An Invariant Test for Equality of Two Large Scale Covariance Matrices

TL;DR

This paper introduces a new invariant test for comparing two large covariance matrices, demonstrating higher power than existing likelihood ratio tests through asymptotic analysis and numerical experiments.

Contribution

It proposes a novel eigenvalue sum-based test for covariance matrix equality, improving power over previous methods in high-dimensional settings.

Findings

The new test is asymptotically normal as dimensions grow.

It outperforms existing tests in numerical simulations.

The method is robust to eigenvalues near 0 or 1.

Abstract

In this work, we are motivated by the recent work of Zhang et al. (2019) and study a new invariant test for equality of two large scale covariance matrices. Two modified likelihood ratio tests (LRTs) by Zhang et al. (2019) are based on the sum of log of eigenvalues (or 1- eigenvalues) of the Beta-matrix. However, as the dimension increases, many eigenvalues of the Beta-matrix are close to 0 or 1 and the modified LRTs are greatly influenced by them. In this work, instead, we consider the simple sum of the eigenvalues (of the Beta-matrix) and compute its asymptotic normality when all $n_1, n_2, p$ increase at the same rate. We numerically show that our test has higher power than two modified likelihood ratio tests by Zhang et al. (2019) in all cases both we and they consider.