Metrika

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Preservation of increasing convex/concave order under the formation of parallel/series system of dependent components

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Abstract

This paper investigates sufficient conditions for preservation property of the increasing convex order and the increasing concave order under the taking of maximum and minimum of statistically dependent random variables, respectively. As applications, we develop the preservation of NBUC and NBU(2) aging properties respectively under the parallel and series systems of components with statistically dependent lifetimes. Some copulas are presented as illustrations on statistical dependence structure satisfying the sufficient condition as well.

Keywords

Copula Increasing concave order Increasing convex order Parallel systems NBUC NBU(2) Series systems 

Notes

Acknowledgements

The authors would like to thank two anonymous reviewers for their insightful comments, which improve the integrity of this manuscript through bringing into view those related research in the literature. The first author’s research is supported by Scientific Research Foundation of Tianjin University of Commerce (R160106), Science and Technology Development Foundation of Tianjin (2017KJ176), and the third level of Tianjin 131 Innovative Talent Training Project.

References

  1. Balakrishnan N, Lai CD (2009) Continuous bivariate distributions. Springer, New YorkMATHGoogle Scholar
  2. Balakrishnan N, Belzunce F, Sordo MA, Suárez-Llorens A (2012) Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data. J Multivar Anal 105:45–54MathSciNetCrossRefMATHGoogle Scholar
  3. Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt, Rinehart and Winston, New YorkMATHGoogle Scholar
  4. Belzunce F, Franco M, Ruiz J, Ruiz MC (2001) On partial orderings between coherent systems with different structures. Probab Eng Inf Sci 15:273–293MathSciNetCrossRefMATHGoogle Scholar
  5. Belzunce F, Suárez-Llorens A, Sordo MA (2012) Comparisons of increasing directionally convex transformations of random vectors with a common copula. Insur Math Econ 50:385–390MathSciNetCrossRefMATHGoogle Scholar
  6. Block HW, Savits TH, Shaked M (1985) A concept of negative dependence using stochastic ordering. Stat Probab Lett 3:81–86MathSciNetCrossRefMATHGoogle Scholar
  7. Cai J, Wei W (2012) On the invariant properties of notions of positive dependence and copulas under increasing transformations. Insur Math Econ 50:43–49MathSciNetCrossRefMATHGoogle Scholar
  8. Cai J, Wu Y (1997) A note on the preservation of the NBUC class under formation of parallel systems with dissimilar components. Microelectron Reliab 37:359–360CrossRefGoogle Scholar
  9. Cao J, Wang Y (1991) The NBUC and NWUC classes of life distributions. J Appl Probab 1991:473–479MathSciNetCrossRefMATHGoogle Scholar
  10. Denuit M, Dhaene J, Goovaerts M, Kaas R (2005) Actuarial theory for dependent risks. Wiley, ChichesterCrossRefMATHGoogle Scholar
  11. Deshpande JV, Kochar SC, Harshinder S (1986) Aspects of positive ageing. J Appl Probab 23:748–758MathSciNetCrossRefMATHGoogle Scholar
  12. Durante F, Papini PL (2009) Componentwise concave copulas and their asymmetry. Kybernetika 45:1003–1011MathSciNetMATHGoogle Scholar
  13. Durante F, Sempi C (2006) On the characterization of a class of binary operations on bivariate distribution functions. Publ Math Debr 69:47–63MathSciNetMATHGoogle Scholar
  14. Esary JD, Proschan F (1963) Relationship between system failure rate and component failure rates. Technometrics 5:183–189MathSciNetCrossRefGoogle Scholar
  15. Franco M, Ruiz JM, Ruiz MC (2001) On closure of the IFR(2) and NBU(2) classes. J Appl Probab 38:235–241MathSciNetCrossRefMATHGoogle Scholar
  16. Gumbel EJ (1960) Bivariate exponential distribuitons. J Am Stat Assoc 55:698–707CrossRefMATHGoogle Scholar
  17. Hendi MI, Mashhour AF, Montasser MA (1993) Closure of the NBUC class under formation of parallel systems. J Appl Probab 30:975–978MathSciNetCrossRefMATHGoogle Scholar
