An exponential characterization based on a type II censored sample

Let S-i,S-n = Sigma(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 less than or equal to i less than or equal to n of a random sample of n lifetimes X(1),...,X(n). Let r be a fixed integer satisfying 2 less than or equal to r less than or equal to n, n greater than or equal to 3. The problem that the vector (S-i,S-n/S-r,S-n,...,S-r-1,S-n/S-r,S-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 greater than or equal to 3. The conjecture has been partially proved by Leslie and van Eeden (1993) under the condition r greater than or equal to(2/3)n+1. Xu and Yang (1995) proved recently that it holds for r greater than or equal to 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 greater than or equal to 4, and it is true for r greater than or equal to 2 if an additional distributional assumption of HNBUE (or HNWUE) is imposed on X(1).