Maximum likelihood characterization of distributions

TL;DR

This paper develops a unified framework for characterizing distributions via maximum likelihood estimators, extending classical results like Gauss's theorem to broader one-parameter group families, and introduces new tools and concepts such as equivalence classes.

Contribution

It provides general MLE characterization theorems for one-parameter group families, introduces tools for identifying MLE-characterizable families, and defines the minimal sample size needed for such characterizations.

Findings

New characterization theorems for one-parameter group families

Tools for determining MLE-characterizability of distributions

Introduction of equivalence classes for MLE characterizations

Abstract

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.