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June 2008 Adaptive estimation of and oracle inequalities for probability densities and characteristic functions
Sam Efromovich
Ann. Statist. 36(3): 1127-1155 (June 2008). DOI: 10.1214/009053607000000965

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.

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Sam Efromovich. "Adaptive estimation of and oracle inequalities for probability densities and characteristic functions." Ann. Statist. 36 (3) 1127 - 1155, June 2008. https://doi.org/10.1214/009053607000000965

Information

Published: June 2008
First available in Project Euclid: 26 May 2008

zbMATH: 1360.62118
MathSciNet: MR2418652
Digital Object Identifier: 10.1214/009053607000000965

Subjects:
Primary: 62G05
Secondary: 62G20

Keywords: Blockwise shrinkage , equivalence , infinite support , infinitely differentiable , mean integrated squared error , minimax , nonparametric , not absolutely integrable

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 3 • June 2008
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