New Characterizations of the Gamma Distribution via Independence of Two Statistics by Using Anosov's Theorem

TL;DR

This paper extends the characterization of gamma distributions by using Anosov's theorem to relate the independence of the sample mean and a broad class of other statistics, revealing new properties and parallels with normal distribution characterizations.

Contribution

It introduces a general theorem and multiple corollaries that characterize gamma distributions through the independence of the sample mean and various homogeneous statistics.

Findings

New characterization results for gamma distributions.

Parallel between gamma and normal distribution characterizations.

Broader class of statistics used in gamma distribution characterization.

Abstract

Available in the literature are properties which characterize the gamma distribution via independence of two appropriately chosen statistics. Well-known is the classical result when one of the statistics is the sample mean and the other one the sample coefficient of variation. In this paper, we elaborate on a version of Anosov's theorem which allows to establish a general result, Theorem 1, and a series of seven corollaries providing new characterization results for gamma distributions. We keep the sample mean as one of involved statistics, while now the second one can be taken from a quite large class of homogeneous feasible definite statistics. It is relevant to mention that there is an interesting parallel between the new characterization results for gamma distributions and recent characterization results for the normal distribution.