## Bernoulli

- Bernoulli
- Volume 13, Number 1 (2007), 169-174.

### On Gauss’s characterization of the normal distribution

Adelchi Azzalini and Marc G. Genton

#### Abstract

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.

#### Article information

**Source**

Bernoulli Volume 13, Number 1 (2007), 169-174.

**Dates**

First available in Project Euclid: 30 March 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.bj/1175287727

**Digital Object Identifier**

doi:10.3150/07-BEJ5166

**Mathematical Reviews number (MathSciNet)**

MR2307401

**Zentralblatt MATH identifier**

1111.62012

**Keywords**

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

#### Citation

Azzalini, Adelchi; Genton, Marc G. On Gauss’s characterization of the normal distribution. Bernoulli 13 (2007), no. 1, 169--174. doi:10.3150/07-BEJ5166. http://projecteuclid.org/euclid.bj/1175287727.