Skip to main content
SpringerLink
Log in

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

  • Published:
Metrika Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. A function \(h:[0,1]\mapsto [0,1]\) is a distortion function if it is non-decreasing, \(h(0)=0\) and \(h(1)=1\).

References

  • Balakrishnan N, Lai CD (2009) Continuous bivariate distributions. Springer, New York

    MATH  Google Scholar 

  • 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 105:45–54

    Article  MathSciNet  MATH  Google Scholar 

  • Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt, Rinehart and Winston, New York

    MATH  Google Scholar 

  • Belzunce F, Franco M, Ruiz J, Ruiz MC (2001) On partial orderings between coherent systems with different structures. Probab Eng Inf Sci 15:273–293

    Article  MathSciNet  MATH  Google Scholar 

  • Belzunce F, Suárez-Llorens A, Sordo MA (2012) Comparisons of increasing directionally convex transformations of random vectors with a common copula. Insur Math Econ 50:385–390

    Article  MathSciNet  MATH  Google Scholar 

  • Block HW, Savits TH, Shaked M (1985) A concept of negative dependence using stochastic ordering. Stat Probab Lett 3:81–86

    Article  MathSciNet  MATH  Google Scholar 

  • Cai J, Wei W (2012) On the invariant properties of notions of positive dependence and copulas under increasing transformations. Insur Math Econ 50:43–49

    Article  MathSciNet  MATH  Google Scholar 

  • Cai J, Wu Y (1997) A note on the preservation of the NBUC class under formation of parallel systems with dissimilar components. Microelectron Reliab 37:359–360

    Article  Google Scholar 

  • Cao J, Wang Y (1991) The NBUC and NWUC classes of life distributions. J Appl Probab 1991:473–479

    Article  MathSciNet  MATH  Google Scholar 

  • Denuit M, Dhaene J, Goovaerts M, Kaas R (2005) Actuarial theory for dependent risks. Wiley, Chichester

    Book  MATH  Google Scholar 

  • Deshpande JV, Kochar SC, Harshinder S (1986) Aspects of positive ageing. J Appl Probab 23:748–758

    Article  MathSciNet  MATH  Google Scholar 

  • Durante F, Papini PL (2009) Componentwise concave copulas and their asymmetry. Kybernetika 45:1003–1011

    MathSciNet  MATH  Google Scholar 

  • Durante F, Sempi C (2006) On the characterization of a class of binary operations on bivariate distribution functions. Publ Math Debr 69:47–63

    MathSciNet  MATH  Google Scholar 

  • Esary JD, Proschan F (1963) Relationship between system failure rate and component failure rates. Technometrics 5:183–189

    Article  MathSciNet  Google Scholar 

  • Franco M, Ruiz JM, Ruiz MC (2001) On closure of the IFR(2) and NBU(2) classes. J Appl Probab 38:235–241

    Article  MathSciNet  MATH  Google Scholar 

  • Gumbel EJ (1960) Bivariate exponential distribuitons. J Am Stat Assoc 55:698–707

    Article  MATH  Google Scholar 

  • Hendi MI, Mashhour AF, Montasser MA (1993) Closure of the NBUC class under formation of parallel systems. J Appl Probab 30:975–978

    Article  MathSciNet  MATH  Google Scholar 

  • Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London

    Book  MATH  Google Scholar 

  • Kochar SC, Li X, Shaked M (2002) The total time on test transform and the excess wealth stochastic orders of distributions. Adv Appl Probab 34:826–845

    Article  MathSciNet  MATH  Google Scholar 

  • Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions. Wiley, New York

    Book  MATH  Google Scholar 

  • Lai CD, Xie M (2006) Stochastic ageing and dependence for reliability. Springer, New York

    MATH  Google Scholar 

  • Li Y (2004) Closure of NBU(2) class under the formation of parallel systems. Stat Probab Lett 67:57–63

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Kochar SC (2001) Some new results involving the NBU(2) class of life distributions. J Appl Probab 38:242–247

    Article  MathSciNet  MATH  Google Scholar 

  • Li H, Li X (2013) Stochastic orders in reliability and risk. Springer, New York

    Book  Google Scholar 

  • Li C, Li X (2017) Aging and ordering properties of multivariate lifetimes with Archimedean dependence structures. Commun Stat Theory Methods 46:874–891

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Qiu G (2007) Some preservation results of NBUC aging property with applications. Stat Pap 48:581–594

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Li Z, Jing B (2000) Some results about the NBUC class of life distributions. Stat Probab Lett 46:229–237

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Li Z, Jing B (2003) Erratum: some results about NBUC class of life distributions. Stat Probab Lett 61:235–236

    Article  Google Scholar 

  • McNeil AJ, Nešlehová J (2009) Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_1\)-norm symmetric distributions. Ann Stat 37:3059–3097

    Article  MATH  Google Scholar 

  • Müller A, Stoyan D (2001) Stochastic comparison of random vectors with a common copula. Math Oper Res 26:723–740

    Article  MathSciNet  MATH  Google Scholar 

  • Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, West Sussex

    MATH  Google Scholar 

  • Nanda AK, Jain K, Singh H (1998) Preservation of some partial orderings under the formation of coherent systems. Stat Probab Lett 39:123–131

    Article  MathSciNet  MATH  Google Scholar 

  • 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 29:264–278

    Article  MathSciNet  MATH  Google Scholar 

  • 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 30:444–454

    Article  MathSciNet  Google Scholar 

  • Nelsen RB (2006) An introduction to copulas. Springer, New York

    MATH  Google Scholar 

  • Pellerey F, Petakos KI (2002) Closure property of the NBUC class under formation of parallel systems. IEEE Trans Reliab 51:452–454

    Article  Google Scholar 

  • Prékopa A (1971) Logarithmic concave measures with application to stochastic programming. Acta Sci Math (Szeged) 32:301–315

    MathSciNet  MATH  Google Scholar 

  • Samaniego FJ (1985) On closure of the IFR class under formation of coherent systems. IEEE Trans Reliab 34:69–72

    Article  MATH  Google Scholar 

  • Shaked M (1977) A familiy of concepts of dependence for bivariate distributions. J Am Stat Assoc 72:642–650

    Article  MATH  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York

    Book  MATH  Google Scholar 

  • 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 40:273–276

    Article  MATH  Google Scholar 

  • Sordo MA, Navarro J, Sarabia JM (2014) Distorted Lorenz curves: models and comparisons. Soc Choice Welf 42:761–780

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. School of Science, Tianjin University of Commerce, Tianjin, 300134, China

    Chen Li

  2. Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ, 07030, USA

    Xiaohu Li

Authors
  1. Chen Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaohu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chen Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, C., Li, X. Preservation of increasing convex/concave order under the formation of parallel/series system of dependent components. Metrika 81, 445–464 (2018). https://doi.org/10.1007/s00184-018-0651-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-018-0651-6

Keywords

Search

Navigation