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Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms

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  • Mathew, Thomas
  • Nordström, Kenneth

Abstract

For a normally distributed random matrixYwith a general variance-covariance matrix[Sigma]Y, 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 of[Sigma]Yunder which the distribution ofY'QYis Wishart. Assuming[Sigma]Ypositive 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 of[Sigma]Yis 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,Sankhya31, 19-22). Application of the results to multivariate components of variance models is briefly indicated.

Suggested Citation

  • Mathew, Thomas & Nordström, Kenneth, 1997. "Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 129-143, April.
    • Handle: RePEc:eee:jmvana:v:61:y:1997:i:1:p:129-143
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    References listed on IDEAS

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    1. Mathew, Thomas, 1989. "MANOVA in the multivariate components of variance model," Journal of Multivariate Analysis, Elsevier, vol. 29(1), pages 30-38, April.
    2. Wong, C. S. & Wang, T. H., 1993. "Multivariate Versions of Cochran's Theorems II," Journal of Multivariate Analysis, Elsevier, vol. 44(1), pages 146-159, January.
    3. Wong, Chi Song & Masaro, Joe & Wang, Tonghui, 1991. "Multivariate versions of Cochran's theorems," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 154-174, October.
    4. Robert Boik, 1988. "The mixed model for multivariate repeated measures: validity conditions and an approximate test," Psychometrika, Springer;The Psychometric Society, vol. 53(4), pages 469-486, December.
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    2. Ye, Rendao & Wang, Tonghui & Gupta, Arjun K., 2014. "Distribution of matrix quadratic forms under skew-normal settings," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 229-239.
    3. Akhil Vaish & N. Rao Chaganty, 2008. "Nonnegative definite solutions to matrix equations with applications to multivariate test statistics," Statistical Papers, Springer, vol. 49(1), pages 87-99, March.
    4. Frank Kleibergen & Lingwei Kong & Zhaoguo Zhan, 2023. "Identification Robust Testing of Risk Premia in Finite Samples," Journal of Financial Econometrics, Oxford University Press, vol. 21(2), pages 263-297.
    5. Hu, Jianhua, 2008. "Wishartness and independence of matrix quadratic forms in a normal random matrix," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 555-571, March.
    6. Masaro, Joe & Wong, Chi Song, 2010. "Wishart-Laplace distributions associated with matrix quadratic forms," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1168-1178, May.
    7. Phil D. Young & Joshua D. Patrick & Dean M. Young, 2023. "A Brief Derivation of Necessary and Sufficient Conditions for a Family of Matrix Quadratic Forms to Have Mutually Independent Non-Central Wishart Distributions," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 478-484, February.

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