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

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.