  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.
