Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series A, Pt. 3, pp. 367--393

MULTIWAVELETS AND SIGNAL DENOISING^{*}

By

SAM EFROMOVICH, *University of New Mexico, Albuquerque*

*SUMMARY. *Despite their better approximation and data compression properties,
multiwavelets are less known to practitioners than uniwavelets and
they are rarely used in statistical applications.
The reason is that calculation of
multiwavelet empirical
coefficients is essentially more complicated than the familiar
orthogonal uniwavelet discrete transform, in particular
a special prefiltering of observations is required.
As a result, even for the case of a signal contaminated by white
Gaussian noise,
empirical multiwavelet coefficients
are contaminated by a nonstationary correlated noise.
On the top of this, different types of prefilters and multiwavelets
lead to different noise distributions.
Thus to make multiwavelets convenient for statistical applications,
denoising should be robust and self--learning.
Also, statistical applications should be found where multiwavelets
justify the additional efforts involved in their use.
In this article a new adaptive procedure of denoising
and recovering derivatives
based on a vaguelette--vaguelette approach is developed.
Asymptotic optimality of the estimator is established, and
results of numerical
simulations are presented that justify the use of the multiwavelet estimator.

*AMS (1991) subject classification. *Primary 62G05; secondary 62G07.

*Key words and phrases. *Adaptation, asymptotics,
colored noise, cristina biwavelets, derivative estimation,
ill-posed problem, learning,
oracle inequality, regression, robustness, small datasets,
vaguelette.