On optimality of the empirical distribution function for the estimation of the invariant distribution function of a diffusion process
Results previously obtained on the problem of efficiency of empirical distribution functions under different conditions [see, e.g., \textit{Yu.A. Kutoyants} and \textit{I. Negri}, On \(L_{2}\) efficiency of an empiric distribution for ergodic diffusion processes. Theory Probab. Appl. 46, No. 1, 140--146 (2001; Zbl 0990.62073)] are rewritten here in a form that they are comparable. The paper starts with the statement of the problem, the statistical problem and the conditions used throughout the text. The principal properties of the empirical distribution functions are presented, then some results on the efficiency of the empirical distribution function as estimators of invariant distribution functions when the efficiency is evaluated with respect to three risk functions are studied. In the last section of the paper an example is given to illustrate the fact that there is a huge class of processes for which the empirical distribution function is an efficient estimator for the invariant distribution function in the three different described set-ups.
scientific article; zbMATH DE number 5939882