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Recent attention has focussed on possible improvements in performance of estimators which might flow from using the smoothed bootstrap. We point out that in a great many problems, such as those involving functions of vector means, any such improvements will be only second-order effects. However, we argue that substantial and significant improvements can occur in problems where local properties of underlying distributions play a decisive role. This situation often occurs in estimating the variance of an estimator defined in an L¹ setting; we illustrate in the special case of the variance of a quantile estimator. There we show that smoothing appropriately can improve estimator convergence rate from n^-1/4 for the unsmoothed bootstrap to n^- (1/2) +, for arbitrary > 0. We provide a concise description of the smoothing parameter which optimizes the convergence rate.
Hall et al. (Thu,) studied this question.