Generalized heterogeneous hypergeometric functions and the distribution of the largest eigenvalue of an elliptical Wishart matrix

TL;DR

This paper introduces generalized hypergeometric functions with two matrix arguments to derive exact eigenvalue distributions of singular elliptical Wishart matrices, enabling precise analysis of their largest eigenvalues.

Contribution

It defines new generalized hypergeometric functions with two matrix arguments and applies them to derive exact eigenvalue distributions for elliptical Wishart matrices.

Findings

Exact distributions of eigenvalues derived

Distribution of largest eigenvalue computed numerically

Applicable to matrix-variate t and Kotz-type models

Abstract

In this study, we derive the exact distributions of eigenvalues of a singular Wishart matrix under an elliptical model. We define generalized heterogeneous hypergeometric functions with two matrix arguments and provide convergence conditions for these functions. The joint density of eigenvalues and the distribution function of the largest eigenvalue for a singular elliptical Wishart matrix are represented by these functions. Numerical computations for the distribution of the largest eigenvalue were conducted under the matrix-variate $t$ and Kotz-type models.