Wavelets, Approximation, and Statistical Applications
Lecture notes in statisticsPublished 1 January 1998
Wolfgang Karl Härdle, Gérard Kerkyacharian, Dominique Picard, A. B. Tsybakov
Citations758
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
Abstract
The mathematical theory of ondelettes (wavelets) was developed by Yves Meyer and many collaborators about 10 years ago. It was designed for ap proximation of possibly irregular functions and surfaces
Keywords
Computer ScienceMathematicsEngineering
Journal of Political EconomyThe Pricing of Options and Corporate Liabilities
29,429 Citations1973Fischer Black, Myron S. Scholes
IEEE Transactions on Pattern Analysis and Machine IntelligenceA theory for multiresolution signal decomposition: the wavelet representation
21,081 Citations1989Stéphane Mallat
It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/Sup j/ can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.
The Journal of FinanceCAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK*
17,396 Citations1964William F. Sharpe
Density Estimation for Statistics and Data Analysis
15,768 Citations2018B.W. Silverman
Computers in PhysicsTen Lectures on Wavelets
15,386 Citations1992Ingrid Daubechies, Christopher Heil
This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
Communications on Pure and Applied MathematicsOrthonormal bases of compactly supported wavelets
8,187 Citations1988Ingrid Daubechies
This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.
BiometrikaIdeal spatial adaptation by wavelet shrinkage
7,813 Citations1994David L. Donoho, Iain M. Johnstone
A new principle for spatially-adaptive estimation: selective wavelet reconstruction with an oracle inequality is described and a practical spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coefficients is developed.
Introduction to Fourier Analysis on Euclidean Spaces.
5,325 Citations1971Elias M. Stein, Guido Weiss
Journal of the American Statistical AssociationAdapting to Unknown Smoothness via Wavelet Shrinkage
4,346 Citations1995David L. Donoho, Iain M. Johnstone
Interpolation Spaces: An Introduction
3,546 Citations2011Jöran Bergh, Jörgen Löfström
Springer series in statisticsExtremes and Related Properties of Random Sequences and Processes
2,831 Citations1983M. R. Leadbetter, Georg Lindgren +1 more
IEEE Signal Processing MagazineWavelets and signal processing
2,760 Citations1991Olivier Rioul, Martin Vetterli
A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes, which includes nonstationary signal analysis, scale versus frequency,Wavelet analysis and synthesis, scalograms, wavelet frames and orthonormal bases, the discrete-time case, and applications of wavelets in signal processing.
The Annals of StatisticsEstimation of the Mean of a Multivariate Normal Distribution
2,719 Citations1981Charles Stein
Choice Reviews OnlineApplied nonparametric regression
2,613 Citations1991
Applied Nonparametric Regression is the first book to bring together in one place the techniques for regression curve smoothing involving more than one variable and argues that all smoothing methods are based on a local averaging mechanism and can be seen as essentially equivalent to kernel smoothing.
The Journal of FinanceSECURITY PRICES, RISK, AND MAXIMAL GAINS FROM DIVERSIFICATION*
2,130 Citations1965John Lintner
Lecture notes in statisticsTranslation-Invariant De-Noising
1,920 Citations1995Ronald R. Coifman, David L. Donoho
A reconstruction subject to far weaker Gibbs phenomena than thresholding based De-Noising using the traditional orthogonal wavelet transform is produced.
Communications on Pure and Applied MathematicsFast wavelet transforms and numerical algorithms I
1,777 Citations1991Gregory Beylkin, Ronald R. Coifman +1 more
The algorithms presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators, and indicate that many previously intractable problems become manageable with the techniques presented here.
Journal of the Royal Statistical Society Series B (Statistical Methodology)Wavelet Shrinkage: Asymptopia?
1,758 Citations1995David L. Donoho, Iain M. Johnstone +2 more
A method for curve estimation based on n noisy data: translate the empirical wavelet coefficients towards the origin by an amount √(2 log n) /√n and draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves.
American Mathematical MonthlyAn Introduction to Harmonic Analysis.
