  We propose methodology for estimation of sparse precision matrices and
statistical inference for their low-dimensional parameters in a
high-dimensional setting 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 $\sqrt{n}/\log p$ and
spectrum uniformly bounded, under a sub-Gaussian tail assumption on the margins
of the true underlying distribution. Consequently, our results lead to
uniformly valid confidence regions for low-dimensional parameters of the
precision 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.
