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February 2013 A Cramér moderate deviation theorem for Hotelling’s $T^{2}$-statistic with applications to global tests
Weidong Liu, Qi-Man Shao
Ann. Statist. 41(1): 296-322 (February 2013). DOI: 10.1214/12-AOS1082

Abstract

A Cramér 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.

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Weidong Liu. Qi-Man Shao. "A Cramér moderate deviation theorem for Hotelling’s $T^{2}$-statistic with applications to global tests." Ann. Statist. 41 (1) 296 - 322, February 2013. https://doi.org/10.1214/12-AOS1082

Information

Published: February 2013
First available in Project Euclid: 26 March 2013

zbMATH: 1347.62032
MathSciNet: MR3059419
Digital Object Identifier: 10.1214/12-AOS1082

Subjects:
Primary: 62E20
Secondary: 60F10 , 62H15

Keywords: brain structure , Cramér moderate deviation , FDR , gene selection , global tests , Hotelling’s $T^{2}$-statistic , simultaneous hypothesis tests

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 1 • February 2013
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