Exact adaptive pointwise estimation on Sobolev classes of densities

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 $x_{0}\in \mathbb R$, 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 Bn. A sharp adaptive estimator is obtained, and the explicit asymptotical constant, associated to its rate of convergence is found