Improved rates for Wasserstein deconvolution with ordinary smooth error in dimension one

Jérôme Dedecker (1) , Aurélie Fischer (2) , Bertrand Michel (3)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 LSTA - Laboratoire de Statistique Théorique et Appliquée

This paper deals with the estimation of a probability measure on the real line from data observed with an additive noise. We are interested in rates of convergence for the Wasserstein metric of order $p\geq 1$. The distribution of the errors is assumed to be known and to belong to a class of supersmooth or ordinary smooth distributions. We obtain in the univariate situation an improved upper bound in the ordinary smooth case and less restrictive conditions for the existing bound in the supersmooth one. In the ordinary smooth case, a lower bound is also provided, and numerical experiments illustrating the rates of convergence are presented.

Mots clés: Deconvolution; Wasserstein metrics; ordinary smooth distributions; minimax rates

Domaines: Théorie [stat.TH]; Statistiques [math.ST]

Jérôme Dedecker, Aurélie Fischer, Bertrand Michel. Improved rates for Wasserstein deconvolution with ordinary smooth error in dimension one. 2014. ⟨hal-00971316v2⟩