P(dS)Qthen 2 ∇F=PQ +(tr P)Q, where dF is the differential of F.For these properties see Neudecker (2001). Further εis the expectation operator. We choose F1=Imand F2=Sp,(p>0). By Haff’s FI we get: εΩ−1Sp=2ε∇Sp+(n−m−1) εSp−1, because ∇I=0. For definition and computation of ∇(·) see also Neudecker (2000). Clearly dSp= p−1  k=0 Sk(dS)Sp−k−1 where S0=Im. Hence 2∇Sp=(m+p)Sp−1+ p−1  k=1 tr SkSp−k−1. Then Ω−1εSp=(n+p−1) εSp−1+ε p−1  k=1 tr SkSp−k−1 or equivalently εSp=(n+p−1)ΩεSp−1+Ωε p−1  k=1 tr SkSp−k−1. 2 Discussion and application (positive powers) The second recursion formula is not really suited for finding εSp, because this requires knowledge of εtr SkSp−k−1for 1 ≤k≤p−1. This expression has to be found by other methods. See Neudecker (2001). The formula is, however, suited for establishing Loewner orderings. Relevant is that ΩεSp=εSpΩ>0 (positive definite). This follows from the first recursion formula, which yields the identity Ω−1εSp=εSpΩ−1 and further the positive definiteness of the two expressions. Preand postmultiplication of the identity by matrix Ωyields: ΩεSp=εSpΩ.AsΩand εSpare two commuting positive definite matrices, they are simultaneously diagonizable by an orthogonal similarity transformation. Hence TΩT=Λand TεSpT=M, say, and consequently Heinz Neudecker 203 ΩεSp=TΛTTMT =TΛMT>0 (positive definite), as both Λand Mare diagonal positive definite. See e.g. Hadley (1972, ex. 7-14). We further conclude that Ω−1εSp>(n+p−1)εSp−1or equivalently (εS)−1εSp> n−1(n+p−1)εSp−1as εS=nΩ(p≥2). The first inequality follows from the second recursion formula. Repeated substitution finally yields the inequality εSp>(εS)p 3 The recursion formula (negative powers) With F1=Imand F2=S−p,wegetbyHaff’s FI Ω−1εS−p=2ε∇S−p+(n−m−1) εS−(p+1) (1) For p=1,2,...,n−m−2, we clearly have dS−p=− p  k=1 S−k(dS)Sk−(p+1) Hence 2∇S−p= p  k=1 S−(p+1) − p  k=1 tr S−kSk−(p+1) =−pS−(p+1) − p  k=1 tr S−kSk−(p+1) Insertion in (1) leads to the recursion formula Ω−1εS−p=(n−m−p−1)εS−(p+1) − p  k=1 εtr S−kSk−(p+1) (2) or equivalently εS−p=(n−m−p−1)ΩεS−(p+1) −Ω p  k=1 εtr S−kSk−(p+1) (3) 204 A recursion formula for expected negative and positive powers of the central Wishart distribution 4 Discussion and application (negative powers) Because of the symmetry of the RHS of (2) we have Ω−1εS−p=(εS−p)Ω−1.(4) We shall prove that Ω−1εS−p>0 (positive definite). Proof. From the commutativity property (4) we conclude that εS−p=TΛpTsay and Ω−1=TMT where Tis orthogonal and Mand Λpare diagonal positive definite. Hence Ω−1 2=TM 1 2T and Ω−1 2(εS−p)Ω−1 2=TM 1 2TTΛpTTM 1 2T =TM 1 2ΛpM1 2T=TMΛpT=TMT TΛpT =Ω −1εS−p Hence Ω−1εS−p>0asΩ−1 2(εS−p)Ω−1 2>0 Because εS−(p+1) >0and p  k=1 εtr S−kSk−(p+1) >0 we conclude from (2) that Ω−1εS−p<(n−m−p−1)εS−(p+1) (5) when p<n−m−1. Or equivalently εS−1εS−p<(n−m−1)−1(n−m−p−1)εS−(p+1) <εS−(p+1) (6) as εS−1=(n−m−1)−1Ω−1 and (n−m−1)−1(n−m−p−1) <1. The inequality (6) ultimately leads by successive substitution to the inequality εS−p>εS−1p(p≥2) (7) Heinz Neudecker 205 As εS−1=n(n−m−1)−1(εS)−1>(εS)−1, we can get the inequality εS−p>εS−1p>(εS)−p(8) It is known that εSp>(εS)pp≥2 (9) Seetheendofsection2. Combining (8) and (9) we finally have the inequality εS−p>εS−1p>(εS)−p>(εSp)−1(10) 5 References Hadley, G. (1972). Linear Algebra. Addison-Wesley Publishing Company Inc., Reading. Neudecker, H. (2000). «A note on the matrix Haffian».Q¨uestii´o, 24, 419-424. Neudecker, H. (2001). «Some applications of the matrix Haffian in connection with differentiable matrix functions of a central Wishart variate».Q¨uestii´o, 25, 187-210.