A Characteristic Property of the Exponential Distribution

Abstract

Let $X$ be a nonnegative random variable with probability distribution function $F$. Suppose $X_{i,n} (i = 1,\cdots, n)$ is the $i$th smallest order statistics in a random sample of size $n$ from $F$. A necessary and sufficient condition for $F$ to be exponential is given which involves the identical distribution of the random variables $X$ and $(n - i) (X_{i+1,n} - X_{i,n})$ for some $i$ and $n$, $(1 \leqq i < n)$.