, Volume 81, Issue 4, pp 445–464 | Cite as

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

  • Chen Li
  • Xiaohu Li


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.


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



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.


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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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