Hypothesis testing of covariance matrices in high-dimensional elliptical models

Abstract

We propose a new two-sample test for high-dimensional covariance matrices under elliptical distributions. The unbiased and consistent estimators of the functions of covariance matrices are proposed in high-dimensional elliptical models. This allows us to introduce a modified statistic that accounts for these possible dependencies in the elliptical data. We derive the asymptotic normality of the proposed test statistic under the high-dimensional null hypothesis. The power of the proposed test is also investigated. In the aspects of methodology and theory, we extend the single covariance matrix study of Xu et al. (Citation2025) to the setting of two-sample covariance testing. We conduct simulations to confirm our asymptotic results.