Finite-sample bounds for the multivariate Behrens-Fisher distribution with proportional covariances

TL;DR

This paper confirms a conjecture on stochastic bounds for the multivariate Behrens-Fisher test statistic, extending previous univariate results and enabling more accurate hypothesis testing with controlled Type I error.

Contribution

It proves a conjecture on stochastic bounds for the multivariate Behrens-Fisher distribution and generalizes previous univariate bounds to the multivariate case.

Findings

Confirmed conjecture on stochastic bounds under null hypothesis

Extended stochastic ordering results to arbitrary finite dimensions

Provided a testing procedure with strong Type I error control

Abstract

The Behrens-Fisher problem is a well-known hypothesis testing problem in statistics concerning two-sample mean comparison. In this article, we confirm one conjecture in Eaton and Olshen (1972), which provides stochastic bounds for the multivariate Behrens-Fisher test statistic under the null hypothesis. We also extend their results on the stochastic ordering of random quotients to the arbitrary finite dimensional case. This work can also be seen as a generalization of Hsu (1938) that provided the bounds for the univariate Behrens-Fisher problem. The results obtained in this article can be used to derive a testing procedure for the multivariate Behrens-Fisher problem that strongly controls the Type I error.