## Bernoulli

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

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

#### 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

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

Digital Object Identifier
doi:10.3150/07-BEJ5166

Mathematical Reviews number (MathSciNet)
MR2307401

Zentralblatt MATH identifier
1111.62012

#### 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. https://projecteuclid.org/euclid.bj/1175287727