On non-central distribution of the matrix ratio

TL;DR

This paper derives the distribution of the ratio between a non-central mean matrix and a sample covariance matrix, extending matrix-variate distributions and relating to the non-central Student's t distribution.

Contribution

It introduces a new derivation for the distribution of a matrix ratio involving non-central means, expanding the understanding of matrix-variate distributions.

Findings

Derived the distribution of the matrix ratio involving non-central means.

Connected matrix-variate distribution with the confluent hypergeometric function ${}_1F_1$.

Considered extensions to other matrix-variate distributions.

Abstract

We derive the distribution of the ratio of a non-central mean matrix and a sample covariance matrix. This aligns with the confluent term ${}_1F_1$ in the non-central uni-variate Student's $t$. Some extensions of matrix-variate distributions are considered.