Block-diagonal test for high-dimensional covariance matrices

The structure testing of a high-dimensional covariance matrix plays an important role in financial stock analyses, genetic series analyses, and many other fields. Testing that the covariance matrix is block-diagonal under the high-dimensional setting is the main focus of this paper. Several test procedures that rely on normality assumptions, two-diagonal block assumptions, or sub-block dimensionality assumptions have been proposed to tackle this problem. To relax these assumptions, we develop a test framework based on U-statistics, and the asymptotic distributions of the U-statistics are established under the null and local alternative hypotheses. Moreover, a test approach is developed for alternatives with different sparsity levels. Finally, both a simulation study and real data analysis demonstrate the performance of our proposed methods.