**Sankhya:
The Indian Journal of Statistics**

2003, Volume 65, Pt. 2, 271--283

**Characterizations Of The Gamma
Distribution Via Conditional Moments**

By

CHAO-WEI CHOU, National Sun Yat-Sen University, Taiwan and WEN-JANG HUANG, National University of Kaohsiung, Taiwan

SUMMARY. In this work we characterize two independent non-degenerate positive random variables $X$ and $Y$ to be gamma distributed with the same scale parameter by the assumptions $E(X^{r+1}|X+Y)=a(X+Y)E(X^{r}|X+Y)$ and $E(X^{r+s+1}|X+Y)=b(X+Y)E(X^{r+s}|X+Y)$ for some fixed integer $r$ and $s=2.$ Furthermore, let $A\equiv \{A(t),t\geq 0\}$ be a renewal process with $\{S_{k},k\geq 1\}$ being the sequence of arrival times, under the assumptions $E(S_{k}^{r+1}|A(t)=n)=atE(S_{k}^{r}|A(t)=n)$ and $ E(S_{k}^{r+s+1}|A(t)=n)=btE(S_{k}^{r+s}|A(t)=n)$ for fixed integers $r,k,n,$ where $1\leq k\leq n,$ and $s=2,$ we prove that $A$ has to be a Poisson process. In the case that $s=1$ the above two results were proved by Huang and Su (1997).}On the other hand, recently characterizations of gamma distribution by the so-called dual regression schemes were investigated by Bobecka and Wesolowski (2001). More precisely, they considered the constancy of regressions of $X$ and $Y,$ while independence of $X/(X+Y)$ and $X+Y$ is assumed instead of independence of $X$ and $Y$. They characterized $X$ and $Y$ to be gamma distributed by the assumptions $E(Y^{u}|X)=c$ and $E(Y^{v}|X)=d,$ for $(u,v)=(1,2),(1,-1)$ or $(-1,-2),$ where $c$ and $d$ are constants. As a generalization, we prove that $X$ and $Y$ are gamma distributed with the same scale parameter under the assumptions $E(Y^{r+1}|X)=cE(Y^{r}|X)$ and $E(Y^{r+2}|X)=dE(Y^{r+1}|X),$ for some fixed integer $r,$ where $c$ and $d$ are constants. Note that $(u,v)=(1,2),(1,-1)$ and $(-1,-2)$ corresponds to $r=0,-1$ and $-2,$ respectively.

*AMS (1991) subject classification. *62E10, 60G55.

*Key words and phrases. *Beta distribution, characterization, constancy
of regression, gamma distribution, Poisson process, renewal process.