Weighted bootstrapped kernel density estimators in two-sample problems

A weighted bootstrap method is proposed to approximate the distribution of the $L_p$ ($1\leq p<\infty$) norms of two-sample statistics involving kernel density estimators. Using an approximation theorem of Horv\'ath, Kokoszka and Steineback [(2000) `Approximations for Weighted Bootstrap Processes with an Application', Statistics and Probability Letters, 48, 59-70], that allows one to replace the weighted bootstrap empirical process by a sequence of Gaussian processes, we establish an unconditional bootstrap central limit theorem for such statistics. The proposed method is quite straightforward to implement in practice. Furthermore, through some simulation studies, it will be shown that, depending on the weights chosen, the proposed weighted bootstrap approximation can sometimes outperform both the classical large-sample theory as well as Efron's [(1979) `Bootstrap Methods: Another Look at the Jackknife', Annals of Statistics, 7, 1-26] original bootstrap algorithm.