Test for high-dimensional mean vectors via the weighted $L_2$-norm

TL;DR

This paper introduces a new high-dimensional mean vector test based on the weighted L2-norm, demonstrating its theoretical properties and superior performance in simulations, especially in weakly dense scenarios.

Contribution

The paper proposes a novel weighted L2-norm test for high-dimensional means, with proven asymptotic normality and advantages over existing methods in certain cases.

Findings

Test statistic follows asymptotic normality under null hypothesis

The test performs well in weakly dense signal scenarios

Simulation results show improved size and power

Abstract

In this paper, we propose a novel approach to test the equality of high-dimensional mean vectors of several populations via the weighted $L_2$-norm. We establish the asymptotic normality of the test statistics under the null hypothesis. We also explain theoretically why our test statistics can be highly useful in weakly dense cases when the nonzero signal in mean vectors is present. Furthermore, we compare the proposed test with existing tests using simulation results, demonstrating that the weighted $L_2$-norm-based test statistic exhibits favorable properties in terms of both size and power.