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**Some stochastic comparisons of the conditional coherent systems.**
*(English)*
Zbl 1198.62147

For a coherent system with \(n\) independent and identical components having life times \(X_1,\dots, X_{n}\), the life time can be expressed as \(T=\tau(X_1,\dots, X_{n})\), where \(\tau\) is a structural function. The signature of the system is defined as the probability vector \((p_1,\dots,p_{n})\) with \(p_{i}=\)(number of orderings for which the \(i\) th failure causes the system failure)/\(n!\). The life time distribution of the residual life and the inactivity time of a coherent system with i.i.d. components depend on the system’s structural design solely through its signature. The authors obtain some stochastic comparisons of residual lives and inactivity times, respectively, between two systems in the hazard rate order and the likelihood ratio order.

Reviewer: A. D. Borisenko (Kyïv)

### Keywords:

hazard rate order; inactivity time; likelihood ratio order; residual life; reversed hazard rate order; signature
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\textit{X. Li} and \textit{Z. Zhang}, Appl. Stoch. Models Bus. Ind. 24, No. 6, 541--549 (2008; Zbl 1198.62147)

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### References:

[1] | Nanda, Preservation of some partial orderings under the formation of coherent systems, Statistics and Probability Letters 39 pp 123– (1998) · Zbl 1094.62519 |

[2] | Belzunce, On partial orderings between coherent systems with different structures, Probability in the Engineering and Informational Sciences 15 pp 273– (2001) · Zbl 0984.60026 |

[3] | Navarro, A note on comparisons among coherent systems with dependent components using signatures, Statistics and Probability Letters 72 pp 179– (2005) · Zbl 1068.60026 |

[4] | Kochar, The signature of a coherent system and its application to comparisons among systems, Naval Research Logistics 46 pp 507– (1999) · Zbl 0948.90067 |

[5] | Navarro, Hazard rate ordering of order statistics and systems, Journal of Applied Probability 43 pp 391– (2006) · Zbl 1111.62098 |

[6] | Li, Stochastic comparisons on residual life and inactivity time of series and parallel systems, Probability in the Engineering and Informational Sciences 17 pp 267– (2003) · Zbl 1037.60017 |

[7] | Asadi, Proceedings of the 7th Iranian Statistical Conference 17 pp 55– (2004) |

[8] | Khaledi, Ordering conditional lifetimes of coherent systems, Journal of Statistical Planning and Inference 137 pp 1173– (2007) · Zbl 1111.60012 |

[9] | Li, Some aging properties of the residual life of k-out-of-n system, IEEE Transactions on Reliability 55 pp 535– (2006) |

[10] | Asadi, The mean residual life function of a k-out-of-n structure at the system level, IEEE Transactions on Reliability 2 pp 314– (2006) |

[11] | Navarro, Mean residual lifetimes of consecutive-k-out-of-n systems, Journal of Applied Probability 44 pp 82– (2007) · Zbl 1135.62084 |

[12] | Lanberg, Characterizations of nonparametric classes of life distributions, The Annals of Probability 8 pp 1163– (1980) |

[13] | Belzunce, On aging properties based on the residual life of k-out-of-n systems, Probability in the Engineering and Informational Sciences 13 pp 193– (1999) · Zbl 0973.60103 |

[14] | Li, On the behaviour of some new ageing properties based upon the residual life of k-out-of-n systems, Journal of Applied Probability 39 pp 426– (2002) · Zbl 1003.62089 |

[15] | Li, Aging properties of the residual life length of k-out-of-n systems with independent but non-identical components, Applied Stochastic Models in Business and Industry 20 pp 143– (2004) · Zbl 1060.62115 |

[16] | Bairamov, A residual life function of a system having parallel or series structures, Journal of Statistical Theory and Applications 1 pp 119– (2002) |

[17] | Asadi, A note on the mean residual life function of a parallel system, Communications in Statistics Theory and Methods 34 pp 475– (2005) · Zbl 1062.62228 |

[18] | Asadi, On the mean past lifetime of components of a parallel system, Journal of Statistical Planning and Inference 136 pp 1197– (2006) · Zbl 1088.62120 |

[19] | Hu, Ordering conditional distributions of generalized order statistics, Probability in the Engineering and Informational Sciences 21 pp 401– (2007) · Zbl 1125.60015 |

[20] | Shaked, Stochastic Orders (2007) |

[21] | Esary, Coherent life functions, SIAM Journal on Applied Mathematics 18 pp 810– (1970) · Zbl 0198.24804 |

[22] | Barlow, Statistical Theory of Reliability and Life Testing (1975) |

[23] | Samaniego, On closure of the IFR class under formation of coherent systems, IEEE Transactions on Reliability 34 pp 69– (1985) · Zbl 0585.62169 |

[24] | Karlin, Total Positivity (1968) |

[25] | Hu, Some new results on ordering conditional distributions of generalized order statistics, Statistics 21 pp 401– (2007) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.