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General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation

Bibliographic Details
Journal Title: The Annals of Statistics
Authors and Corporations: Zhang, Cun-Hui
In: The Annals of Statistics, 33, 2005, 1, p. 54-100
Media Type: E-Article
Language: English
published:
Institute of Mathematical Statistics
Subjects:
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rft.atitle General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation
rft.epage 100
rft.genre article
rft.issn 0090-5364
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rft.jtitle The Annals of Statistics
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rft.pub Institute of Mathematical Statistics
rft.date 2005-02-01
x.date 2005-02-01T00:00:00Z
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abstract <p>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.</p>
authors Array ( [rft.aulast] => Zhang [rft.aufirst] => Cun-Hui )
languages eng
url https://www.jstor.org/stable/3448656
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author Zhang, Cun-Hui
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description <p>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.</p>
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spelling Zhang, Cun-Hui 0090-5364 Institute of Mathematical Statistics Wavelet Methods https://www.jstor.org/stable/3448656 <p>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.</p> General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation The Annals of Statistics
spellingShingle Zhang, Cun-Hui, The Annals of Statistics, General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation, Wavelet Methods
title General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation
title_full General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation
title_fullStr General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation
title_full_unstemmed General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation
title_short General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation
title_sort general empirical bayes wavelet methods and exactly adaptive minimax estimation
topic Wavelet Methods
url https://www.jstor.org/stable/3448656