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ORDERING PROPERTIES OF SPACINGS FROM HETEROGENEOUS GEOMETRIC SAMPLES

  • Weiyong Ding (a1), Yiying Zhang (a2) and Peng Zhao (a3)

Abstract

In the reliability context, the geometric distribution is a natural choice to model the lifetimes of some equipment and components when they are measured by the number of completed cycles of operation or strokes, or in case of periodic monitoring of continuous data. This paper aims at investigating how the heterogeneity among the parameters affects some characteristics such as the distribution and hazard rate functions of spacings arising from independent heterogeneous geometric random variables. First, refined representations of the distribution functions are provided for both the second spacing and sample range from heterogeneous geometric sample. Second, stochastic comparisons are carried out on the second spacings and sample ranges for two sets of independent and heterogeneous geometric random variables in the sense of the usual stochastic and hazard rate orderings. The results established here not only fill the gap on the study of stochastic properties of spacings from heterogeneous geometric samples, but also are expected to be applied in the fields of statistics and reliability.

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1. Du, B., Zhao, P., & Balakrishnan, N. (2012). Likelihood ratio and hazard rate orderings of the maxima in two multiple-outlier geometric samples. Probability in the Engineering and Informational Sciences 26: 375391.
2. Jeske, D.R. & Blessinger, T. (2004). Tunable approximations for the mean and variance of the maximum of heterogeneous geometrically distributed random variables. The American Statistician 58(4): 322327.
3. Hardy, G.H., Littlewood, J.E., & Pólya, G. (1929). Some simple inequalities satisfied by convex function. Messenger of Mathematics 58: 145152.
4. Hardy, G.H., Littlewood, J.E., & Pólya, G. (1934). Inequalities. Cambridge: Cambridge University Press.
5. Kochar, S.C. (2012). Stochastic comparisons of order statistics and spacings: A review. ISRN Probability and Statistics, vol. 2012, Article ID 839473, 47 pages.
6. Kochar, S.C. & Xu, M. (2011). Stochastic comparisons of spacings from heterogeneous samples. In Wells, Martin T. and SenGupta, Ashis (eds.), Advances in directional and linear statistics. Physica-Verlag HD, pp. 113129.
7. Lundberg, B. (1955). Fatigue life of airplane structures. Journal of the Aeronautical Science 22: 394.
8. Mao, T. & Hu, T. (2010). Equivalent characterizations on ordering of order statistics and sample ranges. Probability in the Engineering and Informational Sciences 24: 245262.
9. Margolin, B.H. & Winokur, H.S. Jr. (1967). Exact moments of the order statistics of the geometric distribution and their relation to inverse sampling and reliability of redundant systems. Journal of the American Statistical Association 62(319): 915925.
10. Marshall, A.W., Olkin, I., & Arnold, B.C. (2011). Inequalities: Theory of majorization and its applications, 2nd edition. New York: Springer-Verlag.
11. Misra, N. & van der Meulen, E.C. (2003). On stochastic properties of m-spacings. Journal of Statistical Planning and Inference 115(2): 683697.
12. Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer-Verlag.
13. Xu, M. & Hu, T. (2011). Order statistics from heterogeneous negative binomial random variables. Probability in the Engineering and Informational Sciences 25: 435448.

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ORDERING PROPERTIES OF SPACINGS FROM HETEROGENEOUS GEOMETRIC SAMPLES

  • Weiyong Ding (a1), Yiying Zhang (a2) and Peng Zhao (a3)

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