A characterization of normality via convex likelihood ratios

TL;DR

This paper presents a novel characterization of the multivariate normal distribution based on the convexity of likelihood ratios, with implications for statistical test admissibility.

Contribution

It introduces a new criterion for normality using convex likelihood ratios, expanding understanding of Gaussian distributions and related statistical tests.

Findings

f(x + y)/f(x) is convex in x iff f is Gaussian

Provides a new characterization of multivariate normality

Implications for the inadmissibility of certain statistical tests

Abstract

This work includes a new characterization of the multivariate normal distribution. In particular, it is shown that a positive density function $f$ is Gaussian if and only if the $f(x+ y)/f(x)$ is convex in $x$ for every $y$. This result has implications to recent research regarding inadmissibility of a test studied by Moran~(1973).