Maximum likelihood characterization of distributions

Gauss' principle states that the maximum likelihood estimator of the parameter in a location family is the sample mean for all samples of all sample sizes if and only if the family is Gaussian. There exist many extensions of this result in diverse directions. In this paper we propose a unified treatment of this literature. In doing so we define the fundamental concept of minimal necessary sample size at which a given characterization holds. Many of the cornerstone references on this topic are retrieved and discussed in light of our findings, and several new characterization theorems are provided.