On the recovery of functions and their derivatives from imperfect measurements

The problem of the estimation of functions and their derivatives from independent observations is considered. For a wide class of noises, the nearest neighbor method is given for estimating functions and their derivatives when the measurements are corrupted by additive noise. This method is shown to be consistent in the mean square sense and with probability one. The choices of the number of neighbors and the speed of convergence is investigated. Finally, applications to the optimization and to the system identification are discussed.