SMOOTHING NOISY DATA WITH COIFLETS

This paper is concerned with an orthogonal wavelet series regression estimator of an unknown smooth regression function observed with noise on a bounded interval. The method is based on applying results of the recently developed theory of wavelets and uses the specific asymptotic interpolating properties of the wavelet approximation generated by a particular wavelet basis, Daubechie's coiflets. Conditions are given for the estimator to attain optimal convergence rates in the integrated mean square sense as the sample size increases to infinity. Moreover, the estimator is shown to be pointwise consistent and asymptotically normal. The numerical implementation of the estimation procedure relies on the discrete wavelet transform; and the algorithm for smoothing a noisy sample of size n requires order O(n) operations. The general theory is illustrated with simulated and real examples and a comparison with other nonparametric smoothers is made.