Abstract
The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in wavelet applications to statistics and to the analysis of experimental data, with many successes in the efficient analysis, processing, and compression of noisy signals and images. This is a selective review article that attempts to synthesize some recent work on ``nonlinear'' wavelet methods in nonparametric curve estimation and their role on a variety of applications. After a short introduction to wavelet theory, we discuss in detail several wavelet shrinkage and wavelet thresholding estimators, scattered in the literature and developed, under more or less standard settings, for density estimation from i.i.d. observations or to denoise data modeled as observations of a signal with additive noise. Most of these methods are fitted into the general concept of regularization with appropriately chosen penalty functions. A narrow range of applications in major areas of statistics is also discussed such as partial linear regression models and functional index models. The usefulness of all these methods are illustrated by means of simulations and practical examples.
Highlights
Nonparametric regression has been a fundamental tool in data analysis over the past two decades and is still an expanding area of ongoing research
After a short introduction to wavelet theory, we discuss in detail several wavelet shrinkage and wavelet thresholding estimators, scattered in the literature and developed, under more or less standard settings, for density estimation from i.i.d. observations or to denoise data modeled as observations of a signal with additive noise
With the increased applicability of these estimators in nonparametric regression, several new wavelet based curve smoothing procedures have been proposed in the recent literature, and one of the purposes of this review is to present few of them under the general concept of penalized least squares regression
Summary
Nonparametric regression has been a fundamental tool in data analysis over the past two decades and is still an expanding area of ongoing research. During the 1990s, the nonparametric regression literature was arguably dominated by (nonlinear) wavelet shrinkage and wavelet thresholding estimators These estimators are a new subset of an old class of nonparametric regression estimators, namely orthogonal series methods. Donoho and Johnstone (1994) and Donoho et al (1995) have introduced nonlinear wavelet estimators in nonparametric regression through thresholding which typically amounts to term-by-term assessment of estimates of coefficients in the empirical wavelet expansion of the unknown function. With the increased applicability of these estimators in nonparametric regression, several new wavelet based curve smoothing procedures have been proposed in the recent literature, and one of the purposes of this review is to present few of them under the general concept of penalized least squares regression. The practical performance of the methods that are discussed is examined by appropriate simulations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
