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On the distribution of sample scale-free scatter matrices

Author

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  • A. M. Mathai

    (McGill University)

  • Serge B. Provost

    (The University of Western Ontario)

Abstract

This paper addresses certain distributional aspects of a scale-free scatter matrix denoted by R that is stemming from a matrix-variate gamma distribution having a positive definite scale parameter matrix B. Under the assumption that B is a diagonal matrix, a structural representation of the determinant of R is derived; the exact density functions of products and ratios of determinants of matrices possessing such a structure are obtained; a closed form expression is given for the density function of R. Moreover, a novel procedure is utilized to establish that certain functions of the determinant of the sample scatter matrix are asymptotically distributed as chi-square or normal random variables. Then, representations of the density function of R that respectively involve multiple integrals, multiple series and Gauss’ hypergeometric function are provided for the general case of a positive definite scale parameter matrix, and an illustrative numerical example is presented. Cutting-edge mathematical techniques have been employed to derive the results. Naturally, they also apply to the conventional sample correlation matrix which is encountered in various multivariate inference contexts.

Suggested Citation

  • A. M. Mathai & Serge B. Provost, 2024. "On the distribution of sample scale-free scatter matrices," Statistical Papers, Springer, vol. 65(1), pages 121-138, February.
    • Handle: RePEc:spr:stpapr:v:65:y:2024:i:1:d:10.1007_s00362-022-01388-8
    DOI: 10.1007/s00362-022-01388-8
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    References listed on IDEAS

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    5. Kollo, Tõnu & Ruul, Kaire, 2003. "Approximations to the distribution of the sample correlation matrix," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 318-334, May.
    6. Dette, Holger & Dörnemann, Nina, 2020. "Likelihood ratio tests for many groups in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
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