Tuning Parameter Selection in Regularized Estimations of Large Covariance Matrices

TL;DR

This paper investigates the effectiveness of different cross-validation strategies for tuning parameter selection in regularized large covariance matrix estimators, providing practical guidelines based on simulation results.

Contribution

It offers empirical recommendations on the number of folds in cross-validation and compares its performance with bootstrap methods for covariance matrix estimation.

Findings

10-fold cross-validation is best for Frobenius norm accuracy.

2-fold and reverse 3-fold CV are suitable for operator norm.

Cross-validation generally outperforms bootstrap methods in this context.

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.