  18. Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, LondonCrossRefMATHGoogle Scholar
  19. Kochar SC, Li X, Shaked M (2002) The total time on test transform and the excess wealth stochastic orders of distributions. Adv Appl Probab 34:826–845MathSciNetCrossRefMATHGoogle Scholar
  20. Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions. Wiley, New YorkCrossRefMATHGoogle Scholar
  21. Lai CD, Xie M (2006) Stochastic ageing and dependence for reliability. Springer, New YorkMATHGoogle Scholar
  22. Li Y (2004) Closure of NBU(2) class under the formation of parallel systems. Stat Probab Lett 67:57–63MathSciNetCrossRefMATHGoogle Scholar
  23. Li X, Kochar SC (2001) Some new results involving the NBU(2) class of life distributions. J Appl Probab 38:242–247MathSciNetCrossRefMATHGoogle Scholar
  24. Li H, Li X (2013) Stochastic orders in reliability and risk. Springer, New YorkCrossRefGoogle Scholar
  25. Li C, Li X (2017) Aging and ordering properties of multivariate lifetimes with Archimedean dependence structures. Commun Stat Theory Methods 46:874–891MathSciNetCrossRefMATHGoogle Scholar
  26. Li X, Qiu G (2007) Some preservation results of NBUC aging property with applications. Stat Pap 48:581–594MathSciNetCrossRefMATHGoogle Scholar
  27. Li X, Li Z, Jing B (2000) Some results about the NBUC class of life distributions. Stat Probab Lett 46:229–237MathSciNetCrossRefMATHGoogle Scholar
  28. Li X, Li Z, Jing B (2003) Erratum: some results about NBUC class of life distributions. Stat Probab Lett 61:235–236CrossRefGoogle Scholar
  29. McNeil AJ, Nešlehová J (2009) Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_1\)-norm symmetric distributions. Ann Stat 37:3059–3097CrossRefMATHGoogle Scholar
  30. Müller A, Stoyan D (2001) Stochastic comparison of random vectors with a common copula. Math Oper Res 26:723–740MathSciNetCrossRefMATHGoogle Scholar
  31. Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, West SussexMATHGoogle Scholar
  32. Nanda AK, Jain K, Singh H (1998) Preservation of some partial orderings under the formation of coherent systems. Stat Probab Lett 39:123–131MathSciNetCrossRefMATHGoogle Scholar
  33. Navarro J, del Águila Y, Sordo MA, Suárez-Llorens A (2013) Stochastic ordering properties for systems with dependent identically distributed components. Appl Stoch Models Bus Ind 29:264–278MathSciNetCrossRefMATHGoogle Scholar
  34. Navarro J, del Águila Y, Sordo MA, Suárez-Llorens A (2014) Preservation of reliability classes under the formation of coherent systems. Appl Stoch Models Bus Ind 30:444–454MathSciNetCrossRefGoogle Scholar
  35. Nelsen RB (2006) An introduction to copulas. Springer, New YorkMATHGoogle Scholar
  36. Pellerey F, Petakos KI (2002) Closure property of the NBUC class under formation of parallel systems. IEEE Trans Reliab 51:452–454CrossRefGoogle Scholar
  37. Prékopa A (1971) Logarithmic concave measures with application to stochastic programming. Acta Sci Math (Szeged) 32:301–315MathSciNetMATHGoogle Scholar
  38. Samaniego FJ (1985) On closure of the IFR class under formation of coherent systems. IEEE Trans Reliab 34:69–72CrossRefMATHGoogle Scholar
  39. Shaked M (1977) A familiy of concepts of dependence for bivariate distributions. J Am Stat Assoc 72:642–650CrossRefMATHGoogle Scholar
  40. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New YorkCrossRefMATHGoogle Scholar
  41. Singh H, Vijayasree G (1991) Preservation of partial orderings under the formation of \(k\)-out-of-\(n\):\(G\) systems of i.i.d. components. IEEE Trans Reliab 40:273–276CrossRefMATHGoogle Scholar
  42. Sordo MA, Navarro J, Sarabia JM (2014) Distorted Lorenz curves: models and comparisons. Soc Choice Welf 42:761–780MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceTianjin University of CommerceTianjinChina
  2. 2.Department of Mathematical SciencesStevens Institute of TechnologyHobokenUSA

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