PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 123, Number 8, August 1995 ON DISGUISED INVERTED WISHART DISTRIBUTION A. K. GUPTA AND S. OFORI-NYARKO (Communicated by Wei-Yin Loh) Abstract. Let A ~ Wp(n, I) and A = ZZ' where Z is a lower triangular matrix with positive diagonal elements. Further, let B = A~l = W'W have inverted Wishart distribution so that W = Z-1. In this paper we derive the distribution of M = WLW1. It is also shown that a=£±±T'MT ~ Fp,„_p+1 where T ~ A/p(0, Ip) is independent of M . 1. Introduction While deriving the minimax estimator of a normal covariance matrix when additional information is available on some coordinates [2], we were confronted with finding the distribution of a random variable of the type (Z'Z)~X where Z is a lower triangular matrix with positive diagonal elements such that ZZ' = A(p x p) ~ Wp(n,L). Here we first derive the distribution of M - WLW, where W = Z~x. The distribution of T'MT is also derived where indepen- dently T ~ Np(0, Ip). We call M a disguished inverted Wishart variable for reasons explained in §3. Tan and Guttman [3] derived the distribution of a disguised Wishart variable. In §2 we present some preliminary results which are used in the sequel. In §3 main results of the paper are derived. 2. Some preliminary results The following lemmas are needed to derive the distribution of M. The proofs of Lemmas 1.1 and 1.2 are given in [1]. Lemma 2.1. If M = MXM\, where Mx is a p x p lower triangular matrix with positive diagonal elements, then (2.1) J(M^Mx) = 2pf[mp-;)+[ i=i where znil(i) is the ith diagonal element of Mx. Also the transformation N = Received by the editors July 7, 1993. 1991 Mathematics Subject Classification. Primary 62H10; Secondary 62H12. Key words and phrases. Minimax estimation, risk, Jacobian, lower triangular matrix, F-distribu- tion. © 1995 American Mathematical Society 2557 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use