Abstract: Testing the homogeneity of multiple high-dimensional covariance matrices is becoming increasingly critical in multivariate statistical analyses owing to the emergence of big data. Many existing homogeneity tests for covariance matrices focus on two populations, under specific situations, for example, either sparse or dense alternatives. As a result, these methods are not suitable for general cases that include multiple groups. We propose a power-enhancement high-dimensional test for multi-sample comparisons of covariance matrices, which includes homogeneity tests of two matrices as a special case. The proposed tests do not require a distributional assumption, and can handle both sparse and non-sparse structures. Based on random-matrix theory, the asymptotic normality properties of our tests are established under both the null and the alternative hypotheses. Numerical studies demonstrate the substantial gain in power for the proposed method. Furthermore, we illustrate the method using a gene expression data set from a breast cancer study.

Key words and phrases: Asymptotic normality, high-dimensional covariance matrix, homogeneity test, multi-sample comparison, power enhancement.