Open Access
April 2016 Inference in adaptive regression via the Kac–Rice formula
Jonathan E. Taylor, Joshua R. Loftus, Ryan J. Tibshirani
Ann. Statist. 44(2): 743-770 (April 2016). DOI: 10.1214/15-AOS1386


We derive an exact $p$-value for testing a global null hypothesis in a general adaptive regression setting. Our approach uses the Kac–Rice formula [as described in Random Fields and Geometry (2007) Springer, New York] applied to the problem of maximizing a Gaussian process. The resulting test statistic has a known distribution in finite samples, assuming Gaussian errors. We examine this test statistic in the case of the lasso, group lasso, principal components and matrix completion problems. For the lasso problem, our test relates closely to the recently proposed covariance test of Lockhart et al. [Ann. Statist. (2004) 42 413–468].

In a few specific settings, our proposed tests will be less powerful than other previously known (and well-established) tests. However, it should be noted that the real strength of our proposal here is its generality. We provide a framework for constructing valid tests across a wide class of regularized regression problems, and as far as we can tell, such a unified view was not possible before this work.


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Jonathan E. Taylor. Joshua R. Loftus. Ryan J. Tibshirani. "Inference in adaptive regression via the Kac–Rice formula." Ann. Statist. 44 (2) 743 - 770, April 2016.


Received: 1 February 2015; Revised: 1 September 2015; Published: April 2016
First available in Project Euclid: 17 March 2016

zbMATH: 1337.62304
MathSciNet: MR3476616
Digital Object Identifier: 10.1214/15-AOS1386

Primary: 62M40
Secondary: 62J05

Keywords: convex analysis , Gaussian processes , regularized regression

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 2 • April 2016
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