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Beschreibung:
The problem of optimal adaptive estimation of a function at a given point from noisy data is considered.Two procedures are proved to be asymptotically optimal for different settings.First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel.We propose a bandwidth selection procedure and prove its optimality in the asymptotic sense. Moreover, this optimality is stated not only among kernel estimators with a variable kernel. The resulting estimator is optimal among all feasible estimators.The important feature of this procedure is that no prior information is used about smoothness properties of the estimated function i.e. the procedure is completely adaptive and "works" for the class of all functions. With it the attainable accuracy of estimation depends on the function itself and it is expressed in terms of "ideal" bandwidth corresponding to this function.The second procedure can be considered as a specification of the first one under the qualitative assumption that the function to be estimated belongs to some Hölder class Σ(β,L) with unknown parameters β, L.This assumption allows to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense.