Electronic Journal of Statistics

On the post selection inference constant under restricted isometry properties

François Bachoc, Gilles Blanchard, and Pierre Neuvial

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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.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 3736-3757.

Received: April 2018
First available in Project Euclid: 20 November 2018

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Zentralblatt MATH identifier

Primary: 62J05: Linear regression 62J15: Paired and multiple comparisons 62F25: Tolerance and confidence regions

Inference post model-selection confidence intervals PoSI constants linear regression high-dimensional inference sparsity restricted isometry property

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Bachoc, François; Blanchard, Gilles; Neuvial, Pierre. On the post selection inference constant under restricted isometry properties. Electron. J. Statist. 12 (2018), no. 2, 3736--3757. doi:10.1214/18-EJS1490. https://projecteuclid.org/euclid.ejs/1542682881

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