LOCAL ADAPTIVITY OF KERNEL ESTIMATES WITH PLUG-IN LOCAL BANDWIDTH SELECTORS

In non-parametric function estimation selection of a smoothing paramet er is one of the most important issues. The performance of smoothing t echniques depends highly on the choice of this parameter. Preferably t he bandwidth should be determined via a data-driven procedure. In this paper we consider kernel estimators in a white noise model, and inves tigate whether locally adaptive plug-in bandwidths can achieve optimal global rates of convergence. We consider various classes of functions : Sobolev classes, bounded variation function classes, classes of conv ex functions and classes of monotone functions. We study the situation s of pilot estimation with oversmoothing and without oversmoothing. Ou r main finding is that simple local plug-in bandwidth selectors can ad apt to spatial inhomogeneity of the regression function as long as the re are no local oscillations of high frequency. We establish the point wise asymptotic distribution of the regression estimator with local pl ug-in bandwidth.