Results 1  10
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34
Regularization of Wavelets Approximations
, 1999
"... this paper, weintroduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hardthresholding and s ..."
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Cited by 85 (7 self)
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this paper, weintroduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hardthresholding and softthresholding estimators of Donoho and Johnstone (1994) are specic members of nonlinear regularized wavelet estimators. They correspond to the lower and upper bound of a class of the penalized leastsquares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established, and are shown to be adaptively minimax. To eciently solve penalized leastsquares problems, Nonlinear Regularized Sobolev Interpolators (NRSI) are proposed as initial estimators, which are shown to have good sampling properties. The NRSI is further ameliorated by Regularized OneStep Estimators (ROSE), which are the onestep estimators of the penalized leastsquares problems using the NRSI as initial estimators. Two other approaches, the graduated nonconvexity algorithm and wavelet networks, are also introduced to handle penalized leastsquares problems. The newly introduced approaches are also illustrated by a few numerical examples. ####### ########## ## ########## ########### ## ############# ## ####### ######################### ##### ######## ##### ## ####### ######## ### ## ########## ########## ## ########### ########## ## ########### ### ######## ## ########## ### ### ####### ########## ## #### ##### ##### ########### ######### ######### ## ###...
Estimation Of A Function With Discontinuities Via Local Polynomial Fit With An Adaptive Window Choice
, 1996
"... . We propose a method of adaptive estimation of a regression function and which is near optimal in the classical sense of the mean integrated error. At the same time, the estimator is shown to be very sensitive to discontinuities or changepoints of the underlying function f or its derivatives. For ..."
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Cited by 27 (2 self)
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. We propose a method of adaptive estimation of a regression function and which is near optimal in the classical sense of the mean integrated error. At the same time, the estimator is shown to be very sensitive to discontinuities or changepoints of the underlying function f or its derivatives. For instance, in the case of a jump of a regression function, beyond the interval of length (in order) n \Gamma1 log n around changepoints the quality of estimation is essentially the same as if locations of jumps were known. The method is fully adaptive and no assumptions are imposed on the design, number and size of jumps. The results are formulated in a nonasymptotic way and can be therefore applied for an arbitrary sample size. 1. Introduction The changepoint analysis which includes sudden, localized changes typically occurring in economics, medicine and the physical sciences has recently found increasing interest, see Muller (1992) for some examples and discussion of the problem. Let...
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...
WaveletBased Nonparametric Bayes Methods
, 1998
"... In this chapter, we will provide an overview of the current status of research involving Bayesian inference in wavelet nonparametric problems. In many statistical applications, there is a need for procedures to (i) adapt to data and (ii) use prior information. The interface of wavelets and the Bayes ..."
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Cited by 17 (2 self)
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In this chapter, we will provide an overview of the current status of research involving Bayesian inference in wavelet nonparametric problems. In many statistical applications, there is a need for procedures to (i) adapt to data and (ii) use prior information. The interface of wavelets and the Bayesian paradigm provide a natural terrain for both of these goals.
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 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.
Nonlinear Wavelet Estimation of TimeVarying Autoregressive Processes
 Bernoulli
, 1998
"... . We consider nonparametric estimation of the parameter functions a i (\Delta) , i = 1; : : : ; p , of a timevarying autoregressive process. Choosing an orthonormal wavelet basis representation of the functions a i , the empirical wavelet coefficients are derived from the time series data as the s ..."
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Cited by 14 (3 self)
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. We consider nonparametric estimation of the parameter functions a i (\Delta) , i = 1; : : : ; p , of a timevarying autoregressive process. Choosing an orthonormal wavelet basis representation of the functions a i , the empirical wavelet coefficients are derived from the time series data as the solution of a least squares minimization problem. In order to allow the a i to be functions of inhomogeneous regularity, we apply nonlinear thresholding to the empirical coefficients and obtain locally smoothed estimates of the a i . We show that the resulting estimators attain the usual minimax L 2 rates up to a logarithmic factor, simultaneously in a large scale of Besov classes. The finitesample behaviour of our procedure is demonstrated by application to two typical simulated examples. 1991 Mathematics Subject Classification. Primary 62M10; secondary 62F10 Key words and phrases. Nonstationary processes, time series, wavelet estimators, timevarying autoregression, nonlinear thresholdi...
