Smoothing Noisy Data with Tapered Coiflets Series
Anestis Antoniadis · OpenGrey (Institut de l'Information Scientifique et Technique) · 1996
This paper is concerned with an orthogonal wavelet series estimator of an unknown smooth regression function observed with noise on a bounded interval. A penalized least-squares approach is adopted and our method uses the specific asymptotic interpolating properties of the wavelet approximation generated by a particular wavelet basis, Daubechie's coiflets. A simple procedure is described to estimate the smoothing parameter of the penalizing functional and conditions are given for the estimator to attain optimal convergence rates in the integrated mean square sense as the sample size increases to infinity. The results are illustrated with simulated and real examples and a comparison with other non-parametric smoothers is made. where f(v) indicates the vth derivative off The asymptotic properties of smoothing splines have been studied by a number of authors, including Wahba (1975, 1990), Rice & Rosenblatt (1981), Rice (1984), Speckman (1985), Nussbaum (1985) and Cox (1988), to cite only a few. The continuous analogue of spline smoothing is called Tikhonov regularization. The estimates of g to be considered in this paper are obtained by such a method of regularization and are based on wavelet decompositions on the interval. Our purpose is to show that wavelets can be used within the framework of regularization methods, thus augmenting rather than supplanting the many popular univariate curving methods. In regularization, a target function g is approximated by a smoother function f To be precise, we must define what we mean by approximate and smoother. To this end, as in DeVore & Lucier (1992), we introduce the error function space g with norm || || and the