Maxisets for Model Selection

Abstract

Abstract. We address the statistical issue of determining the maximal spaces (maxisets) where model selection procedures attain a given rate of convergence. By considering first general dictionaries, then orthonormal bases, we characterize these maxisets in terms of approximation spaces. These results are illustrated by classical choices of wavelet model collections. For each of them, the maxisets are described in terms of functional spaces. We take a special care of the issue of calculability and measure the induced loss of performance in terms of maxisets.

Combining thresholding rules: a new way to improve the performance of wavelet estimators

“…This approach consists of determining the maxiset of a thresholding procedure that is the maximal functional space for which the quadratic risk of the procedure reaches a given rate of convergence. As previously discussed in Cohen, De Vore, Kerkyacharian, and Picard (2001b), Picard (2000, 2002), Autin (2004Autin ( , 2008a, Autin, Le Pennec, Loubes, and Rivoirard (2010), Autin, Freyermuth, and von Sachs (2011a) and Autin, Freyermuth, and von Sachs (2011b), this approach can be successful at differentiating between minimax-equivalent procedures whenever their maxisets are nested. Without such embeddings, the comparison would be impossible.…”