Local superefficiency of data-driven projection density estimators in continuous time

Abstract

We construct a data-driven projection density estimator for continuous time processes. This estimator reaches superoptimal rates over a class F0 of densities that is dense in the family of all possible densities, and a «reasonable» rate elsewhere. The class F0 may be chosen previously by the analyst. Results apply to Rd- Rd-valued processes and to N-valued processes. In the particular case where squareintegrable local time does exist, it is shown that our estimator is strictly better than the local time estimator over F0.