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Bootstrap confidence sets under model misspecification. (English) Zbl 1327.62179
Summary: 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.

62F25 Parametric tolerance and confidence regions
62F40 Bootstrap, jackknife and other resampling methods
62E17 Approximations to statistical distributions (nonasymptotic)
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