ON A CHARACTERIZATION OF THE GAMMA DISTRIBUTION: THE INDEPENDENCE OF THE SAMPLE MEAN AND THE SAMPLE COEFFICIENT OF VARIATION

TEA-YUAN HWANG^{1} AND CHIN-YUAN HU^{2}

1Institute of Statistics, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C.

2Department of Business Education, National Changhua University of Education, Changhua 50058, Taiwan, R.O.C.

(Received January 26, 1996; revised June 24, 1998)

Abstract. Let n ≥ 3 and let X_{1}, …, X_{n} be positive i.i.d. random variables whose common distribution function f has a continuous p.d.f. Using earlier work of the present authors and a method due to Anosov for solving certain integro-functional equations, it is shown that the independence of the sample mean and the sample coefficient of variation is equivalent to that f is a gamma function. While the proof is of methodological interest, this conclusion can also be arrived at without any assumptions by appealing to the Laplace-Stieltjes transform, as in the Concluding Remark (Section 3).

Key words and phrases: Characterization, gamma distribution, coefficient of variation.