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June, 1989 On Smoothing and the Bootstrap
Peter Hall, Thomas J. DiCiccio, Joseph P. Romano
Ann. Statist. 17(2): 692-704 (June, 1989). DOI: 10.1214/aos/1176347135


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.


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Peter Hall. Thomas J. DiCiccio. Joseph P. Romano. "On Smoothing and the Bootstrap." Ann. Statist. 17 (2) 692 - 704, June, 1989.


Published: June, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0672.62051
MathSciNet: MR994260
Digital Object Identifier: 10.1214/aos/1176347135

Primary: 62G05
Secondary: 62G30

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 2 • June, 1989
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