March 7, 2023, 09:30–10:45
Maths Job Market Seminar
We define a general method for finding a quasi-best approximant in sup-norm to a target density belonging to a given model, based on independent samples drawn from distributions which average to the target (which does not necessarily belong to the model). We also provide a general method for selecting among a countable family of such models. These estimators satisfy oracle inequalities in the general setting. The quality of the bounds depends on the volume of sets on which |p-q| is close to its maximum, where p,q belong to the model (or possibly to two different models, in the case of model selection). In particular, using piecewise polynomials on dyadic partitions, we recover optimal rates of convergence for classes of functions with anisotropic smoothness, with optimal dependence on semi-norms measuring the smoothness of the density in the coordinate directions. Moreover, our method adapts to the anisotropic smoothness within a given range (depending on the degree of the polynomials).