Tuning Parameter Selection in Regularized Estimations of Large Covariance Matrices

Abstract: Recently many regularized estimators of large covariance matrices have been proposed, and the tuning parameters in these estimators are usually selected via cross-validation. However, there is no guideline on the number of folds for conducting cross-validation and there is no comparison between cross-validation and the methods based on bootstrap. Through extensive simulations, we suggest 10-fold cross-validation (nine-tenths for training and one-tenth for validation) be appropriate when the estimation accuracy is measured in the Frobenius norm, while 2-fold cross-validation (half for training and half for validation) or reverse 3-fold cross-validation (one-third for training and two-thirds for validation) be appropriate in the operator norm. We also suggest the "optimal" cross-validation be more appropriate than the methods based on bootstrap for both types of norm.