On the Post Selection Inference constant under Restricted Isometry Properties

Abstract

Uniformly valid confidence intervals post model selection in regression can be constructed based on Post-Selection Inference (PoSI) constants. PoSI constants are minimal for orthogonal design matrices, and can be upper bounded in function of the sparsity of the set of models under consideration, for generic design matrices. In order to improve on these generic sparse upper bounds, we consider design matrices satisfying a Restricted Isometry Property (RIP) condition. We provide a new upper bound on the PoSI constant in this setting. This upper bound is an explicit function of the RIP constant of the design matrix, thereby giving an interpolation between the orthogonal setting and the generic sparse setting. We show that this upper bound is asymptotically optimal in many settings by constructing a matching lower bound.

arXiv commentElectronic journal of statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2018

References in corpus (9)

• On asymptotically optimal confidence regions and tests for high-dimensional models
• On the conditions used to prove oracle results for the Lasso
• Valid post-selection inference
• Exact post-selection inference, with application to the lasso
• Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors
• Optimal Inference After Model Selection
• Exact Post Model Selection Inference for Marginal Screening
• Confidence Sets Based on Sparse Estimators Are Necessarily Large
• Spherical Cap Packing Asymptotics and Rank-Extreme Detection

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