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.
Publié le : 1989-06-14
Classification:
Bandwidth,
bootstrap,
kernel,
$L^1$ regression,
mean squared error,
nonparametric density estimation,
quantile,
smoothing,
variance estimation,
62G05,
62G30
@article{1176347135,
author = {Hall, Peter and DiCiccio, Thomas J. and Romano, Joseph P.},
title = {On Smoothing and the Bootstrap},
journal = {Ann. Statist.},
volume = {17},
number = {1},
year = {1989},
pages = { 692-704},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347135}
}
Hall, Peter; DiCiccio, Thomas J.; Romano, Joseph P. On Smoothing and the Bootstrap. Ann. Statist., Tome 17 (1989) no. 1, pp. 692-704. http://gdmltest.u-ga.fr/item/1176347135/