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Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms

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  • Mathew, Thomas
  • Nordström, Kenneth
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 61 (1997)
    Issue (Month): 1 (April)
    Pages: 129-143

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    Handle: RePEc:eee:jmvana:v:61:y:1997:i:1:p:129-143

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    Keywords: complex covariance structure group symmetry covariance model multivariate components of variance model skew-symmetric matrix;

    References

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    1. Mathew, Thomas, 1989. "MANOVA in the multivariate components of variance model," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 29(1), pages 30-38, April.
    2. Robert Boik, 1988. "The mixed model for multivariate repeated measures: validity conditions and an approximate test," Psychometrika, Springer, Springer, vol. 53(4), pages 469-486, December.
    3. Wong, C. S. & Wang, T. H., 1993. "Multivariate Versions of Cochran's Theorems II," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 44(1), pages 146-159, January.
    4. Wong, Chi Song & Masaro, Joe & Wang, Tonghui, 1991. "Multivariate versions of Cochran's theorems," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 39(1), pages 154-174, October.
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    Cited by:
    1. Hu, Jianhua, 2008. "Wishartness and independence of matrix quadratic forms in a normal random matrix," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 99(3), pages 555-571, March.
    2. Mortarino, Cinzia, 2005. "A decomposition for a stochastic matrix with an application to MANOVA," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 92(1), pages 134-144, January.
    3. Akhil Vaish & N. Rao Chaganty, 2008. "Nonnegative definite solutions to matrix equations with applications to multivariate test statistics," Statistical Papers, Springer, Springer, vol. 49(1), pages 87-99, March.
    4. Masaro, Joe & Wong, Chi Song, 2010. "Wishart-Laplace distributions associated with matrix quadratic forms," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 101(5), pages 1168-1178, May.

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