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General empirical Bayes wavelet methods and exactly adaptive minimax estimation. (English) Zbl 1064.62009
Summary: In many statistical problems, stochastic signals can be represented as a sequence of noisy wavelet coefficients. We develop general empirical Bayes methods for the estimation of the 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.

MSC:
62C12 Empirical decision procedures; empirical Bayes procedures
62G08 Nonparametric regression and quantile regression
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62B10 Statistical aspects of information-theoretic topics
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