Skip to main content Accessibility help
×
Home
Hostname: page-component-5cfd469876-kgr8m Total loading time: 0.227 Render date: 2021-06-24T10:33:37.287Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Hazard rate ordering of order statistics and systems

Published online by Cambridge University Press:  14 July 2016

Jorge Navarro
Affiliation:
Universidad de Murcia
Moshe Shaked
Affiliation:
University of Arizona
Corresponding
Rights & Permissions[Opens in a new window]

Abstract

Let X = (X 1, X 2, …, X n ) be an exchangeable random vector, and write X (1:i) = min{X 1, X 2, …, X i }, 1 ≤ in. 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.

Type
Research Papers
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.

References

Baggs, G. E. and Nagaraja, H. N. (1996). Reliability properties of order statistics from bivariate exponential distributions. Commun. Statist. Stoch. Models 12, 611631.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Block, H. and Joe, H. (1997). Tail behavior of the failure rate functions of mixtures. Lifetime Data Anal. 3, 269288.CrossRefGoogle Scholar
Block, H. W., Li, Y. and Savits, T. H. (2003). Initial and final behaviour of failure rate functions for mixtures and systems. J. Appl. Prob. 40, 721740.CrossRefGoogle Scholar
Block, H. W., Savits, T. H. and Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10, 765772.CrossRefGoogle Scholar
Freund, J. E. (1961). A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971977.CrossRefGoogle Scholar
Holt, J. D. (1978). Competing risk analyses with special reference to matched pair experiments. Biometrika 65, 159165.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1975). A vector multivariate hazard rate. J. Multivariate Anal. 5, 5366, 498.CrossRefGoogle Scholar
Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. II. Multivariate reverse rule distributions. J. Multivariate Anal. 10, 499516.CrossRefGoogle Scholar
Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999). The “signature” of a coherent system and its application to comparisons among systems. Naval Res. Logistics 46, 507523.3.0.CO;2-D>CrossRefGoogle Scholar
Korwar, R. (2003). On stochastic orders for the lifetime of a k-out-of-n system. Prob. Eng. Inf. Sci. 17, 137142.CrossRefGoogle Scholar
Li, Y. (2005). Asymptotic baseline of the hazard rate function of mixtures. J. Appl. Prob. 42, 892901.CrossRefGoogle Scholar
Marshall, A. W. (1975). Some comments on the hazard gradient. Stoch. Process. Appl. 3, 293300.CrossRefGoogle Scholar
Navarro, J. and Hernandez, P. J. (2004a). How to obtain bathtub-shaped failure rate models from normal mixtures. Prob. Eng. Inf. Sci. 18, 511531.CrossRefGoogle Scholar
Navarro, J. and Hernandez, P. J. (2004b). Properties of the failure rate functions of generalized mixtures. Tech. Rep., Facultad de Matematicas, Universidad de Murcia.Google Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2004). Properties of coherent systems with dependent components. To appear in Commun. Statist. Theory Meth.Google Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2005). A note on comparisons among coherent systems with dependent components using signatures. Statist. Prob. Lett. 72, 179185.CrossRefGoogle Scholar
Prentice, R. L. et al. (1978). The analysis of failure times in the presence of competing risks. Biometrics 34, 541554.CrossRefGoogle ScholarPubMed
Samaniego, F. J. (1985). On the closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 6972.CrossRefGoogle Scholar
Scarsini, M. and Shaked, M. (1999). Distributions with known initial hazard rate functions. J. Statist. Planning Infer. 78, 3955.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
Shaked, M. and Suarez-Llorens, A. (2003). On the comparison of reliability experiments based on the convolution order. J. Amer. Statist. Assoc. 98, 693702.CrossRefGoogle Scholar
You have Access
16
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Hazard rate ordering of order statistics and systems
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Hazard rate ordering of order statistics and systems
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Hazard rate ordering of order statistics and systems
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *