On skewed Gaussian graphical models

Abstract

We introduce a skewed Gaussian graphical model as an extension to the Gaussian graphical model. One of the appealing properties of the Gaussian distribution is that conditional independence can be fully characterized by the sparseness in the precision matrix. The skewed Gaussian distribution adds a shape parameter to the Gaussian distribution to take into account possible skewness in the data; thus it is more flexible than the Gaussian model. Nevertheless, the appealing property of the Gaussian distribution is retained to a large degree: the conditional independence is still characterized by the sparseness in the parameters, which now include a shape parameter in addition to the precision matrix. As a result, the skewed Gaussian graphical model can be efficiently estimated through a penalized likelihood method just as the Gaussian graphical model. We develop an algorithm to maximize the penalized likelihood based on the alternating direction method of multipliers, and establish the asymptotic normality and variable selection consistency for the new estimator. Through simulations, we demonstrate that our method performs better than the Gaussian and Gaussian copula methods when these distributional assumptions are not satisfied. The method is applied to a breast cancer MicroRNA dataset to construct a gene network, which shows better interpretability than the Gaussian graphical model.