A central limit theorem for Lp transportation cost on the real line with application to fairness assessment in machine learning. (14th October 2019)

• Record Type: Journal Article
• Authors: del Barrio, Eustasio; Gordaliza, Paula; Loubes, Jean-Michel
• Is Part Of: Information and inference. Volume 8:Number 4(2019)
• Page Start: 817
• Page End: 849
• Publication Date: 2019-10-14
• Subjects: optimal transport -- Monge–Kantorovich distance -- central limit theorem -- fair learning; Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaz016
• Languages: English
• ISSNs: 2049-8764

Abstract

Abstract: We provide a central limit theorem for the Monge–Kantorovich distance between two empirical distributions with sizes $n$ and $m$, $\mathcal{W}_p(P_n, Q_m), \ p\geqslant 1, $ for observations on the real line. In the case $p>1$ our assumptions are sharp in terms of moments and smoothness. We prove results dealing with the choice of centring constants. We provide a consistent estimate of the asymptotic variance, which enables to build two sample tests and confidence intervals to certify the similarity between two distributions. These are then used to assess a new criterion of data set fairness in classification.