A model-based approach to density estimation in sup-norm

Building on the l−estimators of Baraud, we define a general method for finding a quasi-best approximant in sup-norm to a target density p⋆ belonging to a given model m, based on independent samples drawn from distributions p⋆i which average to p⋆ (which does not necessarily belong to m). We also provide a general method for selecting among a countable family of such models. Both of these esti- mators satisfy oracle inequalities in the general setting. The quality of the bounds depends on the volume of sets C on which |f| is close to its maximum, where f = p − q for some p, q ∈ m (or p ∈ m and q ∈ m′, in the case of model selection). In particular, using piecewise polynomials on dyadic partitions of Rd, we recover optimal rates of convergence for classes of functions with anisotropic smoothness, with optimal depen- dence on semi-norms measuring the smoothness of p⋆ in the coordinate directions. Moreover, our method adapts to the anisotropic smoothness, as long as it is smaller than 1 plus the degree of the polynomials.