On Gauss's characterization of the normal distribution

Adelchi Azzalini*, Marc G. Genton

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


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 modem 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.

Original languageEnglish (US)
Pages (from-to)169-174
Number of pages6
Issue number1
StatePublished - 2007
Externally publishedYes


  • Cauchy functional equation
  • Characterization property
  • Location family
  • Maximum likelihood
  • Normal distribution
  • Sample mean vector

ASJC Scopus subject areas

  • Statistics and Probability


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