Wavelet regression: a penalty function approach

Wavelet thresholding as introduced by Donoho and Johnstone (1994) provides a simple technique which involves thresholding the output of the Discrete Wavelet Transform (DWT) of the data of interest, as a way of removing noise from the signal. They show that their estimators are asymptotically minimax for a wide range of norms and spaces. This convergence requires strong regularization or thresholding of the coe cients from the DWT. In reality a function may have a small number of discontinuities, but is unlikely to have the worst case behaviour that a minimax estimator counters. Thus, the resulting wavelet estimators for nite sample sizes, tend to have a large number of in ection points characterized by \noise spikes", despite the strong thresholding. The choice of thresholding has been investigated bymany authors, including Nason (1996), Abramovich and Benjamini (1995) and Abramovich et al (1998), using a variety of objectives, none of which directly address this problem. We introduce a new method of wavelet regression which penalizes lack of smoothness in the reconstruction; this simple method is shown to work well in practice, by maintaining sharp changes in the function and suppressing \noise spikes".