Local normal approximations and probability metric bounds for the matrix-variate $T$ distribution and its application to Hotelling's $T$ statistic

TL;DR

This paper develops local density approximations for the matrix-variate T distribution, enabling bounds on probability metrics and extending univariate results to the matrix setting.

Contribution

It introduces local expansions for the matrix-variate T distribution density relative to the normal, extending previous univariate results to the matrix-variate case.

Findings

Derived upper bounds on total variation and Hellinger distances.

Extended univariate Student distribution results to matrix-variate distributions.

Provided tools for assessing approximation quality in high-dimensional statistical models.

Abstract

In this paper, we develop local expansions for the ratio of the centered matrix-variate $T$ density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several probability metrics (such as the total variation and Hellinger distance) between the corresponding induced measures. This work extends some of the results of Shafiei & Saberali (2015) and Ouimet (2022) for the univariate Student distribution to the matrix-variate setting.