par Del Barrio, Eustasio;Cuesta Albertos, Juan;Hallin, Marc 
;Matran, Carlos
Publication Publié, 2018-05
;Matran, CarlosPublication Publié, 2018-05
Travail de recherche/Working paper
| Résumé : | We consider the smooth interpolation problem under cyclical monotonicity constraint. More precisely, consider finite n-tuples X =fx1; : : : ; xng and Y = fy1; : : : ; yng of points in Rd, and assume the existence of a unique bijection T : X ! Y such that f(x; T(x)): x 2 Xg is cyclically monotone: our goal is to define continuous, cyclically mono-tone maps T : Rd ! Rd such that T(xi) = yi, i = 1; : : : ; n, extending a classical result by Rockafellar on the sub differentials of convex functions. Our solutions T are Lipschitz, and we provide a sharp lower bound for the corresponding Lipschitz constants. The problem is motivated by, and the solution naturally applies to, the concept of empirical center-outwarddistribution function in Rd developed in Hallin (2018). Those empirical distribution functions indeed are de_ned at the observations only. Our interpolation provides a smooth extension, as well as a multivariate, outward-continuous, jump function version thereof (the latter naturally generalizes the traditional left-continuous univariate concept); both satisfy a Glivenko-Cantelli property as n ! 1. |
