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On stochastic comparisons of k-out-of-n systems with Weibull components

Part of: Distribution theory - Probability Operations research and management science

Published online by Cambridge University Press:  28 March 2018

Narayanaswamy Balakrishnan ,
Ghobad Barmalzan  and
Abedin Haidari
Narayanaswamy Balakrishnan*
Affiliation:
McMaster University
Ghobad Barmalzan*
Affiliation:
University of Zabol
Abedin Haidari*
Affiliation:
University of Zabol
*
* Postal address: Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, L85 4K1, Canada. Email address: bala@mcmaster.ca
** Postal address: Department of Statistics, University of Zabol, Sistan and Baluchestan, Iran.
** Postal address: Department of Statistics, University of Zabol, Sistan and Baluchestan, Iran.
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Abstract

In this paper we prove that a parallel system consisting of Weibull components with different scale parameters ages faster than a parallel system comprising Weibull components with equal scale parameters in the convex transform order when the lifetimes of components of both systems have different shape parameters satisfying some restriction. Moreover, while comparing these two systems, we show that the dispersive and the usual stochastic orders, and the right-spread order and the increasing convex order are equivalent. Further, some of the known results in the literature concerning comparisons of k-out-of-n systems in the exponential model are extended to the Weibull model. We also provide solutions to two open problems mentioned by Balakrishnan and Zhao (2013) and Zhao et al. (2016).

Keywords

Convex transform orderstar orderusual stochastic orderdispersive orderright-spread orderWeibull distributionk-out-of-n systemmultiple-outlier model

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 216 - 232
Copyright
Copyright © Applied Probability Trust 2018 

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References

Ahmed, A. N., Alzaid, A., Bartoszewicz, J. and Kochar, S. C. (1986). Dispersive and superadditive ordering. Adv. Appl. Prob. 18, 1019–1022. CrossRefGoogle Scholar
Amini-Seresht, E., Qiao, J., Zhang, Y. and Zhao, P. (2016). On the skewness of order statistics in multiple-outlier PHR models. Metrika 79, 817–836. Google Scholar
Arnold, B. C. and Groeneveld, R. A. (1995). Measuring skewness with respect to the mode. Amer. Statistician 49, 34–38. Google Scholar
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. John Wiley, New York. Google Scholar
Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Rev. Mat. Complut. 20, 7–107. CrossRefGoogle Scholar
Balakrishnan, N. and Cohen, A. C. (1991). Order Statistics and Inference: Estimation Methods. Academic Press, Boston, MA. Google Scholar
Balakrishnan, N. and Rao, C. R. (eds) (1998a). Order Statistics: Theory and Methods (Handbook Statist. 16). Elsevier, Amsterdam. Google Scholar
Balakrishnan, N. and Rao, C. R. (eds) (1998b). Order Statistics: Applications (Handbook Statist. 17). Elsevier, Amsterdam. Google Scholar
Balakrishnan, N. and Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Prob. Eng. Inf. Sci. 27, 403–443. Google Scholar
Boland, P. J., El-Neweihi, E. and Proschan, F. (1994). Applications of the hazard rate ordering in reliability and order statistics. J. Appl. Prob. 31, 180–192. Google Scholar
Bon, J.-L. and Pǎltǎnea, E. (2006). Comparisons of order statistics in a random sequence to the same statistics with i.i.d. variables. ESAIM Prob. Statist. 10, 1–10. Google Scholar
Da, G., Xu, M. and Balakrishnan, N. (2014). On the Lorenz ordering of order statistics from exponential populations and some applications. J. Multivariate Anal. 127, 88–97. Google Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ. CrossRefGoogle Scholar
Ding, W., Yang, J. and Ling, X. (2017). On the skewness of extreme order statistics from heterogeneous samples. Commun. Statist. Theory Meth. 46, 2315–2331. 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. CrossRefGoogle Scholar
Fang, L. and Zhang, X. (2012). New results on stochastic comparison of order statistics from heterogeneous Weibull populations. J. Korean Statist. Soc. 41, 13–16. Google Scholar
Fang, L. and Zhang, X. (2013). Stochastic comparison of series systems with heterogeneous Weibull components. Statist. Prob. Lett. 83, 1649–1653. Google Scholar
Fernandez-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. CrossRefGoogle Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd edn. John Wiley, New York. 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. (2006). Weibull distribution: some stochastic comparisons results. J. Statist. Planning Infer. 136, 3121–3129. 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. (2009). Comparisons of parallel systems according to the convex transform order. J. Appl. Prob. 46, 342–352. Google Scholar
Kochar, S. and Xu, M. (2011). On the skewness of order statistics in the multiple-outlier models. J. Appl. Prob. 48, 271–284. Google Scholar
Kochar, S. and Xu, M. (2014). On the skewness of order statistics with applications. Ann. Operat. Res. 212, 127–138. CrossRefGoogle Scholar
MacGillivray, H. L. (1986). Skewness and asymmetry: measures and orderings. Ann. Statist. 14, 994–1011. (Correction: 15 (1987), 884.) Google Scholar
Mao, T. and Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Prob. Eng. Inf. Sci. 24, 245–262. Google Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York. Google Scholar
Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York. CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester. Google Scholar
Murthy, D. N. P., Xie, M. and Jiang, R. (2004). Weibull Models. John Wiley, Hoboken, NJ. Google Scholar
Oja, H. (1981). On location, scale, skewness and kurtosis of univariate distributions. Scand. J. Statist. 8, 154–168. Google Scholar
Pǎltǎnea, E. (2008). On the comparison in hazard rate ordering of fail-safe systems. J. Statist. Planning Infer. 138, 1993–1997. Google Scholar
Pǎltǎnea, E. (2011). Bounds for mixtures of order statistics from exponentials and applications. J. Multivariate Anal. 102, 896–907. Google Scholar
Pledger, P. and Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Optimizing Methods in Statistics, Academic Press, New York, pp. 89–113. Google Scholar
Proschan, F. and Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J. Multivariate Anal. 6, 608–616. Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. Google Scholar
Van Zwet, W. R. (1964). Convex Transformations of Random Variables. Mathematisch Centrum, Amsterdam. Google Scholar
Yu, Y. (2016). On stochastic comparisons of order statistics from heterogeneous exponential samples. Preprint. Available at https://arxiv.org/abs/1607.06564. Google Scholar
Zhao, P., Li, X. and Balakrishnan, N. (2009). Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables. J. Multivariate Anal. 100, 952–962. Google Scholar
Zhao, P. and Balakrishnan, N. (2011a). Dispersive ordering of fail-safe systems with heterogeneous exponential components. Metrika 74, 203–210. CrossRefGoogle Scholar
Zhao, P. and Balakrishnan, N. (2011b). MRL ordering of parallel systems with two heterogeneous components. J. Statist. Planning Infer. 141, 631–638. Google Scholar
Zhao, P. and Balakrishnan, N. (2011c). New results on comparisons of parallel systems with heterogeneous gamma components. Statist. Prob. Lett. 81, 36–44. Google Scholar
Zhao, P. and Balakrishnan, N. (2011d). Some characterization results for parallel systems with two heterogeneous exponential components. Statistics 45, 593–604. CrossRefGoogle Scholar
Zhao, P. and Balakrishnan, N. (2012). Stochastic comparisons of largest order statistics from multiple-outlier exponential models. Prob. Eng. Inf. Sci. 26, 159–182. Google Scholar
Zhao, P., Zhang, Y. and Qiao, J. (2016). On extreme order statistics from heterogeneous Weibull variables. Statistics 50, 1376–1386. Google Scholar