https://hal.science/hal-03890826Deo, NeilNeilDeoDPMMS - Department of Pure Mathematics and Mathematical Statistics - CMS - Faculty of mathematics Centre for Mathematical Sciences [Cambridge] - CAM - University of Cambridge [UK]Randrianarisoa, ThibaultThibaultRandrianarisoaLPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris CitéOn Adaptive Confidence Sets for the Wasserstein DistancesHAL CCSD2023Wasserstein distanceUncertainty QuantificationNonparametric confidence sets[STAT] Statistics [stat][MATH] Mathematics [math]Randrianarisoa, Thibault2022-12-08 18:04:442023-03-24 14:53:292022-12-13 12:06:47enJournal articlesapplication/pdf1In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with diameter measured in Wasserstein distance Wp, p >=1, and for densities with unknown regularity measured on a Besov scale. As sampling domains, we focus on the d-dimensional torus Td, in which case 1<=p<= 2, and Rd, for which p = 1. We identify necessary and sufficient conditions for the existence of adaptive confidencesets with diameters of the order of the regularity-dependent Wp-minimax estimation rate. Interestingly, it appears that the possibility of such adaptation of the diameter depends on the dimension of the underlying space. In low dimensions, d<= 4, adaptation to any regularity is possible. In higher dimensions, adaptation is possible if and only if the underlying regularities belong to some interval of width at least d/(d-4). This contrasts with the usual Lp-theory where, independently of the dimension, adaptation occurs only if regularities lie in a small fixed-width window. For configurations allowing these adaptive sets to exist, we explicitly construct confidence regions via the method of risk estimation. These are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis and methods extend to weak losses such as Sobolev norms with negative smoothness indices.