The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

A. M. Mathai and Serge B. Provost ()
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A. M. Mathai: Department of Mathematics and Statistics, McGill University, Montreal, QC H3A 0G4, Canada
Serge B. Provost: Department of Statistical and Actuarial Sciences, Western University, London, ON N6A 5B7, Canada

Mathematics, 2024, vol. 12, issue 15, 1-24

Abstract: The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and j th largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Keywords: distribution of eigenvalues; matrix-variate gamma distribution; central Wishart distribution; matrix-variate beta distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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