Bootstrap confidence sets under model misspecification

TL;DR

This paper develops a multiplier bootstrap method for likelihood-based confidence sets that remains valid under model misspecification and small to moderate sample sizes, with control over the impact of parameter dimension.

Contribution

It introduces a bootstrap procedure that is robust to model misspecification and provides theoretical guarantees for its validity in finite samples.

Findings

Bootstrap approximation valid if p^3/n is small

Method remains valid under small modelling bias

Conservative adjustment when models are significantly misspecified

Abstract

A multiplier bootstrap procedure for construction of likelihood-based confidence sets is considered for finite samples and a possible model misspecification. Theoretical results justify the bootstrap validity for a small or moderate sample size and allow to control the impact of the parameter dimension $p$: the bootstrap approximation works if $p^3/n$ is small. The main result about bootstrap validity continues to apply even if the underlying parametric model is misspecified under the so-called small modelling bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modelling bias. We illustrate the results with numerical examples for misspecified linear and logistic regressions.