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December 2013 Confidence sets in sparse regression
Richard Nickl, Sara van de Geer
Ann. Statist. 41(6): 2852-2876 (December 2013). DOI: 10.1214/13-AOS1170

Abstract

The problem of constructing confidence sets in the high-dimensional linear model with $n$ response variables and $p$ parameters, possibly $p\ge n$, is considered. Full honest adaptive inference is possible if the rate of sparse estimation does not exceed $n^{-1/4}$, otherwise sparse adaptive confidence sets exist only over strict subsets of the parameter spaces for which sparse estimators exist. Necessary and sufficient conditions for the existence of confidence sets that adapt to a fixed sparsity level of the parameter vector are given in terms of minimal $\ell^{2}$-separation conditions on the parameter space. The design conditions cover common coherence assumptions used in models for sparsity, including (possibly correlated) sub-Gaussian designs.

Citation

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Richard Nickl. Sara van de Geer. "Confidence sets in sparse regression." Ann. Statist. 41 (6) 2852 - 2876, December 2013. https://doi.org/10.1214/13-AOS1170

Information

Published: December 2013
First available in Project Euclid: 17 December 2013

zbMATH: 1288.62108
MathSciNet: MR3161450
Digital Object Identifier: 10.1214/13-AOS1170

Subjects:
Primary: 62J05
Secondary: 62G15

Keywords: Composite testing problem , Detection boundary , high-dimensional inference , quadratic risk estimation

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 6 • December 2013
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