Threshold selection for covariance estimation

Thresholding is a regularization method commonly used for covariance estimation, which provides consistent estimators if the population covariance satisfies certain sparsity condition (Bickel and Levina, 2008a; Cai and Liu, 2011). However, the performance of the thresholding estimators heavily depends on the threshold level. By minimizing the Frobenius risk of the adaptive thresholding estimator for covariances, we conduct a theoretical study for the optimal threshold level, and obtain its analytical expression. A consistent estimator based on this expression is proposed for the optimal threshold level, which is easy to implement in practice and efficient in computation. Numerical simulations and a case study on gene expression data are conducted to illustrate the proposed method.