A parametric registration model for warped distributions with Wasserstein’s distance

We consider a parametric deformation model for distributions. More precisely, we assume we observe JJ samples of random variables which are warped from an unknown distribution template. We tackle in this paper the problem of estimating the individual deformation parameters. For this, we construct a registering criterion based on the Wasserstein distance to quantify the alignment of the distributions. We prove consistency of the empirical estimators.