Central limit theorem and bootstrap procedure for Wassersteins variations with an application to structural relationships between distributions

Wasserstein barycenters and variance-like criteria based on the 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 Wassersteins variation using a bootstrap procedure. We then 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 Wassersteins barycenters for which we prove central limit theorems, including bootstr