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16
Wavelet estimators in nonparametric regression: a comparative simulation study
 Journal of Statistical Software
, 2001
"... OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. ..."
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Cited by 72 (9 self)
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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.
Wavelet Analysis and Its Statistical Applications
, 1999
"... In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this ..."
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Cited by 44 (9 self)
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In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this article is intended to give a relatively accessible introduction to standard wavelet analysis and to provide an up to date review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be e ective, rather than to researchers already familiar with the eld. Given that objective, we do not emphasise mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in...
Exact Risk Analysis of Wavelet Regression
, 1995
"... Wavelets have motivated development of a host of new ideas in nonparametric regression smoothing. Here we apply the tool of exact risk analysis, to understand the small sample behavior of wavelet estimators, and thus to check directly the conclusions suggested by asymptotics. Comparisons between som ..."
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Cited by 25 (2 self)
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Wavelets have motivated development of a host of new ideas in nonparametric regression smoothing. Here we apply the tool of exact risk analysis, to understand the small sample behavior of wavelet estimators, and thus to check directly the conclusions suggested by asymptotics. Comparisons between some wavelet bases, and also between hard and soft thresholding are given from several viewpoints. Our results provide insight as to why the viewpoints and conclusions of Donoho and Johnstone differ from those of Hall and Patil. 1 Introduction In a series of papers, Donoho and Johnstone (1992 [9],1994a [10], 1995 [13]) and Donoho, Johnstone, Kerkyacharian and Picard (1995) [14] developed nonlinear wavelet shrinkage technology in nonparametric regression. For other work relating wavelets and nonparametric estimation, see Doukhan (1988) [15], Kerkyacharian and Picard, (1992) [21], Antoniadis (1994) [1] and Antoniadis, Gregoire and McKeague (1994) [2]. These papers have both introduced a new clas...
Some Uses of Cumulants in Wavelet Analysis
 J. Nonparametric Statistics
, 1996
"... Cumulants are useful in studying nonlinear phenomena and in developing (approximate) statistical properties of quantities computed from random process data. Wavelet analysis is a powerful tool for the approximation and estimation of curves and surfaces. This work considers both wavelets and cumulant ..."
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Cited by 19 (2 self)
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Cumulants are useful in studying nonlinear phenomena and in developing (approximate) statistical properties of quantities computed from random process data. Wavelet analysis is a powerful tool for the approximation and estimation of curves and surfaces. This work considers both wavelets and cumulants, developing some sampling properties of linear wavelet fits to a signal in the presence of additive stationary noise via the calculus of cumulants. Of some concern is the construction of approximate confidence bounds around a fit. Some extensions to spatial processes, irregularly observed processes and long memory processes are indicated.
Estimating The Square Root Of A Density Via Compactly Supported Wavelets
, 1997
"... This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by ..."
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Cited by 19 (6 self)
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This paper addresses the problem of univariate density estimation in a novel way. Our approach falls in the class of so called projection estimators, introduced by
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 ..."
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Cited by 17 (1 self)
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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 Jamesâ€“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.
Universal Near Minimaxity of Wavelet Shrinkage
, 1995
"... We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coefficients towards the origin by an amount p 2 log(n) \Delta oe= p n. The method is nearly minimax for a wide variety of loss functions  e.g. pointwise error, global error measured in L p ..."
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Cited by 13 (3 self)
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We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coefficients towards the origin by an amount p 2 log(n) \Delta oe= p n. The method is nearly minimax for a wide variety of loss functions  e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives  and for a wide range of smoothness classes, including standard Holder classes, Sobolev classes, and Bounded Variation. This is a broader nearoptimality than anything previously proposed in the minimax literature. The theory underlying the method exploits a correspondence between statistical questions and questions of optimal recovery and informationbased complexity. This paper contains a detailed proof of the result announced in Donoho, Johnstone, Kerkyacharian & Picard (1995).
Nonparametric Density Estimation using Wavelets
, 1995
"... Here the problem of density estimation using wavelets is considered. Nonparametric wavelet density estimators have recently been proposed and seem to outperform classical estimators in representing discontinuities and local oscillations. The purpose of this paper is to give a review of di#erent ..."
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Cited by 9 (1 self)
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Here the problem of density estimation using wavelets is considered. Nonparametric wavelet density estimators have recently been proposed and seem to outperform classical estimators in representing discontinuities and local oscillations. The purpose of this paper is to give a review of di#erent types of wavelet density estimators proposed in the literature. Properties, comparisons with classical estimators and applications are stressed. Multivariate extensions are considered. Performances of wavelet estimators are analyzed using a family of normal mixture densities and the Old Faithful Geyser dataset. Key words and phrases: Nonparametric Density Estimation, Wavelets. AMS Subject Classification: 62G07, 42A06. 1 Introduction In nonparametric theory, density estimation is perhaps one of the most investigated topics. Let X 1 , , X n be a sample of size n from an unknown probability density function f . The purpose is to estimate f without any assumption on its form. In this pa...
Deterministic/Stochastic Wavelet Decomposition for Recovery of Signal from Noisy Data
 Technometrics
, 1998
"... In a series of recent papers on nonparametric regression, D. Donoho and I. Johnstone developed wavelet shrinkage methods for recovering unknown piecewisesmooth deterministic signals from noisy data. Wavelet shrinkage based on the Bayesian approach involves specifying a prior distribution on the wav ..."
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Cited by 9 (0 self)
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In a series of recent papers on nonparametric regression, D. Donoho and I. Johnstone developed wavelet shrinkage methods for recovering unknown piecewisesmooth deterministic signals from noisy data. Wavelet shrinkage based on the Bayesian approach involves specifying a prior distribution on the wavelet coefficients, which is usually assumed to have a distribution with zero mean. There is no a priori reason why all prior means should be zero; indeed, one can imagine certain types of signals where this is not a good choice of model. In this paper, we take an empirical Bayes approach where we propose an estimator for the prior mean that is "plugged into" the Bayesian shrinkage formulas. Another way we are more general than previous work is that we assume the underlying signal is composed of a piecewisesmooth deterministic part plus a zeromean stochastic part; that is, the signal may contain a reasonably large number of nonzero wavelet coefficients. Our goal is to predict this signal fr...
PREVENTING THE DIRAC DISASTER: Wavelet Based Density Estimation
 J. Ital. Statist. Soc
, 1998
"... This paper addresses the problem of choosing the optimal number of basis functions in constructing wavelet series density estimators. It is well known that projection estimators tend to overfit the density if the number of basis functions in the orthogonal expansion is too large. In extreme cases t ..."
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Cited by 8 (3 self)
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This paper addresses the problem of choosing the optimal number of basis functions in constructing wavelet series density estimators. It is well known that projection estimators tend to overfit the density if the number of basis functions in the orthogonal expansion is too large. In extreme cases the estimator is close to the Dirac function concentrated at the observations. We propose a roughness measure of wavelet estimators and establish a data driven method for determining the number of levels to be included in the estimate. Our method exploits the idea of Good and Gaskins (1971) who used the Fisher information functional as a roughness penalty measure. The method is demonstrated on simulated data. Key words and phrases: Fisher Information, Wavelets, Density Estimation. AMS Subject Classification: 62G07, 42A06. Abbreviated title: Linear Wavelet Density Estimators. 1 Introduction It is a well known fact that taking too many basis functions in the orthogonal density estimate will...