Universal Near Minimaxity of

We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coefficients towards the origin by an amount v210g(n) . aj.,fii. The method is nearly minimax for a wide variety of loss functions----e.g. pointwise error, global error measured in LP norms, pointwise and global error in estimation of derivatives-and for a wide range of smoothness classes, including standard HOlder 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 ques­ tions 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).