  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.
