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We propose a study of a distribution registration model for generaldeformation functions. In this framework, we provide estimators of thedeformations as well as a goodness of fit test of the model. For this, weconsider a criterion which studies the Fr{\'e}chet mean (or barycenter) of thewarped distributions whose study enables to make inference on the model. Inparticular we obtain the asymptotic distribution and a bootstrap procedure forthe Wasserstein variation.

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