Nonparametric Estimation

Abstract

Below, we consider several problems of estimation concerning diffusion-type observations in situations when it is impossible to describe the underlying uncertainty as a parametric family. We study the asymptotic of the kernel-type estimators of trend coefficients and a “natural” estimator of the trajectory (state) of a deterministic limit system. The rate of convergence of these kernel-type estimators depends on the smoothness of the trend of the nonperturbed system and the optimality of this rate is established in an appropriate sense. For the state estimators, we introduce a low minimax bound on the risk function and then prove that the observed trajectory of the perturbed system is an asymptotically optimal estimator.