## A note on a characterization of the exponential distribution based on a type II censored sample.(English)Zbl 0842.62005

Summary: 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 $S_{i,n}= \sum^i_{j=1} (n-j+ 1) (X_{(j)}- X_{(j- 1)}), \qquad 1\leq i\leq n.$ 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 $$X_{(r)}$$. R. Dufour [Ph.D. dissertation, Univ. Montréal (1982)] conjectured that if the vector of proportions $$(S_{1,n}/ S_{r,n}, \dots, S_{r-1, n}/ S_{r,n})$$ has the distribution of the order statistics of $$r-1$$ uniform $$(0,1)$$ random variables, then $$X_1$$ has an exponential distribution. J. Leslie and C. van Eeden [Ann. Stat. 21, No. 3, 1640-1647 (1993; Zbl 0791.62014)] proved the conjecture provided $$n-r$$ is no larger 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.

### MSC:

 62E10 Characterization and structure theory of statistical distributions 62G30 Order statistics; empirical distribution functions

Zbl 0791.62014
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