Georgia Institute of Technology College of Sciences

Bootstrap is one of the most powerful and common tools in statistical inference. In this talk a multiplier bootstrap procedure is considered for construction of likelihood-based confidence sets. 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, where n is a sample size. The main result about bootstrap validity continues to apply even if the underlying parametric model is misspecified under a 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 conservative: the size of the constructed confidence sets is increased by the modelling bias. The approach is also extended to the problem of simultaneous confidence estimation. A simultaneous multiplier bootstrap procedure is justified for the case of exponentially large number of models. Numerical experiments for misspecified regression models nicely confirm our theoretical results.