Extremal Probabilistic Problems and Hotelling's $T^2$ Test Under a Symmetry Condition

Abstract

We consider the Hotelling $T^2$ statistic for an arbitrary $d$-dimensional sample. If the sampling is not too deterministic or inhomogeneous, then under the zero-means hypothesis the limiting distribution for $T^2$ is $\chi^2_d$. It is shown that a test for the orthant symmetry condition introduced by Efron can be constructed which does not differ essentially from the one based on $\chi^2_d$ and at the same time is applicable not only to large random homogeneous samples but to all multidimensional samples. The main results are not limit theorems, but exact inequalities corresponding to the solutions to certain extremal problems. The following auxiliary result itself may be of interest: $\chi_d - \sqrt{d - 1}$ has a monotone likelihood ratio.