Bootstrap Estimation of Conditional Distributions

Abstract

Techniques are developed for bootstrap estimation of conditional distributions, with application to confidence intervals and hypothesis tests for one parameter, conditional on the value of an estimator of another. Both Monte Carlo and saddlepoint methods for approximating bootstrap distributions are considered, and empirical methods are suggested for implementing these techniques. For example, in the case of Monte Carlo methods, we suggest empirical techniques for selecting both the smoothing parameter, necessary to define the estimator, and the importance resampling probabilities, required for efficient bootstrap simulation. The smoothing parameter depends critically on the number of Monte Carlo simulations, as well as on the data. Both our theoretical and numerical results indicate that pivoting can substantially improve performance.