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Wishartness and independence of matrix quadratic forms in a normal random matrix

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  • Hu, Jianhua
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    Abstract

    Let Y be an nxp multivariate normal random matrix with general covariance [Sigma]Y. The general covariance [Sigma]Y of Y means that the collection of all np elements in Y has an arbitrary npxnp covariance matrix. A set of general, succinct and verifiable necessary and sufficient conditions is established for matrix quadratic forms Y'WiY's with the symmetric Wi's to be an independent family of random matrices distributed as Wishart distributions. Moreover, a set of general necessary and sufficient conditions is obtained for matrix quadratic forms Y'WiY's to be an independent family of random matrices distributed as noncentral Wishart distributions. Some usual versions of Cochran's theorem are presented as the special cases of these results.

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    Bibliographic Info

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

    Volume (Year): 99 (2008)
    Issue (Month): 3 (March)
    Pages: 555-571

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    Handle: RePEc:eee:jmvana:v:99:y:2008:i:3:p:555-571

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    Related research

    Keywords: primary 62.40 secondary 62E15 Cochran's theorem Independence Matrix quadratic form Noncentral Wishart distribution Wishart distribution Wishartness;

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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. 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.
    5. Masaro, Joe & Wong, Chi Song, 2003. "Wishart distributions associated with matrix quadratic forms," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 1-9, April.
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