Algorithm for the product of Jack polynomials and its application to the sphericity test

TL;DR

This paper develops an algorithm to compute the distribution of eigenvalue ratios in singular beta-Wishart matrices, leveraging Jack polynomial products, with applications to sphericity testing.

Contribution

It introduces a novel algorithm for expanding products of Jack polynomials, enabling practical computation of eigenvalue ratio distributions for sphericity tests.

Findings

Derived explicit density and distribution functions for eigenvalue ratios.

Successfully implemented the algorithm for numerical computation.

Enhanced accuracy and efficiency in sphericity testing procedures.

Abstract

In this study, we derive the density and distribution function of a ratio of the largest and smallest eigenvalues of a singular beta-Wishart matrix for the sphericity test. These functions can be expressed in terms of the product of Jack polynomials. We propose an algorithm that expands the product of Jack polynomials by a linear combination of Jack polynomials. Numerical computation for the derived distributions is performed using the algorithm.