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Hazard rate ordering of k-out-of-n systems

Author

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  • Shaked, Moshe
  • George Shanthikumar, J.

Abstract

Let {X1, X2, ..., Xn} be a set of independent continuous random lifetimes and let {Y1, Y2, ..., Yn} be another set of independent continuous random lifetimes. Let X(1) [less-than-or-equals, slant] X(2) [less-than-or-equals, slant] ... [less-than-or-equals, slant] X(n) be the order statistics associated with the Xi's, and let Y(1) [less-than-or-equals, slant] Y(2) [less-than-or-equals, slant] ... [less-than-or-equals, slant] Y(n) be the order statistics associated with the Yi's. Several recent papers have given conditions under which X(k) [less-than-or-equals, slant]hr Y(k), K = 1, 2, ..., n, where '[less-than-or-equals, slant]hr' denotes the hazard rate order. For example, it is known that if the Xi's are independent and identically distributed, and if the Yi's are independent and identically distributed, and if Xi [less-than-or-equals, slant]hr Yi, I = 1, 2, ... n, then X(k) [less-than-or-equals, slant]hr Y(k). In fact, if the Xi's are not necessarily identically distributed and the Yi's are not necessarily identically distributed, but X[alpha] [less-than-or-equals, slant]hr Y[beta], [alpha], [beta] = 1, 2, ..., n, then it is still true that X(k) [less-than-or-equals, slant]hr Y(k), K = 1, 2, ..., n. The purpose of this paper is to obtain an even stronger result. Our proof is different than the other proofs in the literature and is more intuitive. Some applications in reliability theory are given.

Suggested Citation

  • Shaked, Moshe & George Shanthikumar, J., 1995. "Hazard rate ordering of k-out-of-n systems," Statistics & Probability Letters, Elsevier, vol. 23(1), pages 1-8, April.
    • Handle: RePEc:eee:stapro:v:23:y:1995:i:1:p:1-8
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    References listed on IDEAS

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    1. Shanthikumar, J. George & Yao, David D., 1986. "The preservation of likelihood ratio ordering under convolution," Stochastic Processes and their Applications, Elsevier, vol. 23(2), pages 259-267, December.
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    Cited by:

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    2. J. Jarrahiferiz & M. Kayid & S. Izadkhah, 2019. "Stochastic properties of a weighted frailty model," Statistical Papers, Springer, vol. 60(1), pages 53-72, February.

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