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February 2005 General empirical Bayes wavelet methods and exactly adaptive minimax estimation
Cun-Hui Zhang
Ann. Statist. 33(1): 54-100 (February 2005). DOI: 10.1214/009053604000000995


In many statistical problems, stochastic signals can be represented as a sequence of noisy wavelet coefficients. In this paper, we develop general empirical Bayes methods for the estimation of true signal. Our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals. In particular, our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in Besov balls of all smoothness and shape indices, and they are uniformly superefficient in convergence rates in all compact sets in Besov spaces with a finite secondary shape parameter. Furthermore, in classes nested between Besov balls of the same smoothness index, our estimators dominate threshold and James–Stein estimators within an infinitesimal fraction of the minimax risks. More general block empirical Bayes estimators are developed. Both white noise with drift and nonparametric regression are considered.


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Cun-Hui Zhang. "General empirical Bayes wavelet methods and exactly adaptive minimax estimation." Ann. Statist. 33 (1) 54 - 100, February 2005.


Published: February 2005
First available in Project Euclid: 8 April 2005

zbMATH: 1064.62009
MathSciNet: MR2157796
Digital Object Identifier: 10.1214/009053604000000995

Primary: 62C12 , 62C25 , 62G05 , 62G08 , 62G20

Keywords: Adaptation , Besov space , Empirical Bayes , minimax estimation , Nonparametric regression , threshold estimate , ‎wavelet , White noise

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 1 • February 2005
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