Asymptotic behavior of the distributions of eigenvalues for beta-Wishart ensemble under the dispersed population eigenvalues

Ryo Nasuda, Koki Shimizu, Hiroki Hashiguchi

Research output: Contribution to journalArticlepeer-review

Abstract

We propose a Laplace approximation of the hypergeometric function with two matrix arguments expanded by Jack polynomials. This type of hypergeometric function appears in the joint density of eigenvalues of the beta-Wishart matrix for parameters (Formula presented.) where the matrix indicates the cases for reals, complexes, and quaternions, respectively. Using the Laplace approximations, we show that the joint density of the eigenvalues can be expressed using gamma density functions when population eigenvalues are infinitely dispersed. In general, for the parameter (Formula presented.) we also show that the distribution of the eigenvalue can be approximated by gamma distributions through broken arrow matrices. We compare approximated gamma distributions with empirical distributions by Monte Carlo simulation.

Original languageEnglish
JournalCommunications in Statistics - Theory and Methods
DOIs
Publication statusAccepted/In press - 2022

Keywords

  • gamma distribution
  • Haar measure
  • Hypergeometric function of matrix arguments
  • Jack polynomial
  • Laplace approximation

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