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.

Abstract not found

Abstract: This article concerns the canonical empirical Bayes problem of estimating normal means under squared-error loss. General empirical estimators are derived which are asymptotically minimax and optimal. Uniform convergence and the speed of convergence are considered. The general empirical Bayes estimators are compared with the shrinkage estimators of Stein (1956) and James and Stein (1961). Estimation of the mixture density and its derivatives are also discussed.

Abstract: We consider the compound decision problem of estimating a vector of n parameters, known up to a permutation, corresponding to n independent observations, and discuss the difference between two symmetric classes of estimators. The first and larger class is restricted to the set of all permutation invariant estimators. The second class is restricted further to simple symmetric procedures. That is, estimators such that each parameter is estimated by a function of the corresponding observation alone. We show that under mild conditions, the minimal total squared error risks over these two classes are asymptotically equivalent up to essentially O(1) difference.

Asymptotic efficiency of simple decisions for the compound decision problem ∗

We propose a general maximum likelihood empirical Bayes (GM-LEB) method for the estimation of a mean vector based on observations with i.i.d. normal errors. We prove that under mild moment conditions on the unknown means, the average mean squared error (MSE) of the GMLEB is within an infinitesimal fraction of the minimum average MSE among all separable estimators which use a single deterministic estimating function on individual observations, provided that the risk is of greater order than (log n) 5 /n. We also prove that the GMLEB is uniformly approximately minimax in regular and weak ℓp balls when the order of the length-normalized norm of the unknown means is between (log n) κ1 /n

Often environmental hazards are assessed by examining the spatial variation of disease-specific mortality or morbidity rates. These rates, when estimated for small local populations, can have a high degree of random variation or uncertainty associated with them. If those rate estimates are used to prioritize environmental clean-up actions or to allocate resources, then those decisions may be influenced by this high degree ofuncertainty. Unfortunately, the effect of this uncertainty is not to add "random noise " into the decsion-making process, but to systematicaily bias action toward the smaIlest populations where uncertainty is greatest and where extreme high and low rate deviations are most likely to be manifest by chance. We present a statistical procedure for adjusting rate estimates for differences in variability due to differentials in local area population sizes. Such adjustments produce rate estimates for areas that have better properties than the unadjusted rates for use in making statistically based decisions about the entire set of areas. Examples are provided for county variation in bladder, stomach, and lung cancer mortality rates for U.S white males for the period 1970 to 1979.