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**Hazard rate ordering of order statistics and systems.**
*(English)*
Zbl 1111.62098

Let \((X_1,X_2,\dots,X_n)\) be an exchangeable random vector and

\[ {X_{(1:i)}}=\min[X_1,X_2, \dots, X_i],\qquad 1\leq i \leq n. \]

In the paper conditions under which \({X_{(1:i)}}\) decreases in \(i\) with respect to the hazard rate order are obtained. A result involving more general (i.e., not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in this paper. An interesting relationship between these two signatures is derived. The results are illustrated in a series of examples.

\[ {X_{(1:i)}}=\min[X_1,X_2, \dots, X_i],\qquad 1\leq i \leq n. \]

In the paper conditions under which \({X_{(1:i)}}\) decreases in \(i\) with respect to the hazard rate order are obtained. A result involving more general (i.e., not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in this paper. An interesting relationship between these two signatures is derived. The results are illustrated in a series of examples.

Reviewer: Jaroslaw Bartoszewicz (Wrocław)

### MSC:

62N05 | Reliability and life testing |

60E15 | Inequalities; stochastic orderings |

62G30 | Order statistics; empirical distribution functions |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60G09 | Exchangeability for stochastic processes |

62E20 | Asymptotic distribution theory in statistics |

### Keywords:

exchangeable distribution; signature; coherent system; hazard rate order; Freund distribution; hazard gradient; cause specificity; Farlie-Gumbel-Maorgenstern distribution
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\textit{J. Navarro} and \textit{M. Shaked}, J. Appl. Probab. 43, No. 2, 391--408 (2006; Zbl 1111.62098)

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