A statistical analysis of a deformation model with Wasserstein barycenters : estimation procedure and goodness of fit test

We propose a study of a distribution registration model for general deformation functions. In this framework, we provide estimators of the deformations as well as a goodness of fit test of the model. For this, we consider a criterion which studies the Fr{\é}chet mean (or barycenter) of the warped distributions whose study enables to make inference on the model. In particular we obtain the asymptotic distribution and a bootstrap procedure for the Wasserstein variation.