Power-Enhanced Simultaneous Test of High-Dimensional Mean Vectors and Covariance Matrices with Application to Gene-Set Testing

Power-enhanced tests with high-dimensional data have received growing attention in theoretical and applied statistics in recent years. Existing tests possess their respective high-power regions, and we may lack prior knowledge about the alternatives when testing for a problem of interest in practice. There is a critical need of developing powerful testing procedures against more general alternatives. This article studies the joint test of two-sample mean vectors and covariance matrices for high-dimensional data. We first expand the high-power regions of high-dimensional mean tests or covariance tests to a wider alternative space and then combine their strengths together in the simultaneous test. We develop a new power-enhanced simultaneous test that is powerful to detect differences in either mean vectors or covariance matrices under either sparse or dense alternatives. We prove that the proposed testing procedures align with the power enhancement principles introduced by Fan, Liao, and Yao and achieve the accurate asymptotic size and consistent asymptotic power. We demonstrate the finite-sample performance using simulation studies and a real application to find differentially expressed gene-sets in cancer studies. Supplementary materials for this article are available online.