A hybrid approach based on saddlepoint and importance sampling methods for bootstrap tail probability estimation

We propose a simple hybrid method which makes use of both saddlepoint and importance sampling techniques to approximate the bootstrap tail probability of an M-estimator. The method does not rely on explicit formula of the Lugannani-Rice type, and is computationally more efficient than both uniform bootstrap sampling and importance resampling suggested in earlier literature. The method is also applied to construct confidence intervals for smooth functions of M-estimands.