An exponential characterization based on a type II censored sample

Abstract

Let S i, n = ∑ j=1 i ( n − j + 1)( X ( j) − X ( j−1) ) be the total-time-on-test at the ith order statistic X ( i) , 1⩽ i⩽ n of a random sample of n lifetimes X 1, …, X n . Let r be a fixed integer satisfying 2⩽ r⩽ n, n⩾3. The problem that the vector ( S 1, n / S r, n ,…, S r−1, n / S r, n ) has the distribution of the order statistics of r − 1 uniform (0,1) random variables implies that X 1 has an exponential distribution has been studied by Seshadri et al. (1969) for the case r = n. The first complete proof of this case is given by Dufour et al. (1984). Dufour (1982) conjectured that this characterization of exponential distribution holds not only for the complete sample but also for a Type II censored sample, i.e., for r < n and n⩾3. The conjecture has been partially proved by Leslie and van Eeden (1993) under the condition r⩾( 2 3 )n+1 . Xu and Yang (1995) proved recently that it holds for r⩾5, which is without the constraint that the lower bound of r increases with n. This note shows that the conjecture is true for r⩾4, and it is true for r⩾2 if an additional distributional assumption of HNBUE (or HNWUE) is imposed on X 1.