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...

We estimate the common density function of n i.i.d. observations, at a fixed point, over Sobolev classes of functions having regularity B. We prove that the optimal rate of convergence cannot be attained in adaptive estimation, i.e. uniformly over B in some interval B. A logarithmically slower...