## The Annals of Statistics

### Asymptotic normality and optimalities in estimation of large Gaussian graphical models

#### Abstract

The Gaussian graphical model, a popular paradigm for studying relationship among variables in a wide range of applications, has attracted great attention in recent years. This paper considers 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 regression approach is proposed to obtain asymptotically efficient estimation of each entry 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 it is no longer possible to achieve the parametric rate in the estimation of each entry. This lower bound result, which provides an answer to the delicate sample size question, is established with a novel construction of a subset of sparse precision matrices in an application of Le Cam’s lemma. 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 $\ell_{q}$ operator norm and to make inference in latent variables in the graphical model. All of this is achieved under a sparsity condition on the precision matrix and a side condition on the range of its spectrum. This significantly relaxes the commonly imposed uniform signal strength condition on the precision matrix, irrepresentability condition on the Hessian tensor operator of the covariance matrix or the $\ell_{1}$ constraint on the precision matrix. Numerical results confirm our theoretical findings. The ROC curve of the proposed algorithm, Asymptotic Normal Thresholding (ANT), for support recovery significantly outperforms that of the popular GLasso algorithm.

#### Article information

Source
Ann. Statist., Volume 43, Number 3 (2015), 991-1026.

Dates
Revised: October 2014
First available in Project Euclid: 15 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1431695636

Digital Object Identifier
doi:10.1214/14-AOS1286

Mathematical Reviews number (MathSciNet)
MR3346695

Zentralblatt MATH identifier
1328.62342

Subjects
Primary: 62H12: Estimation
Secondary: 62F12: Asymptotic properties of estimators 62G09: Resampling methods

#### Citation

Ren, Zhao; Sun, Tingni; Zhang, Cun-Hui; Zhou, Harrison H. Asymptotic normality and optimalities in estimation of large Gaussian graphical models. Ann. Statist. 43 (2015), no. 3, 991--1026. doi:10.1214/14-AOS1286. https://projecteuclid.org/euclid.aos/1431695636

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#### Supplemental materials

• Supplement to “Asymptotic normality and optimalities in estimation of large Gaussian graphical model”. In this supplement we collect proofs of Theorems 1–3 in Section 2, proofs of Theorems 6, 8 in Section 3 and proofs of Theorems 10–11 as well as Proposition 1 in Section 4.