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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.