Some problems of nonparametric estimation by observations of ergodic diffusion process

Abstract

We consider the problems of the density and distribution function estimation by the observations of diffusion process with ergodic properties. In every problem we first propose a minimax bound on the risk of any estimator and then study the asymptotic behavior of several estimators. It is shown that the empiric distribution function is asymptotically normal and asymptotically efficient (in the minimax sense) estimator of the distribution function. In the density estimation problem, we describe the asymptotic behavior of a kernel-type estimator and one another (unbiased) estimator. Both of them are √ T-consistent, asymptotically normal and asymptotically efficient.