A penalized likelihood approach in covariance graphical model selection

Abstract

One of the major challenges in modern statistics is to investigate the complex relationships and dependencies existing in data. Covariance or correlation matrix estimation that addresses the relationships among random variables attracts a lot of attention due to its ubiquity in data analysis. Of particular interest is to identify zero entries in the covariance matrix, since the zero entry corresponds to marginal independence between two variables. This is referred as covariance graphical model selection. Identifying pairwise independence in this model is helpful to elucidate relations between the variables. We propose a penalized likelihood approach for covariance graphical model selection and a BIC type criterion for the selection of the tuning parameter. An attractive feature of a likelihood based approach is its improved efficiency comparing to banding or thresholding. Another attractive feature of the proposed method is that the positive definiteness of the covariance matrix is explicitly ensured. We showed that the penalized likelihood estimator converge to the true covariance matrix under frobenius norm with explicit rate.