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Comparisons of Parallel Systems According to the Convex Transform Order

Part of: Distribution theory - Probability Nonparametric inference Survival analysis and censored data Sampling theory, sample surveys

Published online by Cambridge University Press:  14 July 2016

Subhash Kochar  and
Maochao Xu
Subhash Kochar*
Affiliation:
Portland State University
Maochao Xu*
Affiliation:
Portland State University
*
∗Postal address: Department of Mathematics and Statistics, Portland State University, Portland, Oregon 97201, USA.
∗∗∗Email address: maochao@pdx.edu
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Abstract

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A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples in the sense of the dispersive order and the right-spread order are established. A sufficient condition is also given for the proportional hazard rate model.

Keywords

Convex transform orderexponential distributionincreasing failure rateparallel system; proportional hazard rateright-spread orderskewness

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 62N05: Reliability and life testing 62G30: Order statistics; empirical distribution functions 62D05: Sampling theory, sample surveys
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 342 - 352
Copyright
Copyright © Applied Probability Trust 2009 

References

Ahmed, A. N., Alzaid, A., Bartoszewicz, J. and Kochar, S. C. (1986). Dispersive and superadditive ordering. Adv. Appl. Prob. 18, 1019–1022.Google Scholar
Arnold, B. C. and Groeneveld, R. A. (1995). Measuring skewness with respect to the mode. Amer. Statistician 49, 34–38.Google Scholar
Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Dykstra, R., Kochar, S. and Rojo, J. (1997). Stochastic comparisons of parallel systems of heterogeneous exponential components. J. Statist. Planning Infer. 65, 203–211.Google Scholar
Fernández-Ponce, J. M., Kochar, S. C. and Muñoz-Perez, J. (1998). Partial orderings of distributions based on right-spread functions. J. Appl. Prob. 35, 221–228.Google Scholar
Khaledi, B.-E. and Kochar, S. (2000). Some new results on stochastic comparisons of parallel systems. J. Appl. Prob. 37, 1123–1128.CrossRefGoogle Scholar
Khaledi, B.-E. and Kochar, S. (2002). Stochastic orderings among order statistics and sample spacings. In Uncertainty and Optimality, ed. Misra, J. C., World Scientific River Edge, NJ, pp. 167–203.Google Scholar
Kochar, S. C. (1989). On extensions of DMRL and related partial orderings of life distributions. Commun. Statist. Stoch. Models 5, 235–245.Google Scholar
Kochar, S. C. and Wiens, D. P. (1987). Partial orderings of life distributions with respect to their aging properties. Naval Res. Logistics 34, 823–829.Google Scholar
Kochar, S. and Xu, M. (2007a). Some recent results on stochastic comparisons and dependence among order statistics in the case of PHR model. J. Iranian Statist. Soc. 6, 125–140.Google Scholar
Kochar, S. and Xu, M. (2007b). Stochastic comparisons of parallel systems when components have proportional hazard rates. Prob. Eng. Inf. Sci. 21, 597–609.Google Scholar
Kochar, S. C., Li, X. and Shaked, M. (2002). The total time on test transform and the excess wealth stochastic order of distributions. Adv. Appl. Prob. 34, 826–845.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Application (Math. Sci. Eng. 143). Academic Press, New York.Google Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
Mitrinović, D. S. (1970). Analytic Inequalities. Springer, Berlin.Google Scholar
Shaked, M. and Shantikumar, J. G. (1998). Two variability orders. Prob. Eng. Inf. Sci. 12, 1–23.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Van Zwet, W. R. (1970). Convex Transformations of Random Variables (Math. Centre Tracts 7), 2nd edn. Mathematical Centre, Amsterdam.Google Scholar