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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
Article
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Abstract

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.

Keywords

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

Mathematics Subject Classification (2010)

Primary: 62N05 Secondary: 60E15 62G30 

Notes

Acknowledgements

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.

References

  1. Aki S, Hirano K (1997) Lifetime distributions of consecutive-k-out-of-n:F systems. Nonlinear Anal 30:555–562MathSciNetCrossRefzbMATHGoogle Scholar
  2. Balakrishnan N, Beutner E, Kamps U (2011) Modeling parameters of a load-sharing system through link functions in sequential order statistics models and associated inference. IEEE Trans Reliab TR-60:605–611CrossRefGoogle Scholar
  3. Bieniek M (2007) Variation diminishing property of densities of uniform generalized order statistics. Metrika 65:297–309MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bieniek M (2008) Projection bounds on expectations of generalized order statistics from DD and DDA families. J Statist Plann Inference 138:971–981MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bieniek M, Burkschat M (2018) On unimodality of the lifetime distribution of coherent systems with failure-dependent component lifetimes. J Appl Probab 55:473–487MathSciNetCrossRefzbMATHGoogle Scholar
  6. Boland P (2001) Signatures of indirect majority systems. J Appl Probab 38:597–603MathSciNetCrossRefzbMATHGoogle Scholar
  7. Burkschat M (2009) Systems with failure-dependent lifetimes of components. J Appl Probab 46:1052–1072MathSciNetCrossRefzbMATHGoogle Scholar
  8. Burkschat M, Navarro J (2013) Dynamic signatures of coherent systems based on sequential order statistics. J Appl Probab 50:272–287MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cramer E (2016) Sequential order statistics. Wiley StatsRef: Statistics Reference Online. 1–7Google Scholar
  10. Cramer E, Kamps U (2001) Sequential k-out-of-n systems. In: Balakrishnan N, Rao CR (eds) Handbook of statistics - advances in reliability, vol 20. Elsevier, Amsterdam, pp 301–372Google Scholar
  11. Cramer E, Kamps U (2003) Marginal distributions of sequential and generalized order statistics. Metrika 58:293–310MathSciNetCrossRefzbMATHGoogle Scholar
  12. Cramer E, Kamps U, Rychlik T (2002) Evaluations of expected generalized order statistics in various scale units. Appl Math 29:285–295MathSciNetzbMATHGoogle Scholar
  13. Cramer E, Kamps U, Rychlik T (2004) Unimodality of uniform generalized order statistics, with applications to mean bounds. Ann Inst Statist Math 56:183–192MathSciNetCrossRefzbMATHGoogle Scholar
  14. Dziubdziela W, Kopociński B (1976) Limiting properties of the k-th record values. Appl Math 15:187–190MathSciNetzbMATHGoogle Scholar
  15. Goroncy A (2014) Bounds on expected generalized order statistics. Statistics 48:593–608MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hollander M, Peña EA (1995) Dynamic reliability models with conditional proportional hazard rates. Lifetime Data Anal 1:377–401MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jasiński K., Navarro J, Rychlik T (2009) Bounds on variances of lifetimes of coherent and mixed systems. J Appl Probab 46:894–908MathSciNetCrossRefzbMATHGoogle Scholar
  18. Kamps U (1995a) A concept of generalized order statistics. J Statist Plann Inference 48:1–23Google Scholar
  19. Kamps U (1995b) A concept of generalized order statistics. Teubner, Stutt-gartGoogle Scholar
  20. Kamps U (2016) Generalized order statistics. Wiley StatsRef: Statistics Reference Online. 1–12Google Scholar
  21. Marichal J-L, Mathonet P, Waldhauser T (2011) On signature-based expressions of system reliability. J Multivariate Anal 102:1410–1416MathSciNetCrossRefzbMATHGoogle Scholar
  22. Marshall AW, Olkin I (2007) Life distributions. Springer, New YorkzbMATHGoogle Scholar
  23. Miziuła P, Rychlik T (2015) Extreme dispersions of semicoherent and mixed system lifetimes. J Appl Probab 52:117–128MathSciNetCrossRefzbMATHGoogle Scholar
  24. Moriguti S (1953) A modification of Schwarz’s inequality with applications to distributions. Ann Math Statist 24:107–113MathSciNetCrossRefzbMATHGoogle Scholar
  25. Navarro J, Balakrishnan N, Samaniego FJ, Bhattacharya D (2008) On the application and extension of system signatures to problems in engineering reliability. Naval Res Logist 55:313–327MathSciNetCrossRefzbMATHGoogle Scholar
  26. Navarro J, Burkschat M (2011) Coherent systems based on sequential order statistics. Naval Res Logist 58:123–135MathSciNetCrossRefzbMATHGoogle Scholar
  27. Rychlik T (1993) Bounds for expectation of L-estimates for dependent samples. Statistics 24:1–7MathSciNetCrossRefzbMATHGoogle Scholar
  28. Rychlik T (2001) Projecting statistical functionals lecture notes in statistics 160. Springer, New YorkCrossRefGoogle Scholar
  29. Samaniego FJ (1985) On closure of the IFR class under formation of coherent systems. IEEE Trans Reliab R-34:69–72CrossRefzbMATHGoogle Scholar

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© The Author(s) 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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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