1,691 Citations1970Guido Weiss, Yitzhak Katznelson
The Annals of StatisticsOptimal Global Rates of Convergence for Nonparametric Regression
1,485 Citations1982Charles J. Stone
Approximation of Functions of Several Variables and Imbedding Theorems
1,202 Citations1975Sergei Mihailovic Nikol’skii
Statistical Estimation: Asymptotic Theory
1,195 Citations2014
Applied and Computational Harmonic AnalysisWavelets on the Interval and Fast Wavelet Transforms
1,038 Citations1993Albert Cohen, Ingrid Daubechies +1 more
The Annals of StatisticsMinimax estimation via wavelet shrinkage
1,028 Citations1998David L. Donoho, Iain M. Johnstone
A nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients is developed, andVariants of this method based on simple threshold nonlinear estimators are nearly minimax.
The Annals of StatisticsFitting time series models to nonstationary processes
938 Citations1997Rainer Dahlhaus
Wavelets: A Tutorial in Theory and Applications
922 Citations1992Charles K. Chui
Orthogonal wavelets daubechies' scaling function on (0.3), D.C. Feauveau wavelet-like local bases wavelets and other bases for fast numerical linear algebra, and some elementary properties of multiresolution analysis of L2 (Rn), W.R. Auscher.
The Annals of StatisticsOptimal Rates of Convergence for Nonparametric Estimators
790 Citations1980Charles J. Stone
The Annals of StatisticsDensity estimation by wavelet thresholding
762 Citations1996David L. Donoho, Iain M. Johnstone +2 more
Applied and Computational Harmonic AnalysisNonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition
613 Citations1995David L. Donoho
This work proposes to solve ill-posed linear inverse problems by nonlinearly "shrinking" the WVD coefficients of the noisy, indirect data by achieving faster rates of convergence for objects known to lie in the bump algebra or in bounded variation than any linear procedure.
Israel Journal of MathematicsOn the subspaces ofL p (p>2) spanned by sequences of independent random variables
529 Citations1970Haskell P. Rosenthal
BiometrikaResidual variance and residual pattern in nonlinear regression
469 Citations1986Théo Gasser, Lothar Sroka +1 more
Choice Reviews OnlineEssential wavelets for statistical applications and data analysis
446 Citations1997
Physics Today<i>Wavelets: An Analysis Tool</i>
446 Citations1996M. Holschneider, Mark Kon
Introduction to wavelet analysis over IR Discretizing and periodizing the half-plane Multi-resolution analysis Fractal analysis and wavelet transforms Group theory as unifying language.
The Annals of StatisticsAsymptotic equivalence of nonparametric regression and white noise
375 Citations1996Lawrence D. Brown, Mark G. Low
Lecture notes in statisticsMinimax Theory of Image Reconstruction
375 Citations1993A. P. Korostelev, A. B. Tsybakov
This book is concerned with a technique for image reconstruction known as the asymptotic minimax approach, which is based on non-parametric regression and change-point analysis, and presents results under the simplest assumptions.
Journal of the Royal Statistical Society Series B (Statistical Methodology)Wavelet Shrinkage Using Cross-Validation
366 Citations1996Guy P. Nason
A modified form of twofold cross-validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2.
From Model Selection to Adaptive Estimation
335 Citations1997Lucien Birgé, Pascal Massart
Many different model selection information criteria can be found in the literature in various contexts including regression and density estimation to select among a given collection of parametric models that model which minimizes an empirical loss plus some penalty term which is proportional to the dimension of the model.
Journal of the American Statistical AssociationApplied Wavelet Analysis with S-PLUS.
321 Citations1997DSS, A. Gregory Bruce +1 more
This book introduces applied wavelet analysis through the S-PLUS software system using a visual data analysis approach, wavelet concepts are explained in a way that is intuitive and easy to understand.
Choice Reviews OnlineWavelet theory and its applications
298 Citations1993
Wavelet Theory Extentions and Ambiguity Functions, Linear Systems Modelling with Wavelet Theory, and Practical Resolution, Gain, and Processing Structures.
The Annals of StatisticsStatistical Estimation and Optimal Recovery
274 Citations1994David L. Donoho
The method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery.
The Annals of StatisticsOptimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors
272 Citations1997Oleg Lepski, Enno Mammen +1 more
The Annals of StatisticsAsymptotic equivalence of density estimation and Gaussian white noise
253 Citations1996Michael Nussbaum
It is shown that an i.i.d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift f l/2 and variance 1/4n -l .
