  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.

Methodology