## 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.

### MSC:

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

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