## The Annals of Statistics

### Uniform asymptotic inference and the bootstrap after model selection

#### Abstract

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

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

Dates
Revised: March 2017
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.aos/1525313082

Digital Object Identifier
doi:10.1214/17-AOS1584

#### Citation

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. https://projecteuclid.org/euclid.aos/1525313082

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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.”.