General empirical Bayes wavelet methods and exactly adaptive minimax estimation
By Cun-Hui Zhang. 2005
66 citations recorded by the source corpus.
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.
Subfields: Computer Networks and Communications, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/05236793ec31c8264330887539baee4d257bff36
How researchers see this paper
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In Computer Networks and Communications
- Guillaume Lecué
- Christophe Chesneau
- Wenhua Jiang
- Sam Efromovich
- Lawrence D. Brown
- Eitan Greenshtein
- Cun-Hui Zhang
- Tommaso Cai
In Statistics and Probability
- Cun-Hui Zhang
- Lawrence D. Brown
- Eitan Greenshtein
- Guillaume Lecué
- Wenhua Jiang
- Christophe Chesneau
- Sam Efromovich
- Peter Buhlmann
Citations in the corpus
- Adapting to Unknown Smoothness by Aggregation of Thresholded Wavelet Estimators (2006, methodology): “In fact, the multi-thresholding estimator has similar minimax performances than the empirical Bayes wavelet methods (see [55] and [32]) and several term-by-term wavelet thresholded estimators defined with a random threshold (see [33] and [7]).”
- NONPARAMETRIC EMPIRICAL BAYES AND COMPOUND DECISION APPROACHES TO ESTIMATION OF A HIGH-DIMENSIONAL VECTOR OF NORMAL MEANS (2009, background): “See, for example, the papers of Zhang (1997, 2005).”
- Bayes procedures for adaptive inference in inverse problems for the white noise model (2012, unknown): “Rigorous results about the asymptotic behavior of empirical Bayes selectors of hyper-parameters in infinite-dimensional problems only exist for a limited number of special problems, see e.g. [ 4,19,20,35,40].”
- Estimation of the density of regression errors (2005, background): “The interested reader can find discussion in [5, 9, 11, 14, 34, 39].”
- General maximum likelihood empirical Bayes estimation of normal means (2009, methodology): “It follows from [3, 38] that a hybrid between the Fourier general EB and universal soft threshold estimators is also adaptive rate minimax in Θp,Cn,n for 1≤O(1)nCp n.”