Probability Theory and Related FieldsMinimax risk overl p -balls forl p -error
221 Citations1994David L. Donoho, Iain M. Johnstone
Theory of Probability and Its ApplicationsAsymptotically Minimax Adaptive Estimation. I: Upper Bounds. Optimally Adaptive Estimates
215 Citations1992O. V. Lepskii
The Annals of StatisticsAdaptive hypothesis testing using wavelets
209 Citations1996Vladimir Spokoiny
Statistics & Probability LettersDensity estimation in Besov spaces
201 Citations1992G. Kerkyacharian, D. Picard
Journal of the American Statistical AssociationWavelet Methods for Curve Estimation
190 Citations1994Anestis Antoniadis, Gérard Gregoire +1 more
Computational Statistics & Data AnalysisAdaptive thresholding of wavelet coefficients
177 Citations1996Felix Abramovich, Yoav Benjamini
Journal of Computational and Graphical StatisticsThe Discrete Wavelet Transform in S
161 Citations1994Guy P. Nason, B. W. Silverman
A software package called wavethresh is introduced that works within the statistical language S to perform one- and two-dimensional discrete wavelet transforms and their inverses and a tutorial introduction to wavelets and the wave Thresh software is provided.
IEEE Transactions on Signal ProcessingWavelet thresholding techniques for power spectrum estimation
160 Citations1994Pierre Moulin
A wavelet thresholding technique is proposed to solve the spectrum estimation problem under specified noise/resolution tradeoffs and it is shown that the wavelet coefficients of the additive noise may be treated as independent random variables.
Wavelets and Multiscale Signal Processing
157 Citations1995Albert Cohen, Robert D. Ryan
The Annals of StatisticsWavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra
149 Citations1997Michael H. Neumann, Rainer von Sachs
A new adaptive wavelet estimator of the time-dependent spectrum of a locally stationary time series is introduced and it is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present.
The Annals of StatisticsFormulae for Mean Integrated Squared Error of Nonlinear Wavelet-Based Density Estimators
145 Citations1995Peter A. Hall, Prakash Patil
The Annals of StatisticsSpline Smoothing in Regression Models and Asymptotic Efficiency in $L_2$
139 Citations1985Michael Nussbaum
Applied and Computational Harmonic AnalysisOn Minimax Wavelet Estimators
124 Citations1996Bernard Delyon, Anatoli Juditsky
ANU Open Research (Australian National University)ON THE MINIMAX OPTIMALITY OF BLOCK THRESHOLDED WAVELET ESTIMATORS
115 Citations1999Peter A. Hall, G. Kerkyacharian +1 more
It is shown that block thresholding also provides extensive adaptivity to many varieties of aberration, including those of chirp and Doppler type, and in particular enjoys those rates without the extraneous logarithmic penalties that are usually suffered by term-by-term thresholding methods.
The Annals of StatisticsFunction estimation via wavelet shrinkage for long-memory data
105 Citations1996Yazhen Wang
A fractional Gaussian noise model is proposed to approximate nonparametric regression with long-range dependence and establish asymptotic results for minimax risks using a wavelet]vaguelette decomposition which decorrelates fractionalGaussian noise.
Theory of Probability and Its ApplicationsNonparametric Estimation of a Density of Unknown Smoothness
104 Citations1986S. Yu. Efroimovich
The Annals of StatisticsPointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes
94 Citations1998A. B. Tsybakov
Statistics & Probability LettersDensity estimation by kernel and wavelets methods: Optimality of Besov spaces
93 Citations1993Gérard Kerkyacharian, Dominique Picard
The Annals of StatisticsInterpolation methods for nonlinear wavelet regression with irregularly spaced design
88 Citations1997Peter Hall, Berwin A. Turlach
Journal of the Royal Statistical Society Series B (Statistical Methodology)On the Choice of Smoothing Parameter, Threshold and Truncation in Nonparametric Regression by Non-Linear Wavelet Methods
88 Citations1996Peter Hall, Prakash Patil
RePEc: Research Papers in EconomicsPractical performance of several data driven bandwidth selectors
87 Citations1992Byeong U. Park, Berwin A. Turlach
This article compares several promising data-driven methods for selecting the bandwidth of a kernel density estimator and believes the plug-in rule is the best of those currently available, but there is still room for improvement.
