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Preservation of distributional properties of component lifetimes by system lifetimes

Author

Listed:
  • Barry C. Arnold

    (University of California)

  • Tomasz Rychlik

    (Polish Academy of Sciences)

  • Magdalena Szymkowiak

    (Poznan University of Technology)

Abstract

We analyze reliability systems with components whose lifetimes are identically distributed, and whose joint distribution admits a Samaniego signature representation of the system lifetime distribution. Our main result is the following. We assume that two systems have the same structure and that the lifetimes of the components of the systems share the same dependence copula. If the first system lifetime precedes (succeeds) its single component lifetime in the convex transform order, and if also the component lifetime of the second system precedes the (succeeds) component lifetime of the first system in the convex transform order then the system-component ordering property is preserved by the second system lifetime, i.e., the system lifetime precedes (succeeds) the component lifetime in the second system also. This allows us to conclude various sufficient and necessary conditions on the system signatures under which the monotone failure rate and density properties of the component lifetimes are inherited by the system lifetime under the condition that the component lifetimes are independent.

Suggested Citation

  • Barry C. Arnold & Tomasz Rychlik & Magdalena Szymkowiak, 2022. "Preservation of distributional properties of component lifetimes by system lifetimes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(4), pages 901-930, December.
    • Handle: RePEc:spr:testjl:v:31:y:2022:i:4:d:10.1007_s11749-022-00808-z
    DOI: 10.1007/s11749-022-00808-z
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    References listed on IDEAS

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    1. Marichal, Jean-Luc & Mathonet, Pierre & Waldhauser, Tamás, 2011. "On signature-based expressions of system reliability," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1410-1416, November.
    2. Sheldon M. Ross & Mehrdad Shahshahani & Gideon Weiss, 1980. "On the Number of Component Failures in Systems Whose Component Lives are Exchangeable," Mathematics of Operations Research, INFORMS, vol. 5(3), pages 358-365, August.
    3. Navarro, Jorge & Rychlik, Tomasz, 2007. "Reliability and expectation bounds for coherent systems with exchangeable components," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 102-113, January.
    4. Jorge Navarro & Francisco J. Samaniego & N. Balakrishnan & Debasis Bhattacharya, 2008. "On the application and extension of system signatures in engineering reliability," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(4), pages 313-327, June.
    5. Okolewski, Andrzej, 2017. "Extremal properties of order statistic distributions for dependent samples with partially known multidimensional marginals," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 1-9.
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