Stochastic Orders for Spacings of Heterogeneous Exponential Random Variables
AbstractWe 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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Bibliographic InfoArticle provided by Elsevier in its journal Journal of Multivariate Analysis.
Volume (Year): 57 (1996)
Issue (Month): 1 (April)
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- Wen, Songqiao & Lu, Qingshu & Hu, Taizhong, 2007. "Likelihood ratio orderings of spacings of heterogeneous exponential random variables," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 743-756, April.
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- N. Balakrishnan & Erhard Cramer, 2008. "Progressive censoring from heterogeneous distributions with applications to robustness," Annals of the Institute of Statistical Mathematics, Springer, vol. 60(1), pages 151-171, March.
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- Dolati, Ali & Genest, Christian & Kochar, Subhash C., 2008. "On the dependence between the extreme order statistics in the proportional hazards model," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 777-786, May.
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