A MODEL-BASED APPROACH TO DENSITY ESTIMATION IN SUP-NORM

Guillaume Maillard (1)

1 Uni.lu - Université du Luxembourg

Building on the −estimators of Baraud (2021), 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 estimators 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 R d , we recover optimal rates of convergence for classes of functions with anisotropic smoothness, with optimal dependence 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.

Domaines: Statistiques [math.ST]

Guillaume Maillard. A MODEL-BASED APPROACH TO DENSITY ESTIMATION IN SUP-NORM. 2022. ⟨hal-03695981⟩