Exact adaptive pointwise estimation on Sobolev classes of densities

Abstract

The subject of this paper is to estimate adaptively the common probability density of n independent, identically distributed random variables. The estimation is done at a fixed point , over the density functions that belong to the Sobolev class Wn (β,L) . We consider the adaptive problem setup, where the regularity parameter β is unknown and varies in a given set B n . A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence is found.