**Sankhya:
The Indian Journal of Statistics**

2003, Volume 65, Pt. 2, 249--258

**Characterizations Of The
Exponential Distribution Via The Residual Lifetime**

By

GWO DONG LIN, Academia Sinica, Taiwan

SUMMARY. This paper clarifies and extends significantly Chong's (1977) characterization result in terms of the residual lifetime. In particular, let $X$ be a positive random variable with finite mean and denote the mean residual lifetime (MRL) $h(x)=E(\hbox{max}\{X-x,0\})$. Then we prove that if the square of the MRL $h(x)$ is proportional to $h(2x)$, $X$ is exponentially distributed. Secondly, a general result concerning the higher-order moments of the residual lifetime is given. Finally, the exponential distribution is characterized by a new relation of its moments.

*AMS (1991) subject classification. *60E05, 62E10.

*Key words and phrases. *Exponential distribution, residual lifetime,
lack-of-memory property.