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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Koki Shimizu & Hiroki Hashiguchi, 2024. "Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix," Mathematics, MDPI, vol. 12(6), pages 1-11, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:6:p:921-:d:1360719
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    References listed on IDEAS

    as
    1. Butler, Ronald W. & Wood, Andrew T.A., 2005. "Laplace approximations to hypergeometric functions of two matrix arguments," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 1-18, May.
    2. Ryo Nasuda & Koki Shimizu & Hiroki Hashiguchi, 2023. "Asymptotic behavior of the distributions of eigenvalues for beta-Wishart ensemble under the dispersed population eigenvalues," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 52(22), pages 7840-7860, November.
    3. Takemura, Akimichi & Sheena, Yo, 2005. "Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 271-299, June.
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