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Eustasio del Barrio, Paula Gordaliza, Jean-Michel Loubes, A central limit theorem for Lp transportation cost on the real line with application to fairness assessment in machine learning, Information and Inference: A Journal of the IMA, Volume 8, Issue 4, December 2019, Pages 817–849, https://doi.org/10.1093/imaiai/iaz016
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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.
