Open Access
May 2014 Maximum likelihood characterization of distributions
Mitia Duerinckx, Christophe Ley, Yvik Swan
Bernoulli 20(2): 775-802 (May 2014). DOI: 10.3150/13-BEJ506

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.

Citation

Download Citation

Mitia Duerinckx. Christophe Ley. Yvik Swan. "Maximum likelihood characterization of distributions." Bernoulli 20 (2) 775 - 802, May 2014. https://doi.org/10.3150/13-BEJ506

Information

Published: May 2014
First available in Project Euclid: 28 February 2014

zbMATH: 06291822
MathSciNet: MR3178518
Digital Object Identifier: 10.3150/13-BEJ506

Keywords: location parameter , maximum likelihood estimator , minimal necessary sample size , one-parameter group family , scale parameter , score function

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability

Vol.20 • No. 2 • May 2014
Back to Top