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May 2020 A Central Limit Theorem for Wasserstein type distances between two distinct univariate distributions
Philippe Berthet, Jean-Claude Fort, Thierry Klein
Ann. Inst. H. Poincaré Probab. Statist. 56(2): 954-982 (May 2020). DOI: 10.1214/19-AIHP990


In this article we study the natural nonparametric estimator of a Wasserstein type cost between two distinct continuous distributions $F$ and $G$ on $\mathbb{R}$. The estimator is based on the order statistics of a sample having marginals $F$, $G$ and any joint distribution. We prove a central limit theorem under general conditions relating the tails and the cost function. In particular, these conditions are satisfied by Wasserstein distances of order $p>1$ and compatible classical probability distributions.

Dans cet article nous étudions l’estimateur non paramétrique naturel d’un coût de type Wasserstein entre deux lois $F$ et $G$ distinctes et continues sur $\mathbb{R}$. Cet estimateur est construit à partir des statistiques d’ordre d’un échantillon d’un couple quelconque de lois marginales $F$ et $G$. Nous démontrons un théorème limite central sous des conditions générales reliant les queues de distribution à la fonction de coût. En particulier, ces conditions sont satisfaites par les distances de Wasserstein d’ordre $p>1$ et les lois classiques compatibles.


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Philippe Berthet. Jean-Claude Fort. Thierry Klein. "A Central Limit Theorem for Wasserstein type distances between two distinct univariate distributions." Ann. Inst. H. Poincaré Probab. Statist. 56 (2) 954 - 982, May 2020.


Received: 2 February 2018; Revised: 6 March 2019; Accepted: 29 March 2019; Published: May 2020
First available in Project Euclid: 16 March 2020

zbMATH: 07199886
MathSciNet: MR4076772
Digital Object Identifier: 10.1214/19-AIHP990

Primary: 60F05, 60F17, 62G20, 62G30

Rights: Copyright © 2020 Institut Henri Poincaré


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Vol.56 • No. 2 • May 2020
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