Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix

TL;DR

This paper introduces new heterogeneous hypergeometric functions with two matrix arguments and derives the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix, advancing statistical eigenvalue analysis.

Contribution

It defines and explores properties of newly introduced heterogeneous hypergeometric functions and applies them to derive the exact distribution of the largest eigenvalue in singular beta-Wishart matrices.

Findings

Exact distribution of the largest eigenvalue for real singular beta-Wishart matrices.

Representation of eigenvalue distributions using heterogeneous hypergeometric functions.

Enhanced methods for eigenvalue analysis in multivariate statistics.

Abstract

This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These functions are newly defined but have already appeared in statistical literature and are useful when dealing with the derivation of certain distributions for the eigenvalues of singular beta-Wishart matrices. The joint density function of the eigenvalues and the distribution of the largest eigenvalue can be expressed in terms of certain heterogeneous hypergeometric functions. Exact computation of the distribution of the largest eigenvalue is conducted here for a real case.