Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms

Thomas Mathew, Kenneth Nordström

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18 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)129-143
Number of pages15
JournalJournal of Multivariate Analysis
Volume61
Issue number1
DOIs
Publication statusPublished - Apr 1 1997

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Keywords

  • Complex covariance structure; group symmetry covariance model; multivariate components of variance model; skew-symmetric matrix

ASJC Scopus subject areas

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

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