Block-Threshold-Adapted Estimators via a maxiset approach

We study the performance of a large collection of block thresholding wavelet estimators, namely the Horizontal Block Thresholding family. In particular, we adopt a maxiset point of view, i.e. we are asking for the maximal functional space for a given estimator to converge in the L2−sense with a chosen rate of convergence. We provide sufficient conditions on the choices of rates and threshold values to ensure large maxisets. By deriving maxiset embeddings, we identify the best estimator of such a family, that is the one associated with the largest maxiset. As a particularity of this paper we propose a refined maxiset approach that models method-dependent threshold values. By a series of simulation studies, we confirm the good performance of the best estimator when comparing to the other members of its family.