Inference for High-Dimensional Covariance Matrices and Thresholding Tests for High-Dimensional Count Distributions

We propose a test for a high-dimensional covariance being banded with possible diverging bandwidth. The test is adaptive to the large p, small n situations without assuming a specific parametric distribution for the data. For covariance estimation, we propose a band width selector for the banding covariance estimator of Bickel and Levina (2008a) by minimizing an empirical estimate of the expected squared Frobenius norms of the estimation error matrix. The ratio consistency of the band width selector to the underlying band width is established. We provide a lower bound for the coverage probability of the underlying band width being contained in an interval around the band width estimate. Extensions to the band width selection for the tapering estimator and threshold level selection for the thresholding covariance estimator are made. We also consider the detection of rare and faint signals in high-dimensional count data. Under Generalized Linear Models, a thresholding statistic based on the maximum likelihood estimators (MLEs) is proposed. A multi-threshold test is constructed by maximizing the standardized thresholding statistic over a set of thresholds. Extensions to Generalized Linear Mixed Models are made, where Gauss quadratures and data cloning are used to approximate the MLEs of such models. Numerical simulations and three case studies are conducted to confirm and demonstrate the proposed approaches