Wishart and chi-square distributions associated with matrix quadratic forms

[摘要] For a normally distributed random matrix Y with a general variance-covariance 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 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 of Sigma(gamma) under which the distribution of Y'QY is Wishart. Assuming Sigma(gamma) positive definite, a necessary and sufficient 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 explicit counterexample is constructed showing that Wishartness of Y'Y need not follow when, for every vector I, l'Y'Yl is distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969, Sankhya A 31, 19-22). Application of the results to multivariate components of variance models is briefly indicated. (C) 1997 Academic Press.

[发布日期] 1997-04-01

[关键词] complex covariance structure;group symmetry covariance model;multivariate components of variance model;skew-symmetric matrix