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Given n observations of a p-dimensional random vector, the covariance matrixand its inverse (precision matrix) are needed in a wide range of applications.Sample covariance (e.g. its eigenstructure) can misbehave when p is comparableto the sample size n. Regularization is often used to mitigate the problem.  In this paper, we proposed an l1-norm penalized pseudo-likelihood estimatefor the inverse covariance matrix. This estimate is sparse due to the l1-normpenalty, and we term this method SPLICE. Its regularization path can becomputed via an algorithm based on the homotopy/LARS-Lasso algorithm.Simulation studies are carried out for various inverse covariance structuresfor p=15 and n=20, 1000. We compare SPLICE with the l1-norm penalizedlikelihood estimate and a l1-norm penalized Cholesky decomposition basedmethod. SPLICE gives the best overall performance in terms of three metrics onthe precision matrix and ROC curve for model selection. Moreover, oursimulation results demonstrate that the SPLICE estimates are positive-definitefor most of the regularization path even though the restriction is notenforced.

A path following algorithm for Sparse Pseudo-Likelihood Inverse Covariance Estimation (SPLICE)