Results 1 
5 of
5
General empirical Bayes wavelet methods and exactly adaptive minimax estimation

, 2005
"... 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 risk ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
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.
A datadriven block thresholding approach to wavelet estimation
, 2005
"... A datadriven block thresholding procedure for wavelet regression is proposed and its theoretical and numerical properties are investigated. The procedure empirically chooses the block size and threshold level at each resolution level by minimizing Stein’s unbiased risk estimate. The estimator is sh ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
A datadriven block thresholding procedure for wavelet regression is proposed and its theoretical and numerical properties are investigated. The procedure empirically chooses the block size and threshold level at each resolution level by minimizing Stein’s unbiased risk estimate. The estimator is sharp adaptive over a class of Besov bodies and achieves simultaneously within a small constant factor of the minimax risk over a wide collection of Besov Bodies including both the dense and sparse cases. The procedure is easy to implement. Numerical results show that it has superior finite sample performance in comparison to the other leading wavelet thresholding estimators.
Tradeoffs Between Global And Local Risks In Nonparametric Function Estimation
"... We investigate the problem of loss adaptation: given a fixed parameter space we want to construct an estimator that adapts to the loss function in the sense that the estimator is optimal both globally and locally at every point. We consider the class of estimator sequences that achieve the minimax r ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
We investigate the problem of loss adaptation: given a fixed parameter space we want to construct an estimator that adapts to the loss function in the sense that the estimator is optimal both globally and locally at every point. We consider the class of estimator sequences that achieve the minimax rate, over a fixed Besov space, for estimating the entire function and establish a lower bound on the performance of any such sequence for estimation of the function at each point. This bound is larger by a logarithmic factor than the usual minimax rate for estimation at a point when the global and local minimax rates of convergence differ. We also consider estimators that achieve optimal minimax rates of convergence at every point and give a lower bound for the maximum global risk. An inequality concerning estimation in a two parameter statistical problem plays a key role in the proof. It can be considered as an generalization of an inequality in Brown and Low (1996b). This may be of independent interest. A particular wavelet estimator is constructed which is globally optimal and which attains the lower bound for the local risk provided by our inequality.
Minimax and Adaptive Inference in Nonparametric Function Estimation
"... Abstract. Since Stein’s 1956 seminal paper, shrinkage has played a fundamental role in both parametric and nonparametric inference. This article discusses minimaxity and adaptive minimaxity in nonparametric function estimation. Three interrelated problems, function estimation under global integrated ..."
Abstract
 Add to MetaCart
Abstract. Since Stein’s 1956 seminal paper, shrinkage has played a fundamental role in both parametric and nonparametric inference. This article discusses minimaxity and adaptive minimaxity in nonparametric function estimation. Three interrelated problems, function estimation under global integrated squared error, estimation under pointwise squared error, and nonparametric confidence intervals, are considered. Shrinkage is pivotal in the development of both the minimax theory and the adaptation theory. While the three problems are closely connected and the minimax theories bear some similarities, the adaptation theories are strikingly different. For example, in a sharp contrast to adaptive point estimation, in many common settings there do not exist nonparametric confidence intervals that adapt to the unknown smoothness of the underlying function. A concise account of these theories is given. The connections as well as differences among these problems are discussed and illustrated through examples. Key words and phrases: Adaptation, adaptive estimation, Bayes minimax,
FOURIER SERIES BASED BANDWIDTH SELECTORS FOR KERNEL DENSITY ESTIMATION
, 2009
"... A class of Fourier series based plugin bandwidth selectors for kernel density estimation is considered in this paper. The proposed datadependent bandwidths are simple to obtain, easy to interpret and consistent for a wide class of compact supported distributions. Some of them present good finite ..."
Abstract
 Add to MetaCart
A class of Fourier series based plugin bandwidth selectors for kernel density estimation is considered in this paper. The proposed datadependent bandwidths are simple to obtain, easy to interpret and consistent for a wide class of compact supported distributions. Some of them present good finite sample comparative performances against the classical twostage direct plugin method or the least squares crossvalidation method, being good alternatives to these classical methods. Finally, we argue that the flexibility of the proposed class of bandwidths makes it suitable for the family approach to density estimation.