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Hazard rate ordering of order statistics and systems

Part of: Special processes Distribution theory - Probability

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: jorgenav@um.es
∗∗Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. Email address: shaked@math.arizona.edu
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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.

References

Baggs, G. E. and Nagaraja, H. N. (1996). Reliability properties of order statistics from bivariate exponential distributions. Commun. Statist. Stoch. Models 12, 611–631.Google 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, 269–288.Google 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, 721–740.Google Scholar
Block, H. W., Savits, T. H. and Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10, 765–772.Google Scholar
Freund, J. E. (1961). A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971–977.Google Scholar
Holt, J. D. (1978). Competing risk analyses with special reference to matched pair experiments. Biometrika 65, 159–165.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1975). A vector multivariate hazard rate. J. Multivariate Anal. 5, 53–66, 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, 499–516.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, 507–523.3.0.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, 137–142.Google Scholar
Li, Y. (2005). Asymptotic baseline of the hazard rate function of mixtures. J. Appl. Prob. 42, 892–901.CrossRefGoogle Scholar
Marshall, A. W. (1975). Some comments on the hazard gradient. Stoch. Process. Appl. 3, 293–300.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, 511–531.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, 179–185.CrossRefGoogle Scholar
Prentice, R. L. et al. (1978). The analysis of failure times in the presence of competing risks. Biometrics 34, 541–554.Google Scholar
Samaniego, F. J. (1985). On the closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72.Google Scholar
Scarsini, M. and Shaked, M. (1999). Distributions with known initial hazard rate functions. J. Statist. Planning Infer. 78, 39–55.Google 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, 693–702.CrossRefGoogle Scholar