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

TL;DR

This paper introduces a new method for estimating sparse high-dimensional precision matrices, achieving optimal rates and valid confidence regions without restrictive assumptions, demonstrated through simulations and real data.

Contribution

It proposes a novel estimator that attains minimax rates and provides valid confidence regions for low-dimensional parameters in high-dimensional settings.

Findings

Estimator achieves minimax rates in supremum norm

Provides Gaussian limiting distribution for low-dimensional components

Enables variable selection without irrepresentability conditions

Abstract

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.