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Preservation of distributional properties of component lifetimes by system lifetimes

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.

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Acknowledgements

The authors thank three anonymous referees whose valuable comments allowed them to improve the presentation of the paper. The third author has been partially supported by PUT under Grant 0211/SBAD/0121.

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Correspondence to Tomasz Rychlik.

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Arnold, B.C., Rychlik, T. & Szymkowiak, M. Preservation of distributional properties of component lifetimes by system lifetimes. TEST (2022). https://doi.org/10.1007/s11749-022-00808-z

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Keywords

  • Coherent system
  • Mixed system
  • Convex transform order
  • Monotone failure rate
  • Monotone density
  • Diagonally dependent copula

Mathematics Subject Classification

  • 60E15
  • 62N05