Maxiset for Density Estimation

The problem of density estimation on R is concerned. Adopting the maxiset point of view, the aim of this paper is threefold. Firstly, we prove that the maxiset of any elitist rule is contained in the intersection of a Besov space and a weak Besov space. Secondly, we provide an adaptive procedure for which the maxiset is the largest one among elitist rules (ideal maxiset). Thirdly, we point out the significance of data-driven thresholds in estimation by comparing the maxisets of this last procedure with the procedure using data-driven thresholds proposed by Juditsky and Lambert-Lacroix.