Joint test for homogeneity of high-dimensional means and covariance matrices using maximum-type statistics

Several tests for simultaneously checking homogeneity of means and covariance matrices, based on the classical likelihood ratio and sum-of-squares type test statistics, have been proposed in the literature. Despite their usefulness, they tend to have unsatisfactory size performance for either nonnormal high-dimensional data or strongly spiked eigenvalue models. This article proposes a novel high-dimensional nonparametric test using the maximum-type statistics. The limiting null distribution of the proposed test statistic is derived to provide p-values. It turns out that under appropriate conditions, the test is particularly powerful against sparse alternatives. Since the unknown correlations among the data sometimes pose a great challenge toward the accurate p-value calculation of the test, we further use a parametric bootstrap technique to achieve size accuracy. We also prove that the proposed simulation-based testing procedure is asymptotically valid in terms of size and power even if the dimension of the data is much larger than the sample size. We demonstrate the effectiveness of our methods through extensive simulation studies and a real data application.