Hypothesis testing on compound symmetric structure of high-dimensional covariance matrix

Abstract

Testing the population covariance matrix is an important topic in multivariate statistical analysis. Owing to the difficulty of establishing the central limit theorem for the test statistic based on sample covariance matrix, in most of the existing literature, it is assumed that the population covariance matrix is bounded in spectral norm as the dimension tends to infinity. Four test statistics are proposed for testing the compound symmetric structure of the population covariance matrix, which has an unbounded spectral norm as the dimension tends to infinity. Three of these tests maintain high power against different kinds of dense alternatives. The asymptotic properties of these statistics are constructed under the null hypothesis and a specific alternative hypothesis. Moreover, extensive simulation studies and an analysis of real data are conducted to evaluate the performance of our proposed tests. The simulation results show that the proposed tests outperform existing methods.