A Cram\'{e}r moderate deviation theorem for Hotelling's $T^2$-statistic with applications to global tests

TL;DR

This paper establishes a Cramér moderate deviation theorem for Hotelling's T^2-statistic under finite moments, enabling accurate large-scale mean vector tests and proposing a new global test with improved approximation for small samples.

Contribution

It introduces a novel moderate deviation theorem for Hotelling's T^2 under finite moments and develops a new global test with an intermediate approximation for better accuracy.

Findings

The theorem holds under finite (3+δ)th moments.

The number of tests can be as large as e^{o(n^{1/3})} without losing calibration.

The proposed test performs well with small sample sizes and real data applications.

Abstract

A Cramer moderate deviation theorem for Hotelling's $T^2$-statistic is proved under a finite $(3+\delta)$th moment. The result is applied to large scale tests on the equality of mean vectors and is shown that the number of tests can be as large as $e^{o(n^{1/3})}$ before the chi-squared distribution calibration becomes inaccurate. As an application of the moderate deviation results, a global test on the equality of m mean vectors based on the maximum of Hotelling's $T^2$-statistics is developed and its asymptotic null distribution is shown to be an extreme value type I distribution. A novel intermediate approximation to the null distribution is proposed to improve the slow convergence rate of the extreme distribution approximation. Numerical studies show that the new test procedure works well even for a small sample size and performs favorably in analyzing a breast cancer dataset.