Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix

Koki Shimizu () and Hiroki Hashiguchi
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Koki Shimizu: Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
Hiroki Hashiguchi: Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Mathematics, 2024, vol. 12, issue 6, 1-11

Abstract: This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations.

Keywords: hypergeometric functions; laplace approximation; spiked covariance model (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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