Adaptive estimation of and oracle inequalities for probability densities and characteristic functions

Abstract

The theory of adaptive estimation and oracle inequalities for the case of Gaussian-shift–finite-interval experiments has made significant progress in recent years. In particular, sharp-minimax adaptive estimators and exact exponential-type oracle inequalities have been suggested for a vast set of functions including analytic and Sobolev with any positive index as well as for Efromovich–Pinsker and Stein blockwise-shrinkage estimators. Is it possible to obtain similar results for a more interesting applied problem of density estimation and/or the dual problem of characteristic function estimation? The answer is “yes.” In particular, the obtained results include exact exponential-type oracle inequalities which allow to consider, for the first time in the literature, a simultaneous sharp-minimax estimation of Sobolev densities with any positive index (not necessarily larger than 1/2), infinitely differentiable densities (including analytic, entire and stable), as well as of not absolutely integrable characteristic functions. The same adaptive estimator is also rate minimax over a familiar class of distributions with bounded spectrum where the density and the characteristic function can be estimated with the parametric rate.