The paper introduces a new test for testing structures of covariances for high dimensional vectors and the data dimension can be much larger than the sample size. Under proper normalization, central and noncentral limit theorems are established. The asymptotic theory is attained without imposing any explicit restriction between data dimension and sample size. To facilitate the related statistical inference, we propose the balanced Rademacher weighted differencing scheme, which is also the delete-half jackknife, to approximate the distribution of the proposed test statistics. We also develop a new testing procedure for substructures of precision matrices. The simulation results show that the tests outperform the exiting methods both in terms of size and power. Our test procedure is applied to a colorectal cancer dataset.
"Test for high dimensional covariance matrices." Ann. Statist. 48 (6) 3565 - 3588, December 2020. https://doi.org/10.1214/20-AOS1943