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February 2007 On Gauss’s characterization of the normal distribution
Adelchi Azzalini, Marc G. Genton
Bernoulli 13(1): 169-174 (February 2007). DOI: 10.3150/07-BEJ5166


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.


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Adelchi Azzalini. Marc G. Genton. "On Gauss’s characterization of the normal distribution." Bernoulli 13 (1) 169 - 174, February 2007.


Published: February 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1111.62012
MathSciNet: MR2307401
Digital Object Identifier: 10.3150/07-BEJ5166

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

Rights: Copyright © 2007 Bernoulli Society for Mathematical Statistics and Probability


Vol.13 • No. 1 • February 2007
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