Confidence sets in sparse regression
Richard Nickl, Sara van de Geer
The Annals of Statistics · 2013 · Volume 41 · Issue 6
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.
Keywords
62G15, 62J05, Composite testing problem, Detection boundary, high-dimensional inference, quadratic risk estimation
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