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

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

2008-06-18

arxiv.org/abs/0806.2909v1

Annals of Statistics 2008, Vol. 36, No. 3, 1127-1155

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/009053607000000965 the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053607000000965

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.

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