Modified Pillai's trace statistics for two high-dimensional sample covariance matrices

TL;DR

This paper introduces modified Pillai's trace statistics for testing the equality of two high-dimensional covariance matrices, allowing for smaller sample sizes and no restrictions on covariance structure, with proven asymptotic properties.

Contribution

The paper proposes two new modified Pillai's trace statistics suitable for high-dimensional settings, with asymptotic distributions that are universal and applicable even when sample sizes are smaller than dimensions.

Findings

Proposed statistics have asymptotic distributions under null hypothesis.

Sample size can be smaller than the dimension.

Method validated through simulations and real data analysis.

Abstract

The goal of this study was to test the equality of two covariance matrices by using modified Pillai's trace statistics under a high-dimensional framework, i.e., the dimension and sample sizes go to infinity proportionally. In this paper, we introduce two modified Pillai's trace statistics and obtain their asymptotic distributions under the null hypothesis. The benefits of the proposed statistics include the following: (1) the sample size can be smaller than the dimensions; (2) the limiting distributions of the proposed statistics are universal; and (3) we do not restrict the structure of the population covariance matrices. The theoretical results are established under mild and practical assumptions, and their properties are demonstrated numerically by simulations and a real data analysis.