Asymptotic Normality and Optimality in the Estimation of Large Gaussian Graphical Model

• Activity Number: 336
• Type: Invited
• Date/Time: Tuesday, August 11, 2015 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Zhao Ren and Tingni Sun and Cun-Hui Zhang* and Harrison Zhou
• Companies: University of Pittsburgh and University of Maryland and Rutgers University and Yale University
• Keywords: statistical inference ; graphical model ; precision matrix ; sample size ; asymptotic efficiency ; optimal convergence rate

Abstract

We consider a fundamental question: When is it possible to estimate low-dimensional parameters at parametric square-root rate in a large Gaussian graphical model? A novel approach is proposed to obtain asymptotically efficient estimation of individual entries of a precision matrix under a sparseness condition relative to the sample size. When the precision matrix is not sufficiently sparse, or equivalently the sample size is not sufficiently large, a lower bound is established to show that the parametric rate is no longer attainable. Moreover, the proposed estimator is proven to have optimal convergence rate when the parametric rate cannot be achieved, under a minimal sample requirement. The proposed estimator is applied to test the presence of an edge in the Gaussian graphical model or to recover the support of the entire model, to obtain adaptive rate-optimal estimation of the entire precision matrix as measured by the matrix operator norms, and to make inference in latent variables in the graphical model. All these are achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum.