The Annals of Statistics

On Smoothing and the Bootstrap

Peter Hall, Thomas J. DiCiccio, and Joseph P. Romano

Full-text: Open access


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 $L^1$ 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 $n^{-(1/2) + \varepsilon}$, for arbitrary $\varepsilon > 0$. We provide a concise description of the smoothing parameter which optimizes the convergence rate.

Article information

Ann. Statist., Volume 17, Number 2 (1989), 692-704.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G30: Order statistics; empirical distribution functions

Bandwidth bootstrap kernel $L^1$ regression mean squared error nonparametric density estimation quantile smoothing variance estimation


Hall, Peter; DiCiccio, Thomas J.; Romano, Joseph P. On Smoothing and the Bootstrap. Ann. Statist. 17 (1989), no. 2, 692--704. doi:10.1214/aos/1176347135.

Export citation