Adaptive estimation on anisotropic Hölder spaces Part I. Fully adaptive case

Abstract

In this paper, we consider the following problem: We want to estimate a noisy signal. The main problem is to find a "good" estimator. To do that, we propose a new criterion to chose, among all possible estimators, the best one. This criterion is useful to define the best family of rates of convergence for any adaptive problem (in a minimax sense). Then, we construct an adaptive estimator (over a family of anisotropic Hölder spaces) for the pointwise loss and we prove its optimality in our sense. This estimator looks like the Lepski's procedure