This paper explores a class of empirical Bayes methods for level-dependent threshold selection in wavelet shrinkage. The prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy-tailed density. The mixing weight, or sparsity parameter, for each level of the transform is chosen by marginal maximum likelihood. If estimation

We study the performances of an adaptive procedure based on a convex combination, with data-driven weights, of term-by-term thresholded wavelet estimators. For the bounded regression model, with random uniform design, and the nonparametric density model, we show that the resulting estimator is optimal in the minimax sense over all Besov balls under the L 2 risk, without any logarithm factor. 1

Estimation of the density of regression errors is a fundamental issue in regression analysis and it is typically explored via a parametric approach. This article uses a nonparametric approach with the mean integrated squared error (MISE) criterion. It solves a long-standing problem, formulated two decades ago by Mark Pinsker, about estimation of a nonparametric error density in a nonparametric regression setting with the accuracy of an oracle that knows the underlying regression errors. The solution implies that, under a mild assumption on the differentiability of the design density and regression function, the MISE of a data-driven error density estimator attains minimax rates and sharp constants known for the case of directly observed regression errors. The result holds for error densities with finite and infinite supports. Some extensions of this result for more general heteroscedastic models with possibly dependent errors and predictors are also obtained; in the latter case the marginal error density is estimated. In all considered cases a blockwise-shrinking Efromovich– Pinsker density estimate, based on plugged-in residuals, is used. The obtained results imply a theoretical justification of a customary practice in applied regression analysis to consider residuals as proxies for underlying regression errors. Numerical and real examples are presented and discussed, and the S-PLUS software is available. 1. Introduction. A

The effects of spatial scale in disease mapping are well-recognized, in that the information conveyed by such maps varies with scale. Here we provide an inferential framework, in the context of tract count data, for describing the distribution of relative risk simultaneously across a hierarchy of multiple scales. In particular, we offer a multiscale extension of the canonical standardized mortality ratio (SMR), consisting of Bayesian posterior-based strategies for both estimation and characterization of uncertainty. As a result, a hierarchy of informative disease and confidence maps can be produced, without the need to first try to identify a single appropriate scale of analysis. We explore the behavior of the proposed methodology in a small simulation study, and we illustrate its usage through an application to data on gastric cancer in Tuscany. Copyright c ○ 2004 John Wiley & Sons, Ltd. 1.

We consider a joint processing of n independent sparse regression problems. Eachis based on a sample (yi1,xi1)...,(yim,xim) of m i.i.d. observationsfrom yi1 = x T i1 βi+εi1, yi1 ∈ R, xi1 ∈ R p, i = 1,...,n, and εi1 ∼ N(0,σ 2), say. p is large enough so that the empirical risk minimizer is not consistent. We consider three possible extensions of the lasso estimator to deal with this problem, the lassoes, the group lasso and the RING lasso, each utilizing a different assumption how these problems are related. For each estimator we give a Bayesian interpretation, and we present both persistency analysis and non-asymptotic error bounds based on restricted eigenvalue- type assumptions. “...and only a star or two set sparsedly in the vault of heaven; and you will find a sight as stimulating as the hoariest summit of the Alps. ” R. L. Stevenson 1

We consider a joint processing of n independent similar sparse regression problems. Each is based on a sample (yi1, xi1)..., (yim, xim) of m i.i.d. observations from yi1 = x T i1 βi + εi1, yi1 ∈ R, xi1 ∈ R p, and εi1 ∼ N(0, σ 2), say. The dimension p is large enough so that the empirical risk minimizer is not feasible. We consider, from a Bayesian point of view, three possible extensions of the lasso. Each of the three estimators, the lassoes, the group lasso, and the RING lasso, utilizes different assumptions on the relation between the n vectors β1,..., βn. “... and only a star or two set sparsedly in the vault of heaven; and you will find a sight as stimulating as the hoariest summit of the Alps. ” R. L. Stevenson 1

It is well known that it is possible to achieve adaptation for “free” in function estimation under a global loss. It is unclear, however, why and how the adaptability is achieved. In this article we show that adaptability is achieved through information pooling. It is first shown that separable rules, which figure prominently in wavelet and other orthogonal series methods, lack adaptability; they are necessarily not rate-adaptive. A sharp lower bound on the cost of adaptation for separable rules is obtained. We then derive a tight lower bound on the amount of information pooling required for achieving global adaptability. Moreover, in a sharp contrast to the separable rules, it is shown that adaptive nonseparable estimators can be superefficient at every point in the parameter spaces. The results demonstate that information pooling is the key to increase estimation precision and achieve adaptability and even superefficiency.

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