  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.
