Asymptotic Distributions of Trimmed Wasserstein Distances Between the True and the Empirical Distribution Function

Abstract

If the distribution function F has a finite mean, then the Wasserstein distance $d({{F}_{n}},F) = \smallint _{{ - \infty }}^{\infty }|{{F}_{n}}(x) - F(x)|dx$ between F and the corresponding empirical distribution function F _n , based on a sample of size n converges almost surely to zero as n →∞. In [6] del Barrio, Giné and Matrán have provided an exhaustive study of the distributional limit theorems associated with this law of large numbers. Nothing can be said about d(F _n , F) = ∞ almost surely for all n ≥ 1 if F has no finite mean. In the present paper we modify d(F _n , F) into a finite quantity for all F by an adaptation of the notion of trimming from statistics, and study the asymptotic distributions of these trimmed Wasserstein distances for appropriate classes of distribution functions F via weighted approximation results for uniform empirical processes.

This work was completed with the support of a NATO Grant.

This work was completed with the support of a NATO Grant, NSA Grant MDA904-02-1-0034 and NSF Grant DMS-0203865.