Test of the Equality of Several High-Dimensional Covariance Matrices: A Normal-Reference Approach

As the field of big data continues to evolve, there is an increasing necessity to evaluate the equality of multiple high-dimensional covariance matrices. Many existing methods rely on approximations to the null distribution of the test statistic or its extreme-value distributions under stringent conditions, leading to outcomes that are either overly permissive or excessively cautious. Consequently, these methods often lack robustness when applied to real-world data, as verifying the required assumptions can be arduous. In response to these challenges, we introduce a novel test statistic utilizing the normal-reference approach. We demonstrate that the null distribution of this test statistic shares the same limiting distribution as a chi-square-type mixture under certain regularity conditions, with the latter reliably estimable from data using the three-cumulant matched chi-square-approximation. Additionally, we establish the asymptotic power of our proposed test. Through comprehensive simulation studies and real data analysis, our proposed test demonstrates superior performance in terms of size control compared to several competing methods.