Superefficiency of a projection density estimator

The author constructs a projection density estimator with a data-driven truncation index. This estimator reaches the superoptimal rates 1/n in mean integrated square error and {ln ln(n)/n}(1/2) in uniform almost sure convergence over a given subspace which is dense in the class of all possible densities; the rate of the estimator is quasi-optimal everywhere else. The subspace in question may be chosen a priori by the statistician.