Bootstrap confidence sets under a model misspecification

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 consistency for 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 consistency continues to apply even if the underlying parametric model is misspecified under the so called Small Modeling 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 modeling bias. We illustrate the results with numerical examples for misspecified linear and logistic regressions.