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Hazard rate ordering of order statistics and systems

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  14 July 2016

Jorge Navarro  and
Moshe Shaked
Jorge Navarro*
Affiliation:
Universidad de Murcia
Moshe Shaked*
Affiliation:
University of Arizona
*
∗Postal address: Facultad de Matematicas, Universidad de Murcia, 30100 Murcia, Spain. Email address: [email protected]
∗∗Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. Email address: [email protected]
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Abstract

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Let X = (X1, X2, …, Xn) be an exchangeable random vector, and write X(1:i) = min{X1, X2, …, Xi}, 1 ≤ i ≤ n. In this paper we obtain conditions under which X(1:i) decreases in i in the hazard rate order. A result involving more general (that is, 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 the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

Keywords

Exchangeable distributionsignaturecoherent systemhazard rate orderstochastic orderFreund distributionhazard gradientcause specificityFarlie-Gumbel-Morgenstern distribution

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 2 , June 2006 , pp. 391 - 408
Copyright
© Applied Probability Trust 2006 

Footnotes

Partially supported by the Ministerio de Ciencia y Tecnologia under grant BFM2003-02947 and Fundacion Seneca under grant 00698/PI/04.

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