We propose methodology for estimation of sparse precision matrices and
statistical inference for their low-dimensional parameters in a high-dimensional set-
ting where the number of parameters p can be much larger than the sample size. We
show that the novel estimator achieves minimax rates in supremum norm and the low-
dimensional components of the estimator have a Gaussian limiting distribution. These
results √ hold uniformly over the class of precision matrices with row sparsity of small
order n/ log p and spectrum uniformly bounded, under a sub-Gaussian tail assump-
tion on the margins of the true underlying distribution. Consequently, our results lead
to uniformly valid confidence regions for low-dimensional parameters of the preci-
sion matrix. Thresholding the estimator leads to variable selection without imposing
irrepresentability conditions. The performance of the method is demonstrated in a
simulation study and on real data.

• Honest confidence regions and optimality in high-dimensional precision matrix estimation