  A famous characterization theorem due to C.F. Gauss states that the maximum
likelihood estimator (MLE) 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, most of them
focussing on location and scale families. In this paper, we propose a unified
treatment of this literature by providing general MLE characterization theorems
for one-parameter group families (with particular attention on location and
scale parameters). In doing so, we provide tools for determining whether or not
a given such family is MLE-characterizable, and, in case it is, 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 the light of our findings, and several new
characterization theorems are provided. Of particular interest is that one part
of our work, namely the introduction of so-called equivalence classes for MLE
characterizations, is a modernized version of Daniel Bernoulli's viewpoint on
maximum likelihood estimation.
