One Bootstrap suffices to generate sharp uniform bounds in functional estimation

Deheuvels, Paul. "One Bootstrap suffices to generate sharp uniform bounds in functional estimation." Kybernetika 47.6 (2011): 855-865. <http://eudml.org/doc/196965>.

@article{Deheuvels2011,
abstract = {We consider, in the framework of multidimensional observations, nonparametric functional estimators, which include, as special cases, the Akaike–Parzen–Rosenblatt kernel density estimators ([1, 18, 20]), and the Nadaraya–Watson kernel regression estimators ([16, 22]). We evaluate the sup-norm, over a given set $\{\bf I\}$, of the difference between the estimator and a non-random functional centering factor (which reduces to the estimator mean for kernel density estimation). We show that, under suitable general conditions, this random quantity is consistently estimated by the sup-norm over $\{\bf I\}$ of the difference between the original estimator and a bootstrapped version of this estimator. This provides a simple and flexible way to evaluate the estimator accuracy, through a single bootstrap. The present work generalizes former results of Deheuvels and Derzko [4], given in the setup of density estimation in $\mathbb \{R\}$.},
author = {Deheuvels, Paul},
journal = {Kybernetika},
keywords = {nonparametric functional estimation; density estimation; regression estimation; bootstrap; resampling methods; confidence regions; empirical processes; density estimation; nonparametric functional estimation; regression estimation; bootstrap; resampling methods; confidence regions; empirical processes},
language = {eng},
number = {6},
pages = {855-865},
publisher = {Institute of Information Theory and Automation AS CR},
title = {One Bootstrap suffices to generate sharp uniform bounds in functional estimation},
url = {http://eudml.org/doc/196965},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Deheuvels, Paul
TI - One Bootstrap suffices to generate sharp uniform bounds in functional estimation
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 6
SP - 855
EP - 865
AB - We consider, in the framework of multidimensional observations, nonparametric functional estimators, which include, as special cases, the Akaike–Parzen–Rosenblatt kernel density estimators ([1, 18, 20]), and the Nadaraya–Watson kernel regression estimators ([16, 22]). We evaluate the sup-norm, over a given set ${\bf I}$, of the difference between the estimator and a non-random functional centering factor (which reduces to the estimator mean for kernel density estimation). We show that, under suitable general conditions, this random quantity is consistently estimated by the sup-norm over ${\bf I}$ of the difference between the original estimator and a bootstrapped version of this estimator. This provides a simple and flexible way to evaluate the estimator accuracy, through a single bootstrap. The present work generalizes former results of Deheuvels and Derzko [4], given in the setup of density estimation in $\mathbb {R}$.
LA - eng
KW - nonparametric functional estimation; density estimation; regression estimation; bootstrap; resampling methods; confidence regions; empirical processes; density estimation; nonparametric functional estimation; regression estimation; bootstrap; resampling methods; confidence regions; empirical processes
UR - http://eudml.org/doc/196965
ER -