Title: Sharp optimality for high dimensional covariance testing
Authors: Yumou Qiu - Iowa State University (United States) [presenting]
Abstract: The theoretical limit of testing a high-dimensional covariance being diagonal is developed by deriving the sharp detection boundary as a function of signal proportion and signal strength under alternative hypotheses. The detection boundary gives the exact minimal signal strength that can be detected by some test under the sparse and faint signal regime, which is the most challenging setting for signal detection. We develop an optimal test by multi-level thresholding that can achieve the detection boundary. The optimality means the proposed test is powerful as long as the signal strength is above the detection boundary. We establish the asymptotic distribution of the thresholding statistic under non-Gaussian data. A novel $U$-statistic composition is developed in conjunction with the matrix blocking and the coupling techniques to handle the complex dependence among sample covariances. We show that the existing tests are non-optimal, and the proposed tests are more powerful than those existing tests. Simulation studies are conducted to demonstrate the utility of the proposed test.