Optimal pointwise adaptive methods in nonparametric estimation

Abstract

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 $\beta \leq 2$.