WISHART AND CHI-SQUARE DISTRIBUTIONS ASSOCIATED WITH MATRIX QUADRATIC-FORMS

For a normally distributed random matrix Y with a general variance-cov ariance matrix Sigma(gamma), and for a nonnegative definite matrix Q, necessary and sufficient conditions are derived for the Wishartness of Y'QY. The conditions resemble those obtained by Wong, Masaro, and Wan g (1991, J. Multivariate Anal. 39, 154-174) and Wong and Wang (1993, J . Multivariate Anal. 44, 146-159), but are verifiable and are obtained by elementary means. An explicit characterization is also obtained fo r the structure of Sigma(gamma) under which the distribution of Y'QY i s Wishart. Assuming Sigma(gamma) positive definite, a necessary and su fficient condition is derived for every univariate quadratic from l'Y' QYl to be distributed as a multiple of a chi-square. For the case Q = I-n, the corresponding structure of Sigma(gamma) is identified. An exp licit counterexample is constructed showing that Wishartness of Y'Y ne ed not follow when, for every vector I, l'Y'Yl is distributed as a mul tiple of a chi-square, complementing the well-known counterexample by Mitra (1969, Sankhya A 31, 19-22). Application of the results to multi variate components of variance models is briefly indicated. (C) 1997 A cademic Press.