Open Access
Translator Disclaimer
June, 1995 A Note on a Characterization of the Exponential Distribution Based on a Type II Censored Sample
Jian-Lun Xu, Grace L. Yang
Ann. Statist. 23(3): 769-773 (June, 1995). DOI: 10.1214/aos/1176324621

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.

Citation

Download Citation

Jian-Lun Xu. Grace L. Yang. "A Note on a Characterization of the Exponential Distribution Based on a Type II Censored Sample." Ann. Statist. 23 (3) 769 - 773, June, 1995. https://doi.org/10.1214/aos/1176324621

Information

Published: June, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0842.62005
MathSciNet: MR1345199
Digital Object Identifier: 10.1214/aos/1176324621

Subjects:
Primary: 62E10
Secondary: 62E15

Rights: Copyright © 1995 Institute of Mathematical Statistics

JOURNAL ARTICLE
5 PAGES


SHARE
Vol.23 • No. 3 • June, 1995
Back to Top