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Abstract
Thresholding is a regularization method commonly used for covariance estimation (Bickel and Levina, 2008, Cai and Liu, 2011), which provides consistent estimators in high-dimensional settings if the population covariance satisfies certain sparsity conditions. However, the performance of those estimators heavily depends on the threshold level. By minimizing the Frobenius risk of the adaptive thresholding covariance estimator, we conduct a theoretical study for the optimal threshold level, and obtain its analytical expression under a general setting of n and p. 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. Based on the concepts developed in the theoretical study, another two efficient numerical methods are proposed for estimating the threshold level. These methods are more flexible and precise. As a result, they provide more precise and stable threshold levels by correctly adjusting to the true covariance structure, which enhances applicability in practice. Additional numerical simulations and a case study on different gene expression data are conducted to compare all proposed methods.
