New characterizations of the Gamma distribution via independence of two statistics by using Anosov’s theorem

Lin G. D.^a, J. M. Stoyanov<sup>b

a</sup> Institute of Statistical Science, Academia Sinica, Nankang, Taipei, Taiwan (ROC)
^b Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

Abstract: There 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 is the sample coefficient of variation. In this paper, we elaborate on a version of Anosov's theorem, which allows us to establish a general result and a series of seven corollaries, providing new characterization results for gamma distributions. We keep the sample mean as one of involved statistics, while the other can be taken from a quite large class of homogeneous feasible definite statistics. We discuss an interesting parallel between the new characterization results for gamma distributions and recent characterization results for the normal distribution.

Keywords: characterization of the gamma distribution, sample mean, sample coefficient of variation, order statistics, feasible statistics, sample size, Gini coefficient, Anosov's theorem.

Received: 22.07.2022
Revised: 22.12.2022
Accepted: 22.02.2023

DOI: 10.4213/tvp5593

 English version: Theory of Probability and its Applications, 2025, 69:4, 592–604

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