Two-Sample Test for High-Dimensional Covariance Matrices: a normal-reference approach

TL;DR

This paper introduces a normal-reference test for comparing high-dimensional covariance matrices, which maintains accurate size control and outperforms existing methods in practical scenarios.

Contribution

A novel normal-reference test for high-dimensional covariance matrices that approximates the null distribution using a chi-square-type mixture, improving applicability and accuracy.

Findings

The proposed test controls size better than existing methods.

Simulation studies show superior performance in real data.

The test has established asymptotic power under local alternatives.

Abstract

Testing the equality of the covariance matrices of two high-dimensional samples is a fundamental inference problem in statistics. Several tests have been proposed but they are either too liberal or too conservative when the required assumptions are not satisfied which attests that they are not always applicable in real data analysis. To overcome this difficulty, a normal-reference test is proposed and studied in this paper. It is shown that under some regularity conditions and the null hypothesis, the proposed test statistic and a chi-square-type mixture have the same limiting distribution. It is then justified to approximate the null distribution of the proposed test statistic using that of the chi-square-type mixture. The distribution of the chi-square-type mixture can be well approximated using a three-cumulant matched chi-square-approximation with its approximation parameters consistently estimated from the data. The asymptotic power of the proposed test under a local alternative is also established. Simulation studies and a real data example demonstrate that in terms of size control, the proposed test outperforms the existing competitors substantially.