A Note on a Characterization of the Exponential Distribution Based on a Type II Censored Sample

Abstract

Let $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$ be the order statistics of a random sample of $n$ lifetimes. The total-time-on-test statistic at $X_{(i)}$ is defined by $S_{i,n} = \sum^i_{j = 1}(n - j + 1)(X_{(j)} - X_{(j - 1)}), 1 \leq i \leq n$. A type II censored sample is composed of the $r$ smallest observations and the remaining $n - r$ lifetimes which are known only to be at least as large as $X_{(r)}$. Dufour conjectured that if the vector of proportions $(S_{1,n}/S_{r,n}, \ldots, S_{r - 1,n}/S_{r,n})$ has the distribution of the order statistics of $r - 1$ uniform(0, 1) random variables, then $X_1$ has an exponential distribution. Leslie and van Eeden proved the conjecture provided $n - r$ is no longer than $(1/3)n - 1$. It is shown in this note that the conjecture is true in general for $n \geq r \geq 5$. If the random variable under consideration has either NBU or NWU distribution, then it is true for $n \geq r \geq 2, n \geq 3$. The lower bounds obtained here do not depend on the sample size.