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

  • Narayanaswamy Balakrishnan (a1), Ghobad Barmalzan (a2) and Abedin Haidari (a2)


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).


Corresponding author

* Postal address: Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, L85 4K1, Canada. Email address:
** Postal address: Department of Statistics, University of Zabol, Sistan and Baluchestan, Iran.
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Ahmed, A. N., Alzaid, A., Bartoszewicz, J. and Kochar, S. C. (1986). Dispersive and superadditive ordering. Adv. Appl. Prob. 18, 10191022.
Amini-Seresht, E., Qiao, J., Zhang, Y. and Zhao, P. (2016). On the skewness of order statistics in multiple-outlier PHR models. Metrika 79, 817836.
Arnold, B. C. and Groeneveld, R. A. (1995). Measuring skewness with respect to the mode. Amer. Statistician 49, 3438.
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. John Wiley, New York.
Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Rev. Mat. Complut. 20, 7107.
Balakrishnan, N. and Cohen, A. C. (1991). Order Statistics and Inference: Estimation Methods. Academic Press, Boston, MA.
Balakrishnan, N. and Rao, C. R. (eds) (1998a). Order Statistics: Theory and Methods (Handbook Statist. 16). Elsevier, Amsterdam.
Balakrishnan, N. and Rao, C. R. (eds) (1998b). Order Statistics: Applications (Handbook Statist. 17). Elsevier, Amsterdam.
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, 403443.
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, 180192.
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, 110.
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, 8897.
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ.
Ding, W., Yang, J. and Ling, X. (2017). On the skewness of extreme order statistics from heterogeneous samples. Commun. Statist. Theory Meth. 46, 23152331.
Dykstra, R., Kochar, S. and Rojo, J. (1997). Stochastic comparisons of parallel systems of heterogeneous exponential components. J. Statist. Planning Infer. 65, 203211.
Fang, L. and Zhang, X. (2012). New results on stochastic comparison of order statistics from heterogeneous Weibull populations. J. Korean Statist. Soc. 41, 1316.
Fang, L. and Zhang, X. (2013). Stochastic comparison of series systems with heterogeneous Weibull components. Statist. Prob. Lett. 83, 16491653.
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, 221228.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd edn. John Wiley, New York.
Khaledi, B.-E. and Kochar, S. (2000). Some new results on stochastic comparisons of parallel systems. J. Appl. Prob. 37, 11231128.
Khaledi, B.-E. and Kochar, S. (2006). Weibull distribution: some stochastic comparisons results. J. Statist. Planning Infer. 136, 31213129.
Kochar, S. C. and Wiens, D. P. (1987). Partial orderings of life distributions with respect to their aging properties. Naval Res. Logistics 34, 823829.
Kochar, S. and Xu, M. (2009). Comparisons of parallel systems according to the convex transform order. J. Appl. Prob. 46, 342352.
Kochar, S. and Xu, M. (2011). On the skewness of order statistics in the multiple-outlier models. J. Appl. Prob. 48, 271284.
Kochar, S. and Xu, M. (2014). On the skewness of order statistics with applications. Ann. Operat. Res. 212, 127138.
MacGillivray, H. L. (1986). Skewness and asymmetry: measures and orderings. Ann. Statist. 14, 9941011. (Correction: 15 (1987), 884.)
Mao, T. and Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Prob. Eng. Inf. Sci. 24, 245262.
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.
Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York.
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.
Murthy, D. N. P., Xie, M. and Jiang, R. (2004). Weibull Models. John Wiley, Hoboken, NJ.
Oja, H. (1981). On location, scale, skewness and kurtosis of univariate distributions. Scand. J. Statist. 8, 154168.
Pǎltǎnea, E. (2008). On the comparison in hazard rate ordering of fail-safe systems. J. Statist. Planning Infer. 138, 19931997.
Pǎltǎnea, E. (2011). Bounds for mixtures of order statistics from exponentials and applications. J. Multivariate Anal. 102, 896907.
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. 89113.
Proschan, F. and Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J. Multivariate Anal. 6, 608616.
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
Van Zwet, W. R. (1964). Convex Transformations of Random Variables. Mathematisch Centrum, Amsterdam.
Yu, Y. (2016). On stochastic comparisons of order statistics from heterogeneous exponential samples. Preprint. Available at
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, 952962.
Zhao, P. and Balakrishnan, N. (2011a). Dispersive ordering of fail-safe systems with heterogeneous exponential components. Metrika 74, 203210.
Zhao, P. and Balakrishnan, N. (2011b). MRL ordering of parallel systems with two heterogeneous components. J. Statist. Planning Infer. 141, 631638.
Zhao, P. and Balakrishnan, N. (2011c). New results on comparisons of parallel systems with heterogeneous gamma components. Statist. Prob. Lett. 81, 3644.
Zhao, P. and Balakrishnan, N. (2011d). Some characterization results for parallel systems with two heterogeneous exponential components. Statistics 45, 593604.
Zhao, P. and Balakrishnan, N. (2012). Stochastic comparisons of largest order statistics from multiple-outlier exponential models. Prob. Eng. Inf. Sci. 26, 159182.
Zhao, P., Zhang, Y. and Qiao, J. (2016). On extreme order statistics from heterogeneous Weibull variables. Statistics 50, 13761386.
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