Bayesian Inference with Wavelets: Density Estimation
 Duke University. Submitted
, 1995
"... We propose a prior probability model in the wavelet coefficient space. The proposed model implements wavelet coefficient thresholding by full posterior inference in a coherent probability model. We introduce a prior probability model with mixture priors for the wavelet coefficients. The prior includ ..."
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Cited by 13 (3 self)
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We propose a prior probability model in the wavelet coefficient space. The proposed model implements wavelet coefficient thresholding by full posterior inference in a coherent probability model. We introduce a prior probability model with mixture priors for the wavelet coefficients. The prior includes a positive prior probability mass at zero which leads to a posteriori thresholding and generally to a posteriori shrinkage on the coefficients. We discuss an efficient posterior simulation scheme to implement inference in the proposed model. The discussion is focused on the density estimation problem. However, the introduced prior probability model on the wavelet coefficient space and the Markov chain Monte Carlo scheme are general. Keywords: Mixture priors; Model choice; Posterior simulation; Wavelet decomposition. 1 1 Introduction Wavelet bases provide a basis for the space of L 2 functions. This basis is conceptually appealing because of the correspondence to nested approximation...
Choice of Wavelet Smoothness, Primary Resolution and Threshold in Wavelet Shrinkage
 Statistics and Computing
, 2001
"... This article introduces a fast crossvalidation algorithm that performs wavelet shrinkage on data sets of arbitrary size and irregular design and also simultaneously selects good values of the primary resolution and number of vanishing moments. We demonstrate the utility of our method by suggesting ..."
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Cited by 13 (4 self)
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This article introduces a fast crossvalidation algorithm that performs wavelet shrinkage on data sets of arbitrary size and irregular design and also simultaneously selects good values of the primary resolution and number of vanishing moments. We demonstrate the utility of our method by suggesting alternative estimates of the conditional mean of the wellknown Ethanol data set. Our alternative estimates outperform the KovacSilverman method with a global variance estimate by 25% because of the careful selection of number of vanishing moments and primary resolution. Our alternative estimates are simpler than, and competitive with, results based on the KovacSilverman algorithm equipped with a local variance estimate. We include a detailed simulation study that illustrates how our crossvalidation method successfully picks good values of the primary resolution and number of vanishing moments for unknown functions based on Walsh functions (to test the response to changing pri...
On Choosing A NonInteger Resolution Level When Using Wavelet Methods
, 1997
"... . In curve estimation using wavelet methods it is common to select the resolution level to be an integer, so as to exploit the computational advantages of the pyramid or cascade algorithm. This choice, however, can produce a noticeable amount of either oversmoothing or undersmoothing. Its analogue f ..."
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Cited by 13 (3 self)
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. In curve estimation using wavelet methods it is common to select the resolution level to be an integer, so as to exploit the computational advantages of the pyramid or cascade algorithm. This choice, however, can produce a noticeable amount of either oversmoothing or undersmoothing. Its analogue for estimation by kernel methods is to restrict the bandwidth to be an integer power of 1 2 , which would seldom be acceptable. In this note we quantify the advantages of noninteger resolution levels. KEYWORDS. Bandwidth, curve estimation, density estimation, dyadic expansion, mean squared error, kernel estimator, nonparametric regression. SHORT TITLE. Smoothing wavelet estimators. AMS (1991) SUBJECT CLASSIFICATION. Primary 62G07, Secondary 62G30. 1 Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia. 2 Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. 1. INTRODUCTION Wavelet methods offer excellent adaptivit...