On the characterization of maximum likelihood estimators for location-scale families

Given maximum likelihood equations for location and scale parameters, one determines conditions under which there exists a uniquely defined parametric statistical model, whose location and scale maximum likelihood estimators are the given ones. The constructive approach is exemplified at several kinds of mean estimators including the mean, mean square, mean mean and stretched power mean. The possible extension of the method to more general situations is discussed and illustrated at the sample median maximum likelihood estimator.