Central Limit Theorem and bootstrap procedure for Wasserstein's variations with application to structural relationships between distributions

Eustasio del Barrio (1) , Paula Gordaliza (1) , Hélène Lescornel (2) , Jean-Michel Loubes (2)

1 UVa - Universidad de Valladolid [Valladolid]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

Wasserstein barycenters and variance-like criterion using Wasserstein distance are used in many problems to analyze the homogeneity of collections of distributions and structural relationships between the observations. We propose the estimation of the quantiles of the empirical process of the Wasserstein's variation using a bootstrap procedure. Then we use these results for statistical inference on a distribution registration model for general deformation functions. The tests are based on the variance of the distributions with respect to their Wasserstein's barycenters for which we prove central limit theorems, including bootstrap versions.

Mots clés: AMS subject classifications: Primary; 62H10; Wasserstein variation; 62E20 Keywords: Wasserstein distance; secondary; Central Limit Theorem; goodness of fit

Domaines: Statistiques [math.ST]

Eustasio del Barrio, Paula Gordaliza, Hélène Lescornel, Jean-Michel Loubes. Central Limit Theorem and bootstrap procedure for Wasserstein's variations with application to structural relationships between distributions. Journal of Multivariate Analysis, 2019, 169, pp.341-362. ⟨10.1016/j.jmva.2018.09.014⟩. ⟨hal-01269785v4⟩