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General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation
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Zeitschriftentitel: | The Annals of Statistics |
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In: | The Annals of Statistics, 33, 2005, 1, S. 54-100 |
Medientyp: | E-Article |
Sprache: | Englisch |
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Institute of Mathematical Statistics
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Schlagwörter: |
author_facet |
Zhang, Cun-Hui Zhang, Cun-Hui |
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author |
Zhang, Cun-Hui |
spellingShingle |
Zhang, Cun-Hui The Annals of Statistics General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation Wavelet Methods |
author_sort |
zhang, cun-hui |
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 |
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Institute of Mathematical Statistics, 2005 |
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Institute of Mathematical Statistics, 2005 |
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2005 |
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Institute of Mathematical Statistics |
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The Annals of Statistics |
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55 |
title |
General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation |
title_unstemmed |
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 |
publishDate |
2005 |
physical |
54-100 |
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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author | Zhang, Cun-Hui |
author_facet | Zhang, Cun-Hui, Zhang, Cun-Hui |
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container_start_page | 54 |
container_title | The Annals of Statistics |
container_volume | 33 |
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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imprint | Institute of Mathematical Statistics, 2005 |
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institution | DE-15, DE-14, DE-Ch1, DE-D13 |
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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 |
title_unstemmed | General Empirical Bayes Wavelet Methods and Exactly Adaptive Minimax Estimation |
topic | Wavelet Methods |
url | https://www.jstor.org/stable/3448656 |