Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms

Abstract

For a normally distributed random matrixYwith a general variance-covariance matrixΣY, and for a nonnegative definite matrixQ, necessary and sufficient conditions are derived for the Wishartness ofY′QY. The conditions resemble those obtained by Wong, Masaro, and Wang (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 for the structure ofΣYunder which the distribution ofY′QYis Wishart. AssumingΣYpositive definite, a necessary and sufficient condition is derived for every univariate quadratic fromlY′QYlto be distributed as a multiple of a chi-square. For the caseQ=In, the corresponding structure ofΣYis identified. An explicit counterexample is constructed showing that Wishartness ofY′Yneed not follow when, for every vectorl, l′Y′Ylis distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969,Sankhyā31, 19-22). Application of the results to multivariate components of variance models is briefly indicated. © 1997 Academic Press.