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

TL;DR

This paper introduces a new power-enhanced simultaneous testing method for high-dimensional mean vectors and covariance matrices, improving detection capabilities across various alternative scenarios in statistical and gene-set analysis.

Contribution

It develops a novel joint testing procedure that extends high-power regions and combines strengths to detect differences in means or covariances under diverse conditions.

Findings

The proposed test achieves accurate asymptotic size and consistent power.

Simulation studies show superior finite-sample performance.

Application to gene-set testing identifies biologically relevant differences.

Abstract

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 paper studies the joint test of two-sample mean vectors and covariance matrices for high-dimensional data. We first expand the high-power region 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 et al. (2015) 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. Our findings in the empirical study are supported by the biological literature.