Maximum Likelihood Inference for a Large Precision Matrix

• Activity Number: 47
• Type: Invited
• Date/Time: Sunday, August 9, 2015 : 4:00 PM to 5:50 PM
• Sponsor: IMS
• Author(s): Yunzhang Zhu* and Xiaotong Shen
• Companies: The Ohio State University and University of Minnesota
• Keywords: Graphical models ; $L_0$-regularization ; $(p,n)$-asymptotics

Abstract

Gaussian graphical model concerns estimating conditional dependence relationships among a large number of Gaussian random variables. Critical component in understanding these conditional dependence relationships lies in the estimation and inference of a large precision matrix. In this talk, we propose maximum likelihood inference for the large precision matrix for Gaussian graphical model in a high-dimensional situation. In particular, we obtain conditions for the asymptotic normality of the constrained maximum likelihood estimate given known sparsity pattern, when the matrix dimension and number of nonzero elements may tend to infinity with the sample size. On this ground, we derive the asymptotic distribution of the constrained maximum likelihood subject to the $L_0$-sparsity constraint of nonzero elements of the precision matrix. Most importantly, we derive the corresponding asymptotic distributions of the likelihood ratio statistics, which is the chi-square distribution when the co-dimension is finite and a normal distribution (after proper scaling) when the co-dimension grows with the sample size slowly.