✦ LIBER ✦

Stochastic Orders for Spacings of Heterogeneous Exponential Random Variables

✍ Scribed by Subhash C Kochar; Ramesh Korwar

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 489 KB
Volume: 57
Category: Article
ISSN: 0047-259X
DOI: 10.1006/jmva.1996.0022

No coin nor oath required. For personal study only.

✦ Synopsis

We obtain some new results on normalized spacings of independent exponential random variables with possibly different scale parameters. It is shown that the density functions of the individual normalized spacings in this case are mixtures of exponential distributions and, as a result, they are log-convex (and, hence, DFR). G. Pledger and F. Proschan (Optimizing Methods in Statistics (J. S. Rustagi, Ed.), pp. 89 113, Academic Press, New York, 1971), have shown, with the help of a counterexample, that in a sample of size 3 the survival function of the last spacing is not Schur convex. We show that, however, this is true for the second spacing for all sample sizes. G. Pledger and F. Proschan (ibid.) also prove that the spacings are stochastically larger when the scale parameters are unequal than when they are all equal. We strengthen this result from stochastic ordering to likelihood ratio ordering. Some new results on dispersive ordering between the normalized spacings have also been obtained.

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