The Annals of Statistics

On Smoothing and the Bootstrap

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

Full-text: Open access

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

Permanent link to this document
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
links.jstor.org

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

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

Citation

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


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