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.

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

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Cited by in corpus (9)

• Self-consistent method for density estimation
• Adaptive estimation of and oracle inequalities for probability densities and characteristic functions
• Model selection for density estimation with L2-loss
• Pointwise adaptive estimation of a multivariate density under independence hypothesis
• Adaptation to lowest density regions with application to support recovery
• Minimax Density Estimation on Sobolev Spaces With Dominating Mixed Smoothness
• On adaptive minimax density estimation on $R^d$
• Adaptive confidence sets in L^2
• On the usefulness of Meyer wavelets for deconvolution and density estimation