Shrinkage Tuning Parameter Selection in Precision Matrices Estimation

TL;DR

This paper introduces efficient methods for selecting the shrinkage parameter in sparse precision matrix estimation, improving accuracy and computational efficiency over traditional cross-validation techniques.

Contribution

It develops a generalized approximate cross-validation method and employs a Bayesian information criterion for better precision matrix estimation.

Findings

Proposed methods outperform traditional cross-validation in accuracy.

Generalized approximate cross-validation reduces computational cost.

Bayesian information criterion effectively identifies nonzero entries.

Abstract

Recent literature provides many computational and modeling approaches for covariance matrices estimation in a penalized Gaussian graphical models but relatively little study has been carried out on the choice of the tuning parameter. This paper tries to fill this gap by focusing on the problem of shrinkage parameter selection when estimating sparse precision matrices using the penalized likelihood approach. Previous approaches typically used K-fold cross-validation in this regard. In this paper, we first derived the generalized approximate cross-validation for tuning parameter selection which is not only a more computationally efficient alternative, but also achieves smaller error rate for model fitting compared to leave-one-out cross-validation. For consistency in the selection of nonzero entries in the precision matrix, we employ a Bayesian information criterion which provably can identify the nonzero conditional correlations in the Gaussian model. Our simulations demonstrate the general superiority of the two proposed selectors in comparison with leave-one-out cross-validation, ten-fold cross-validation and Akaike information criterion.