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

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.