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Power-enhanced tests with high-dimensional data have received growingattention in theoretical and applied statistics in recent years. Existing testspossess their respective high-power regions, and we may lack prior knowledgeabout the alternatives when testing for a problem of interest in practice.There is a critical need of developing powerful testing procedures against moregeneral alternatives. This paper studies the joint test of two-sample meanvectors and covariance matrices for high-dimensional data. We first expand thehigh-power region of high-dimensional mean tests or covariance tests to a wideralternative space and then combine their strengths together in the simultaneoustest. We develop a new power-enhanced simultaneous test that is powerful todetect differences in either mean vectors or covariance matrices under eithersparse or dense alternatives. We prove that the proposed testing proceduresalign with the power enhancement principles introduced by Fan et al. (2015) andachieve the accurate asymptotic size and consistent asymptotic power. Wedemonstrate the finite-sample performance using simulation studies and a realapplication to find differentially expressed gene-sets in cancer studies. Ourfindings in the empirical study are supported by the biological literature.

Power-enhanced simultaneous test of high-dimensional mean vectors and covariance matrices with application to gene-set testing