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 x0 E R, over the density functions that belong to the Sobolev class Wn (B,L). We consider the adaptive problem setup, where the regularity parameter B is unknown and varies in a given set B. A sharp adaptive estimator is obtained, andthe explicit asymptotical constant, associated to its rate of convergence is found.