Band Width Selection for High Dimensional Covariance Matrix Estimation

The banding estimator of Bickel and Levina (2008a) and its tapering version of Cai, Zhang and Zhou (2010), are important high dimensional covariance estimators. Both estimators require choosing a band width parameter. We propose a band width selector for the banding covariance estimator 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 also 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. Numerical simulations and a case study on sonar spectrum data are conducted to confirm and demonstrate the proposed band width and threshold estimation approaches.