Bootstrap confidence sets under model misspecification

A multiplier bootstrap procedure for construction of likelihood-based condence sets is considered for nite samples and a possible model misspecication. Theoretical results justify the bootstrap consistency for a small or moderate sample size and allow to control the impact of the parameter dimension p: the bootstrap approximation works if p3=n is small. The main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecied under the so called Small Modeling Bias condition. In the case when the true model deviates signicantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed condence sets is increased by the modeling bias. We illustrate the results with numerical examples for misspecied constant and logistic regressions.