On adaptive estimation of nonlinear functionals

The known asymptotically efficient estimates of a nonlinear functional of an unknown function are crucially depended on prior information about a smoothness of this function. We show on example of filtering that this information is not necessary at all whenever the asymptotically efficient estimation is possible, that is, when the smoothness is sufficiently large. On the other side, if no prior information about the smoothness is given then there does not exist adaptive estimator which achieves the optimal nonadaptive rates for risk convergence.