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An adaptable generalization of Hotelling’s \(T^2\) test in high dimension. (English) Zbl 1451.62083
There is an investigation on the two-sample testing problem for high dimensional means in connection with the Hotelling’s \(T^2\) approach. A test statistic based upon the regularized Hotelling’s \(T^2\), RTH, introduced by L. S. Chen et al. [J. Am. Stat. Assoc. 106, No. 496, 1345–1360 (2011; Zbl 1234.62082)] for the one-sample case, is proposed and extensively discussed. The authors provide a Bayesian framework to analyze the power of the RTH and construct a new composite test by combining the RTH statistics corresponding to a set of optimally chosen regularization parameters. The new composite testing procedure will be named “adaptable RTH”, shortly ARTH. The weak convergence of the considered stochastic process to a Gaussian limit is proved. The asymptotic behavior of the test by relaxing the assumption of Gaussian to sub-Gaussian is investigated. The regularized Hotelling’s \(T^2\), RTH, test is introduced and largely discussed in the second section of the article while the adaptable ARTH is presented in the third section. The calibration of type I error probability and extension to a general class of sub-Gaussian distributions are presented in the fourth and fifth section. Simulation results are shown in the sixth section and a practical application to a breast cancer data set in the seventh section. In the eighth section, we find a short discussion on obtained results. The ninth section is devoted to the proofs of main results and is followed by a short appendix containing some auxiliary results. One reports that additional simulation results and detailed proofs of the main results are contained in a supplementary material in doi:10.1214/19-AOS1869SUPP.

MSC:
62J10 Analysis of variance and covariance (ANOVA)
62H15 Hypothesis testing in multivariate analysis
62H20 Measures of association (correlation, canonical correlation, etc.)
60B20 Random matrices (probabilistic aspects)
60G15 Gaussian processes
15B52 Random matrices (algebraic aspects)
62P10 Applications of statistics to biology and medical sciences; meta analysis
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