Weighted bootstrapping of U-statistics

Abstract

We introduce a generalized bootstrap procedure for U-statistics. The idea is to reweight the terms of the original U-statistic by stochastic weights. Specific choices of the weights lead to well-known resampling schemes including Efron's bootstrap and Bayesian bootstrap. We establish the asymptotic consistency of the weighted bootstrap for U-statistics and studentized U-statistics. Our results extend the recent work by Mason and Newton (1992) on weighted bootstrapping of means.