Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms

AbstractFor 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