Statistical Inference and Optimalities in Estimation of Gaussian Graphical Model

• Activity Number: 136
• Type: Contributed
• Date/Time: Monday, August 5, 2013 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Zhao Ren*+ and Harrison Zhou and Tingni Sun and Cun-Hui Zhang
• Companies: Yale University and Yale University and University of Pennsylvania and Rutgers University
• Keywords: graphical model ; asymptotic efficiency ; precision matrix ; minimax lower bound ; optimal rate of convergence ; latent graphical model

Abstract

Gaussian graphical model has a wide range of applications. The study of Gaussian graphical model had attracted a lot of attention recently. This paper considers a fundamental question: when is it possible to obtain statistical inference for estimation of Gaussian Graphical Model? A novel regression approach is proposed to obtain asymptotically efficient estimation of each entry when the precision matrix is sufficiently sparse. When the precision matrix is not sfficiently sparse, a lower bound is established to show that it is no longer possible to achieve the parametric rate estimation of each entry through a novel construction of a subset of sparse precision matrices and Le Cam's Lemma. The asymptotic normality result is applied to do adaptive support recovery, to obtain adaptive rate-optimal estimation of the precision matrix under various matrix lq norms, and to do inference and estimation for latent variable graphical models, without the irrepresentable condition and the l1 constraint of the precision matrix which are commonly required in literature.