Uniform Asymptotic Inference and the Bootstrap After Model Selection

TL;DR

This paper extends the validity of a model selection inference method to nonnormal errors in fixed dimensions, proposes a conservative bootstrap approach, and discusses limitations in high-dimensional settings.

Contribution

It proves the asymptotic validity of Tibshirani et al.'s method under nonnormal errors and fixed dimensions, introduces a bootstrap version, and analyzes its limitations in high dimensions.

Findings

Asymptotic validity holds for fixed d and nonnormal errors.

Bootstrap version is asymptotically conservative and often produces shorter intervals.

Uniform validity fails when the dimension d grows with the sample size.

Abstract

Recently, Tibshirani et al. (2016) 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.