On Gauss’s characterization of the normal distribution. (English) Zbl 1111.62012

Azzalini, Adelchi; Genton, Marc G.

Bernoulli 13, No. 1, 169-174 (2007).

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.

MSC:

62E10	Characterization and structure theory of statistical distributions
62F10	Point estimation
62H12	Estimation in multivariate analysis

Keywords:

characterization property; Cauchy functional equation; location family; maximum likelihood; normal distribution; sample mean vector