Precision Matrix Estimation with ROPE

It is known that the accuracy of the maximum likelihood based covariance and precision matrix estimates can be improved by penalized log-likelihood estimation. In this article we propose a Ridge-type Operator for the Precision matrix Estimation, ROPE for short, to maximize a penalized likelihood function where the Frobenius norm is used as the penalty function. We show that there is an explicit closed form representation of a shrinkage estimator for the precision matrix when using a penalized log-likelihood, which is analogous to ridge regression in a regression context. The performance of the proposed method is illustrated by a simulation study and real data applications. Computer codes used in the example analyses as well as other supplementary materials for this article are available online.