  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 Wasserstein's
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 Wasserstein's barycenters for which we prove central
limit theorems, including bootstrap versions.
