Comparisons for series and parallel systems with discrete Weibull components via separate comparisons of parameters

Document Type : Original Scientific Paper

Author

Department of Statistics, University of Zabol, Sistan and Baluchestan, Iran

Abstract

‎In this paper‎, ‎we obtain the usual stochastic order of series and parallel systems comprising heterogeneous discrete Weibull (DW) components‎. ‎Suppose X1,...,Xn and Y1,...,Yn denote the independent component¢s lifetimes of two systems such that Xi ~ DW(bi‎, ‎p i) and Yi ~ DW(b*i‎, ‎p *i), i=1,...,n. We obtain the usual stochastic order between series systems‎ ‎when the vector b is switched to the vector b* with respect to the majorization order‎, ‎and when the vector log (1-p) is switched to the vector log (1-p*) in the sense of the weak supermajorization order‎. ‎We also discuss the usual stochastic order between series systems by using the unordered majorization between the vectors log(1-p) and log(1-p*), and the p-majorization order between the parameters b and b*. It is also shown that the usual stochastic order between parallel systems comprising heterogeneous discrete Weibull components when the vector -log p is switched to the vector -logp* in the sense of the weak supermajorization order‎. ‎These results enable us to find some lower bounds for the survival functions of a series and parallel systems consisting of independent heterogeneous discrete Weibull components.

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References

Balakrishnan, N., Barmalzan, G. and Haidari, A. (2018). On stochastic comparisons of k-out-of-n systems with Weibull components. Journal of Applied Probability, 55, 216–232.
Balakrishnan, N., Haidari, A. and Masoumifard, K. (2015). Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Transactions on Reliability, 64, 333–348.
Belzunce, F., Martínez-Riquelme, C. and Mulero, J. (2016). An Introduction to Stochastic Orders. Academic Press, London.
Cheng, K.W. (1977). Majorization: Its extensions and preservation theorems. Tech. Rep. No. 121. Department of Statistics, Stanford University, Stanford, CA.
Jeske, D.R. and Blessinger, T. (2004). Tunable approximations for the mean and variance of heterogeneous geometrically distributed random variables. The American Statistician, 58, 322–327.
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions - Vol. 1, Second edition. John Wiley & Sons, New York.
Haidari, A., Payandeh Najafabadi, A.T. and Balakrishnan, N. (2019). Comparisons between parallel systems with exponentiated generalized gamma components. Communications in Statistics: Theory and Methods, 48, 1316–1332.
Kundu, A., Chowdhury, S., Nanda, A. and Hazra, N. (2016). Some results on majorization and their applications. Journal of Computational and Applied Mathematics, 301, 161–177.
Lai, C.D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer Science & Business Media.
Margolin, B.H. and Winokur, H.S. (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, 915–925.
Marshall, A.W., Olkin, I. and Arnold, B.C. (2011). Inequalities: Theory of Majorization and its Applications. Springer-Verlag, New York.
Murthy, D.N.P., Xie, M. and Jiang, R. (2004). Weibull Models. John Wiley & Sons, Hoboken, New Jersey.
Nakagawa, T. and Osaki, S. (1975). The discrete Weibull distribution. IEEE Transactions on Reliability, 24, 300–301.
Parker, D.S. and Ram, P. (1977). Greed and Majorization. Tech. Report, Department of Computer Science, University of California, Los Angeles.
Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders. Springer-Verlag, New York.
Xu, M. and Hu, T. (2011). Order statistics from heterogeneous negative binomial random variables. Probability in the Engineering and Informational Sciences, 25, 435–448.
Zhao, P., Zhang, Y. and Qiao, J. (2016). On extreme order statistics from heterogeneous Weibull variables. Statistics, 50, 1376–1386.
Journal of Statistical Modelling: Theory and Applications
Volume 1, Issue 1 - Serial Number 1
January 2020
Pages 53-64

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History

  • Receive Date: 26 October 2019
  • Revise Date: 18 December 2019
  • Accept Date: 22 December 2019

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