Maximal Spaces with Given Rate of Convergence for Thresholding Algorithms

In recent years, various nonlinear methods have been proposed and deeply investigated in the context of nonparametric estimation: shrinkage methods [21], locally adaptive bandwidth selection [16] and wavelet thresholding [7]. One way of comparing the performances of two different method is to fix a class of functions to be estimated and to measure the estimation rate achieved by each method over this class. In this context, most of these methods have been proved to achieve minimax rate for a given loss function, over various classes modelled by the unit balls of function spaces: Hölder, Sobolev and more generally Besov and Triebel-Lizorkin spaces. It should be noted that the choice of such a class is quite subjective. Moreover it happens very often that the minimax properties can be extended (without deteriorating the rate of convergence) to larger spaces (see e.g. [11]). It is thus natural to address the following question: given an estimation method and a prescribed estimation rate for a given loss function, what is the maximal space, over which this rate is achieved ? If it exists, such a space will appear as naturally linked with the method under consideration. The goal of this paper is to discuss the existence and the nature of maximal spaces in the context of nonlinear methods based on thresholding (or shrinkage) procedures. Before going further, some remarks should be made: