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 bandwidth. The resulting estimator is asymptotically optimal among all feasible estimators. The important feature of this procedure is that it is fully adaptive and it "works" for a very wide class of functions obeying a mild regularity restriction. With it the attainable accuracy of estimation depends on the function itself and is expressed in terms of the "ideal adaptive bandwidth" corresponding to this function and a given kernel.
¶ The second procedure can be considered as a specialization of the first one under the qualitative assumption that the function to be estimated belongs to some Hölder class $\Sigma (\beta, L)$ with unknown parameters $\beta, L$. This assumption allows us to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense in any range of adaptation with $\beta \leq 2$.