The Annals of Statistics

Bootstrap confidence sets under model misspecification

Vladimir Spokoiny and Mayya Zhilova

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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.

Article information

Ann. Statist., Volume 43, Number 6 (2015), 2653-2675.

Received: November 2014
Revised: June 2015
First available in Project Euclid: 7 October 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F25: Tolerance and confidence regions 62F40: Bootstrap, jackknife and other resampling methods
Secondary: 62E17: Approximations to distributions (nonasymptotic)

Likelihood-based bootstrap confidence set finite sample size multiplier/weighted bootstrap Gaussian approximation Pinsker’s inequality


Spokoiny, Vladimir; Zhilova, Mayya. Bootstrap confidence sets under model misspecification. Ann. Statist. 43 (2015), no. 6, 2653--2675. doi:10.1214/15-AOS1355.

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Supplemental materials

  • Supplement to “Bootstrap confidence sets under model misspecification”. The supplementary material contains a proof of the square-root Wilks approximation for the bootstrap world, proofs of the main results from Section 2, and results on Gaussian approximation for $\ell_{2}$-norm of a sum of independent vectors.