  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{\'e}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.
