Adapting to Unknown Smoothness via Wavelet Shrinkage

We attempt to recover a function of unknown smoothness from noisy, sampled data. We introduce a procedure, SureShrink, which suppresses noise by thresholding the empirical wavelet coe cients. The thresholding is adaptive: a threshold level is assigned to each dyadic resolution level by the principle of minimizing the Stein Unbiased Estimate of Risk (Sure) for threshold estimates. The computational e ort of the overall procedure is order N log(N) as a function of the sample size N . SureShrink is smoothness-adaptive: if the unknown function contains jumps, the reconstruction (essentially) does also; if the unknown function has a smooth piece, the reconstruction is (essentially) as smooth as the mother wavelet will allow. The procedure is in a sense optimally smoothness-adaptive: it is near-minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet. We know from a previous paper by the authors that traditional smoothing methods { kernels, splines, and orthogonal series estimates { even with optimal choices of the smoothing parameter, would be unable to perform in a near-minimax way over many spaces in the Besov scale. Examples of SureShrink are given: the advantages of the method are particularly evident when the underlying function has jump discontinuities on a smooth background.