On optimality of the empirical distribution function for the estimation of the invariant distribution function of a diffusion process

In this work we present some results on the optimality of the empirical distribution function as estimator of the invariant distribution function of an ergodic diffusion process. The results presented were obtained in different previous works under conditions that are are rewritten in a unified form that make comparable those results. It is well known that the empirical distribution function is an unbiased and uniformly consistent estimator for the invariant distribution function of an ergodic diffusion process. It is also an efficient estimator in the sense that the risk of this estimator attains an asymptotic minimax lower bound. In this paper we review some results on the problem of the efficiency of the empirical distribution function considering three types of risk function. The first one is in a semi-parametric set-up. The second one is the integrated mean square error and the third is based on the sup norm.