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A Brief Derivation of Necessary and Sufficient Conditions for a Family of Matrix Quadratic Forms to Have Mutually Independent Non-Central Wishart Distributions

Author

Listed:
  • Phil D. Young

    (Baylor University)

  • Joshua D. Patrick

    (One Bear Place #97140, Baylor University)

  • Dean M. Young

    (One Bear Place #97140, Baylor University)

Abstract

We provide a new, concise derivation of necessary and sufficient conditions for a real matrix-normally distributed matrix X and we characterize the general matrix-normal covariance structure such that, given the symmetric positive-semidefinite matrices Ai, i = 1,2,...,m, the matrix quadratic forms X ′ A i X ${\textbf {X}}^{\prime }{\textbf {A}}_{i}{\textbf {X}}$ are distributed as independent noncentral Wishart matrices.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:1:d:10.1007_s13171-021-00260-5
    DOI: 10.1007/s13171-021-00260-5
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    References listed on IDEAS

    as
    1. Phil D. Young & Dean M. Young, 2016. "Characterizations of Noncentral Chi-Squared-Generating Covariance Structures for a Normally Distributed Random Vector," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 78(2), pages 231-247, August.
    2. 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.
    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. 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.
    5. 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.
    6. 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.
    7. 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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