Fisher's Combined Probability Test for High-Dimensional Covariance Matrices

Testing large covariance matrices is of fundamental importance in statistical analysis with high-dimensional data. In the past decade, three types of test statistics have been studied in the literature: quadratic form statistics, maximum form statistics, and their weighted combination. It is known that quadratic form statistics would suffer from low power against sparse alternatives and maximum form statistics would suffer from low power against dense alternatives. The weighted combination methods were introduced to enhance the power of quadratic form statistics or maximum form statistics when the weights are appropriately chosen. In this article, we provide a new perspective to exploit the full potential of quadratic form statistics and maximum form statistics for testing high-dimensional covariance matrices. We propose a scale-invariant power-enhanced test based on Fisher's method to combine the p-values of quadratic form statistics and maximum form statistics. After carefully studying the asymptotic joint distribution of quadratic form statistics and maximum form statistics, we first prove that the proposed combination method retains the correct asymptotic size under the Gaussian assumption, and we also derive a new Lyapunov-type bound for the joint distribution and prove the correct asymptotic size of the proposed method without requiring the Gaussian assumption. Moreover, we show that the proposed method boosts the asymptotic power against more general alternatives. Finally, we demonstrate the finite-sample performance in simulation studies and a real application. Supplementary materials for this article are available online.