WAVELET SHRINKAGE - ASYMPTOPIA

Much recent effort has sought asymptotically minimax methods for recov ering infinite dimensional objects-curves, densities, spectral densiti es, images-from noisy data. A now rich and complex body of work develo ps nearly or exactly minimax estimators for an array of interesting pr oblems. Unfortunately, the results have rarely moved into practice, fo r a variety of reasons-among them being similarity to known methods, c omputational intractability and lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data: translate the em pirical wavelet coefficients towards the origin by an amount root(2 lo g n)sigma/root n. The proposal differs from those in current use, is c omputationally practical and is spatially adaptive; it thus avoids sev eral of the previous objections. Further, the method is nearly minimax both for a wide variety of loss functions-pointwise error, global err or measured in L(p)-norms, pointwise and global error in estimation of derivatives-and for a wide range of smoothness classes, including sta ndard Holder and Sobolev classes, and bounded variation. This is a muc h broader near optimality than anything previously proposed: we draw l oose parallels with near optimality in robustness and also with the br oad near eigenfunction properties of wavelets themselves. Finally, the theory underlying the method is interesting, as it exploits a corresp ondence between statistical questions and questions of optimal recover y and information-based complexity.