Smooth Cyclically Monotone Interpolation and Empirical Center-Outward Distribution Functions

Eustasio Del Barrio, Juan Cuesta Albertos, Marc Hallin and Carlos Matran

No 2018-15, Working Papers ECARES from ULB -- Universite Libre de Bruxelles

Abstract: 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.

Pages: 19 p.
Date: 2018-05
New Economics Papers: this item is included in nep-mac
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