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

By Jana Jankov'a, Sara A. van de Geer. Test (Madrid), 2015

80 citations recorded by the source corpus.

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 n/logp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}/\log p$$\end{document} 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.

Subfields: Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a5a5c8959320632b3c34077df426847847e6ca7d

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Citations in the corpus

A modern maximum-likelihood theory for high-dimensional logistic regression (2018, background): “In parallel, there also exists an extensive body of literature on penalized maximum likelihood estimates/procedures for generalized linear models, see [6, 10, 29, 35, 38, 52, 53], for example.”
Semiparametric efficiency bounds for high-dimensional models (2016, methodology): “Thus, after normalization by √n and by the (estimated) standard deviation, it follows that it is asymptotically standard normal and minimax optimal [see Ren et al. (2015), Janková and van de Geer (2017)].”
Change-point detection in high-dimensional covariance structure (2016, methodology): “For each window size n ∈ N, each central point t ∈ Tn and each side S ∈ {l, r} we define a de-sparsified estimator of precision matrix [26] [27] as”