Comparisons of the Expectations of System and Component Lifetimes in the Failure Dependent Proportional Hazard Model

  • Mariusz BieniekEmail author
  • Marco Burkschat
  • Tomasz Rychlik
Open Access


In the failure dependent proportional hazard model, it is assumed that identical components work jointly in a system. At the moments of consecutive component failures the hazard rates of still operating components can change abruptly due to a change of the load acting on each component. The modification of the hazard rate consists in multiplying the original rate by a positive constant factor. Under the knowledge of the system structure and parameters of the failure dependent proportional hazard model, we determine tight lower and upper bounds on the expected differences between the system and component lifetimes, measured in various scale units based on the central absolute moments of the component lifetime. The results are specified for the systems with unimodal Samaniego signatures.


Coherent system Failure dependent proportional hazard model Generalized order statistics Samaniego signature Sharp bound 

Mathematics Subject Classification (2010)

Primary: 62N05 Secondary: 60E15 62G30 



The authors thank to the anonymous referee for mny valuable comments which helped in preparation of the final version of the paper. The first and third authors were supported by National Science Centre of Poland under grant 2015/19/B/ST1/03100.


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OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsMaria Curie Skłodowska UniversityLublinPoland
  2. 2.Institute of StatisticsRWTH Aachen UniversityAachenGermany
  3. 3.Institute of MathematicsPolish Academy of SciencesWarsawPoland

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