We consider, in the framework of multidimensional observations, nonparametric functional estimators, which include, as special cases, the Akaike-Parzen-Rosenblatt kernel density estimators (, , ), and the Nadaraya-Watson kernel regression estimators (, ). We evaluate the sup-norm, over a given set <span class="tex"><b>I</b></span>, 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 <span class="tex"><b>I</b></span> 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 , given in the setup of density estimation in <b>R</b>.
density estimation, nonparametric functional estimation, regression estimation, bootstrap, resampling methods, confidence regions, empirical processes
62G05, 62G08, 62G09, 62G15, 62G20, 62G30