Distribution of ratio of two Wishart matrices and evaluation of cumulative probability by holonomic gradient method

TL;DR

This paper derives the distribution of the ratio of two Wishart matrices with different covariances, expressing its CDF via hypergeometric functions and applying the holonomic gradient method for numerical evaluation, aiding in statistical hypothesis testing.

Contribution

It introduces a new analytical form for the ratio distribution and demonstrates a numerical method for evaluating its CDF, enhancing statistical testing procedures.

Findings

Derived the density function of the Wishart ratio.

Expressed the CDF using hypergeometric functions of matrix argument.

Applied the holonomic gradient method for numerical evaluation.

Abstract

We study the distribution of the ratio of two central Wishart matrices with different covariance matrices. We first derive the density function of a particular matrix form of the ratio and show that its cumulative distribution function can be expressed in terms of the hypergeometric function 2F1 of a matrix argument. Then we apply the holonomic gradient method for numerical evaluation of the hypergeometric function. This approach enables us to compute the power function of Roy's maximum root test for testing the equality of two covariance matrices.