Universal Near Minimaxity of Wavelet Shrinkage

We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coe cients towards the origin by an amount p 2 log(n) = p n. The method is nearly minimax for a wide variety of loss functions { e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives { and for a wide range of smoothness classes, including standard H older classes, Sobolev classes, and Bounded Variation. This is a broader near-optimality than anything previously proposed in the minimax literature. The theory underlying the method exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity. This paper contains a detailed proof of the result announced in Donoho, Johnstone, Kerkyacharian & Picard (1995).