When the score function is the identity function - A tale of characterizations of the normal distribution

Christophe Ley

Econometrics and Statistics, 2023, vol. 26, issue C, 153-160

Abstract: The normal distribution is well-known for several results that it is the only to fulfil. Much less well-known is the fact that many of these characterizations follow from the fact that the derivative of the log-density of the normal distribution is the (negative) identity function. This a priori very simple yet surprising observation allows a deeper understanding of existing characterizations and paves the way for an immediate extension of various seemingly normal-based characterizations to a general density by replacing the (negative) identity function in these results with the derivative of that log-density.

Keywords: Maximum likelihood characterization; Score function; Skew-symmetric distributions; Stein characterization; Variance bounds (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:ecosta:v:26:y:2023:i:c:p:153-160

DOI: 10.1016/j.ecosta.2020.10.001

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