estimation of deformations of densities

Today, Jean-Michel Loubes from Toulouse gave a seminar in Dauphine on the estimation of deformations using Wassertsein distances. This is functional data analysis, where samples from random transforms of the original density are observed towards estimating the baseline (or true) measure

As a neophyte, I found the problem of interest if difficult to evaluate, in particular wrt the identifiability of μ. Esp. when the distribution of the transform φ is unknown. I also wondered about the choice of means over medians, because of the added robustness of the later… In a possible connection with David Dunson’s median estimate of densities. I ran the following simulation based on 150 (centred) location-scale transforms of a normal mixture [in red] with the median of the 150 density estimators [in blue]. It is not such a poor estimate! Now, the problem itself could have implications in ABC where we have replicas of random versions of the ABC density. For instance, DIYABC produces a few copies of the ABC posteriors for the parameters of the model. Jean-Michel also mentioned  connection with transport problems.