Diagonal Likelihood Ratio Test for Equality of Mean Vectors in High-Dimensional Data

TL;DR

This paper introduces a flexible likelihood ratio test for high-dimensional mean vectors that works under diagonal covariance assumptions and does not require strict covariance structure, showing advantages in simulations and real data.

Contribution

The paper develops a novel likelihood ratio test for high-dimensional mean vectors that is flexible and does not require covariance matrices to be diagonal, unlike existing methods.

Findings

Test statistics are sums of log-transformed squared t-statistics.

The method is asymptotically normal under null and alternative hypotheses.

Simulation and real data show improved performance over existing tests.

Abstract

We propose a likelihood ratio test framework for testing normal mean vectors in high-dimensional data under two common scenarios: the one-sample test and the two-sample test with equal covariance matrices. We derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure. In comparison with the diagonal Hotelling's tests, our proposed test statistics display some interesting characteristics. In particular, they are a summation of the log-transformed squared $t$-statistics rather than a direct summation of those components. More importantly, to derive the asymptotic normality of our test statistics under the null and local alternative hypotheses, we do not need the requirement that the covariance matrices follow a diagonal matrix structure. As a consequence, our proposed test methods are very flexible and readily applicable in practice. Simulation studies and a real data analysis are also carried out to demonstrate the advantages of our likelihood ratio test methods.