Topic: A generalization of Hotelling's $T^2$ test in high dimension

Abstract: We propose a two-sample test for detecting the difference between mean vectors in a high-dimensional regime based on a ridge-regularized Hotelling's $T^2$. To choose the regularization parameter, a method is derived that aims at maximizing local power within a class of local alternatives. We also propose a composite test that combines the optimal tests corresponding to a specific collection of local alternatives. Weak convergence of the stochastic process corresponding to the ridge-regularized Hotelling's $T^2$ is established under the assumption of sub-Gaussianity of the observations, and it is used to derive the cut-off values of the proposed test. The performance of the proposed test procedure is illustrated through an application to a breast cancer data set where the goal is to detect the pathways with different DNA copy number alterations across breast cancer subtypes. We also discuss a generalization of this testing framework to high-dimensional MANOVA problems.