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Unformatted text preview: J. Ital. Statist. Soc. (1997) 2. pp. 97-/30 WAVELETS IN STATISTICS: A REVIEW ** A. Antoniadis* University Joseph Fourier Abstract The field of nonparametric function estimation has broadened its appeal in recent years with an array of new tools for statistical analysis. In particular, theoretical and applied research on the field of wavelets has had noticeable influence on statistical topics such as nonparametric regression, nonparametric density estimation, nonparametric discrimina- tion and many other related topics. This is a survey article that attempts to synthetize a broad variety of work on wavelets in statistics and includes some recent developments in nonparametric curve estimation that have been omitted from review articles and books on the subject. After a short introduction to wavelet theory, wavelets are treated in the famil- iar context of estimation of «smooth» functions. Both «linear» and «nonlinearx wavelet estimation methods are discussed and cross-validation methods for choosing the smooth- ing parameters are addressed. Finally, some areas of related research are mentioned, such as hypothesis testing, model selection, hazard rate estimation for censored data, and non- parametric change-point problems. The closing section formulates some promising re- search directions relating to wavelets in statistics. Keywords and phrares: Wavelets, multiresolution analysis, nonparametric curve estima- tion, density estimation, regression, model selection, orthogonal series, thresholding, cross- validation, shrinkage, denoising. 1. Introduction Wavelet theory has provided statisticians with powerful new techniques for non- parametric inference by combining recent advances in approximation theory with insight gained from applied signal analysis. When faced with the problem of recovering a 'piecewise' smooth function when only noise measurements are available, wavelet smoothing methods provide a natural and flexible approach to * Address for correspondence: Laboratoire IMAG-LMC, University Joseph Fourier, BP 53,38041 Grenoble Cedex 9, France. ** Research supported by the IDOPT project (CNRS-INRIA-UJF-INPG). This paper, the following discussions and the reply by A. Antoniadis are the result of a seminar hold in Rome, in September 1997. The editorial board wants to thank ISTAT for its financial support. 97 A. ANTONIADlS the estimation of the true function due their outstanding ability and efficiency to respond to local variations without allowing pathological behavior. This article surveys recent developments and applications of wavelets in non- parametric curve estimation, as well as topics that were omitted from previous review articles and books. Both «linear» and «nonlinear» wavelet estimation meth- ods are presented and the relative advantages and disadvantages of each method are discussed. Our exposition assumes no prior knowledge of the theory of wave- lets, and we briefly develop all the necessary tools, under minimal conditions. Aslets, and we briefly develop all the necessary tools, under minimal conditions....
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