## 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}

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

^{2}Department 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.

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