The Annals of Statistics

Uniform asymptotic inference and the bootstrap after model selection

Ryan J. Tibshirani, Alessandro Rinaldo, Rob Tibshirani, and Larry Wasserman

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Recently, Tibshirani et al. [J. Amer. Statist. Assoc. 111 (2016) 600–620] proposed a method for making inferences about parameters defined by model selection, in a typical regression setting with normally distributed errors. Here, we study the large sample properties of this method, without assuming normality. We prove that the test statistic of Tibshirani et al. (2016) is asymptotically valid, as the number of samples $n$ grows and the dimension $d$ of the regression problem stays fixed. Our asymptotic result holds uniformly over a wide class of nonnormal error distributions. We also propose an efficient bootstrap version of this test that is provably (asymptotically) conservative, and in practice, often delivers shorter intervals than those from the original normality-based approach. Finally, we prove that the test statistic of Tibshirani et al. (2016) does not enjoy uniform validity in a high-dimensional setting, when the dimension $d$ is allowed grow.

Article information

Ann. Statist., Volume 46, Number 3 (2018), 1255-1287.

Received: July 2016
Revised: March 2017
First available in Project Euclid: 3 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F05: Asymptotic properties of tests 62F35: Robustness and adaptive procedures 62J05: Linear regression 62J07: Ridge regression; shrinkage estimators

Post-selection inference selective inference asymptotics bootstrap forward stepwise regression lasso


Tibshirani, Ryan J.; Rinaldo, Alessandro; Tibshirani, Rob; Wasserman, Larry. Uniform asymptotic inference and the bootstrap after model selection. Ann. Statist. 46 (2018), no. 3, 1255--1287. doi:10.1214/17-AOS1584.

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Supplemental materials

  • Supplement to “Uniform asymptotic inference and the bootstrap after model selection”. This document gives additional figures, details, and proofs for the paper “Uniform asymptotic inference and the bootstrap after model selection.”.