## The Annals of Statistics

### On Smoothing and the Bootstrap

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347135

Digital Object Identifier
doi:10.1214/aos/1176347135

Mathematical Reviews number (MathSciNet)
MR994260

Zentralblatt MATH identifier
0672.62051

JSTOR