×

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. MontrĂ©al (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

Citations:

Zbl 0791.62014
PDF BibTeX XML Cite
Full Text: DOI