ADAPTING TO UNKNOWN SMOOTHNESS VIA WAVELET SHRINKAGE

We attempt to recover a function of unknown smoothness from noisy samp led data. We introduce a procedure, SureShrink, that suppresses noise by thresholding the empirical wavelet coefficients. The thresholding i s adaptive: A threshold level is assigned to each dyadic resolution le vel by the principle of minimizing the Stein unbiased estimate of risk (Sure) for threshold estimates. The computational effort of the overa ll procedure is order N.log(N) as a function of the sample size N. Sur eShrink is smoothness adaptive: If the unknown function contains jumps , then the reconstruction (essentially) does also; if the unknown func tion has a smooth piece, then the reconstruction is (essentially) as s mooth as the mother wavelet will allow. The procedure is in a sense op timally smoothness adaptive: It is near minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends o n the choice of mother wavelet. We know from a previous paper by the a uthors that traditional smoothing methods-kernels, splines, and orthog onal series estimates-even with optimal choices of the smoothing param eter, would be unable to perform in a near-minimax way over many space s in the Besov scale. Examples of SureShrink are given. The advantages of the method are particularly evident when the underlying function h as jump discontinuities on a smooth background.