The Annals of Statistics

Confidence sets in sparse regression

Richard Nickl and Sara van de Geer

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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.

Article information

Ann. Statist., Volume 41, Number 6 (2013), 2852-2876.

First available in Project Euclid: 17 December 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J05: Linear regression
Secondary: 62G15: Tolerance and confidence regions

Composite testing problem high-dimensional inference detection boundary quadratic risk estimation


Nickl, Richard; van de Geer, Sara. Confidence sets in sparse regression. Ann. Statist. 41 (2013), no. 6, 2852--2876. doi:10.1214/13-AOS1170.

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