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

TL;DR

This paper investigates Hotelling's T^2 test under symmetry conditions, providing inequalities that extend its applicability to all multidimensional samples and exploring extremal problems related to its distribution.

Contribution

It introduces a symmetry-based test for Hotelling's T^2 that applies broadly and presents exact inequalities as solutions to extremal problems.

Findings

The T^2 statistic converges to chi-squared distribution under certain conditions.

A new symmetry condition-based test is nearly as effective as the classical chi-squared test.

Distributional inequalities are established as solutions to extremal problems.

Abstract

We consider Hotelling's T^2 statistic for an arbitrary d-dimensional sample. If the sampling is not too deterministic or inhomogeneous, then under zero means hypothesis, T^2 tends to \chi^2_d in distribution. We show that a test for the orthant symmetry condition introduced by Efron can be constructed which does not essentially differ from the one based on \chi^2_d and at the same time is applicable not only for large random homogeneous samples but for all multidimensional samples without exceptions. The main assertions have the form of inequalities, not that of limit theorems; these inequalities are exact representing the solutions to certain extremal problems. Let us also mention an auxiliary result which itself may be of interest: \chi_d-(d-1)^{1/2} decreases in distribution in d to its limit N(0,1/2).