Consider the following problem: if the maximum likelihood estimate of a location parameter of a population is given by the sample mean, is it true that the distribution is of normal type? The answer is positive and the proof was provided by Gauss, albeit without using the likelihood terminology. We revisit this result in a modern context and present a simple and rigorous proof. We also consider extensions to a p-dimensional population and to the case with a parameter additional to that of location.
Publié le : 2007-02-14
Classification:
characterization property,
Cauchy functional equation,
location family,
maximum likelihood,
normal distribution,
sample mean vector
@article{1175287727,
author = {Azzalini, Adelchi and Genton, Marc G.},
title = {On Gauss's characterization of the normal distribution},
journal = {Bernoulli},
volume = {13},
number = {1},
year = {2007},
pages = { 169-174},
language = {en},
url = {http://dml.mathdoc.fr/item/1175287727}
}
Azzalini, Adelchi; Genton, Marc G. On Gauss’s characterization of the normal distribution. Bernoulli, Tome 13 (2007) no. 1, pp. 169-174. http://gdmltest.u-ga.fr/item/1175287727/