Journal of the American Statistical AssociationVisual Error Criteria for Qualitative Smoothing
85 Citations1995J. S. Marron, A. B. Tsybakov
Journal of Computational and Graphical StatisticsExact Risk Analysis of Wavelet Regression
79 Citations1998J. S. Marron, Süleyman Adak +3 more
Journal of Time Series AnalysisSPECTRAL DENSITY ESTIMATION VIA NONLINEAR WAVELET METHODS FOR STATIONARY NON‐GAUSSIAN TIME SERIES
77 Citations1996Michael H. Neumann
Computational Statistics & Data AnalysisData dependent wavelet thresholding in nonparametric regression with change-point applications
76 Citations1996Todd Ogden, Emanuel Parzen
HAL (Le Centre pour la Communication Scientifique Directe)Non-parametric estimation of the diffusion coefficient by wavelets methods
74 Citations1992Valentine Genon‐Catalot, Catherine Larédo +1 more
Theory of Probability and Its ApplicationsAsymptotically Minimax Adaptive Estimation. II. Schemes without Optimal Adaptation: Adaptive Estimators
63 Citations1993O. V. Lepskii
TechnometricsXploRe: An Interactive Statistical Computing Environment
62 Citations1997John Hinde, Wolfgang Karl Härdle +3 more
Minimax Bayes, Asymptotic Minimax and Sparse Wavelet Priors
58 Citations1994Iain M. Johnstone
BernoulliOn Wavelet Methods for Estimating Smooth Functions
55 Citations1995Peter A. Hall, Prakash Patil
BernoulliNeo-classical minimax problems, thresholding and adaptive function estimation
53 Citations1996David L. Donoho, Iain M. Johnstone
The scalar minimax results imply: Lepskii's results that it is not possible fully to adapt the unknown degree of smoothness without incurring a performance cost; and that simple thresholding of the empirical wavelet transform gives an estimate of a function at a fixed point which is, to within constants, optimally adaptive to unknown degreeof smoothness.
Applied and Computational Harmonic AnalysisWavelet Smoothing of Evolutionary Spectra by Nonlinear Thresholding
50 Citations1996Rainer von Sachs, Kai Schneider
Lecture notes in statisticsAdaptive Density Estimation
48 Citations1995Gérard Kerkyacharian
This paper presents some results concerning the adaptive estimation of density with wavelet methods using the Donoho-Johnstone-Kerkyacharian-Picard procedure, and investigates their minimax properties.
Universal Near Minimaxity of Wavelet Shrinkage
43 Citations1997David L. Donoho, Iain M. Johnstone +2 more
A method for curve estimation based on n noisy data that translates the empirical wavelet coefficients towards the origin by an amount that is nearly minimax for a wide variety of loss functions and a broader near-optimality than anything previously proposed in the minimax literature.
Statistica NeerlandicaPractical estimation of multivariate densities using wavelet methods
42 Citations1995K. Tribouley
The wavelets method simplifies significantly both the theoretical and the practical computations for estimating multivariate densities using wavelets.
Revista Matemática IberoamericanaProjecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions
34 Citations1994Pierre Gilles Lemarié–Rieusset
Annales de l Institut Henri Poincaré Probabilités et StatistiquesLp adaptive density estimation in a β mixing framework
24 Citations1998K. Tribouley, Gabrielle Viennet
Lecture notes in statisticsForeign Exchange Rates Have Surprising Volatility
24 Citations1996Peter Bossaerts, Wolfgang Karl Härdle +1 more
Érudit documents and data repository (Érudit Consortium, University of Montreal)Trading Patterns, Time Deformation and Stochastic Volatility in Foreign Exchange Markets
24 Citations1995Éric Ghysels, Christian Gouriéroux +1 more
The Annals of StatisticsPerformance of wavelet methods for functions with many discontinuities
20 Citations1996Peter Hall, Ian McKay +1 more
Estimating covariances of locally stationary processes: consistency of best basis methods
20 Citations2002David L. Donoho, Stéphane Mallat +1 more
It is shown that it is possible to obtain an empirical basis from sampled data which is nearly as good as the ideal theoretical basis.
Statistics & Probability LettersA note on the wavelet oracle
10 Citations1999Peter A. Hall, Gérard Kerkyacharian +1 more
…
