Optimal modification of the LRT for the equality of two high-dimensional covariance matrices

TL;DR

This paper introduces an optimally modified likelihood ratio test for assessing the equality of two high-dimensional covariance matrices, effective even when dimensions exceed sample sizes, with robust performance demonstrated through simulations and real data.

Contribution

It proposes a new modified LRT suitable for high-dimensional settings where classical tests fail, and establishes the test under minimal moment conditions.

Findings

Test performs well when dimensions exceed sample sizes

Robust against affine transformations

Effective in real data applications

Abstract

This paper considers the optimal modification of the likelihood ratio test (LRT) for the equality of two high-dimensional covariance matrices. The classical LRT is not well defined when the dimensions are larger than or equal to one of the sample sizes. In this paper, an optimally modified test that works well in cases where the dimensions may be larger than the sample sizes is proposed. In addition, the test is established under the weakest conditions on the moments and the dimensions of the samples. We also present weakly consistent estimators of the fourth moments, which are necessary for the proposed test, when they are not equal to 3. From the simulation results and real data analysis, we find that the performances of the proposed statistics are robust against affine transformations.