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Keywords:
nonparametric functional estimation; density estimation; regression estimation; bootstrap; resampling methods; confidence regions; empirical processes
nonparametric functional estimation; density estimation; regression estimation; bootstrap; resampling methods; confidence regions; empirical processes
Summary:
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 $\mathbb{R}$.
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