Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T07:56:15.112Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Orderings and Ageing Properties of Residual Life Lengths of Live Components in (n-k+1)-Out-Of-n Systems

Part of: Operations research and management science Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Narayanaswamy Balakrishnan ,
Ghobad Barmalzan  and
Abedin Haidari
Narayanaswamy Balakrishnan*
Affiliation:
McMaster University
Ghobad Barmalzan*
Affiliation:
Zabol University
Abedin Haidari*
Affiliation:
Zabol University
*
∗ Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada. Email address: bala@mcmaster.ca
∗∗ Postal address: Department of Statistics, Zabol University, Sistan and Baluchestan, Iran.
∗∗ Postal address: Department of Statistics, Zabol University, Sistan and Baluchestan, Iran.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose that a system consists of n independent and identically distributed components and that the life lengths of the n components are Xi, i = 1, …, n. For k ∈ {1, …, n - 1}, let X(k)1, …, X(k)n-k be the residual life lengths of the live components following the kth failure in the system. In this paper we extend various stochastic ordering results presented in Bairamov and Arnold (2008) on the residual life lengths of the live components in an (n - k + 1)-out-of-n system, and also present a new result concerning the multivariate stochastic ordering of live components in the two-sample situation. Finally, we also characterize exponential distributions under a weaker condition than those introduced in Bairamov and Arnold (2008) and show that some special ageing properties of the original residual life lengths get preserved by residual life lengths.

Keywords

(n-k+1)-out-of-n systemstochastic orderingmultivariate mixturedecreasing hazard ratelive componentorder statistics

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 58 - 68
Copyright
© Applied Probability Trust 

References

An, M. Y. (1998). Logconcavity versus logconvexity: a complete characterization. J. Econom. Theory 80, 350–369.CrossRefGoogle Scholar
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. John Wiley, New York.Google Scholar
Bagai, I. and Kochar, S. C. (1986). On tail-ordering and comparison of failure rates. Commun. Statist. Theory Meth. 15, 1377–1388.CrossRefGoogle Scholar
Bairamov, I. and Arnold, B. C. (2008). On the residual lifelengths of the remaining components in an n - k + 1 out of n system. Statist. Prob. Lett. 78, 945–952.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Bartoszewicz, J. (1986). Dispersive ordering and the total time on test transformation. Statist. Prob. Lett. 4, 285–288.CrossRefGoogle Scholar
Belzunce, F., Mercader, J.-A., Ruiz, J.-M. and Spizzichino, F. (2009). Stochastic comparisons of multivariate mixture models. J. Multivariate Anal. 100, 1657–1669.CrossRefGoogle Scholar
Boland, P. J., El-Neweihi, E. and Proschan, F. (1994). Applications of the hazard rate ordering in reliability and order statistics. J. Appl. Prob. 31, 180–192.CrossRefGoogle Scholar
Boland, P. J., Shaked, M. and Shanthikumar, J. G. (1998). Stochastic ordering of order statistics. In Order Statistics: Theory & Methods (Handbook Statist. 16), North-Holland, Amsterdam, pp. 89–103.Google Scholar
Cramer, E. and Kamps, U. (1996). Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Ann. Inst. Statist. Math. 48, 535–549.CrossRefGoogle Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Hu, T. and Wei, Y. (2001). Stochastic comparisons of spacings from restricted families of distributions. Statist. Prob. Lett. 53, 91–99.CrossRefGoogle Scholar
Hu, T. and Zhuang, W. (2006). Stochastic comparisons of m-spacings. J. Statist. Planning Infer. 136, 33–42.CrossRefGoogle Scholar
Kamps, U. (1995). A Concept of Generalized Order Statistics. Teubner, Stuttgart.CrossRefGoogle Scholar
Khaledi, B.-E. and Kochar, S. (1999). Stochastic orderings between distributions and their sample spacings. II. Statist. Prob. Lett. 44, 161–166.CrossRefGoogle Scholar
Khaledi, B.-E. and Kochar, S. (2000a). On dispersive ordering between order statistics in one-sample and two-sample problems. Statist. Prob. Lett. 46, 257–261.CrossRefGoogle Scholar
Khaledi, B.-E. and Kochar, S. (2000b). Some new results on stochastic comparisons of parallel systems. J. Appl. Prob. 37, 1123–1128.CrossRefGoogle Scholar
Kirmani, S. N. U. A. (1996). On sample spacings from IMRL distributions. Statist. Prob. Lett. 29, 159–166. (Correction: 29 (1997), 159–166.)CrossRefGoogle Scholar
Kochar, S. C. (1999). On stochastic orderings between distributions and their sample spacings. Statist. Prob. Lett. 42, 345–352.CrossRefGoogle Scholar
Lai, C.-D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Google Scholar
Li, X. (2005). A note on expected rent in auction theory. Operat. Res. Lett. 33, 531–534.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
Misra, A. K. and Misra, N. (2012). Stochastic properties of conditionally independent mixture models. J. Statist. Planning Infer. 142, 1599–1607.CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Păltănea, E. (2008). On the comparison in hazard rate ordering of fail-safe systems. J. Statist. Planning Infer. 138, 1993–1993.CrossRefGoogle Scholar
Paul, A. and Gutierrez, G. (2004). Mean sample spacings, sample size and variability in auction-theoretic framework. Operat. Res. Lett. 32, 103–108.CrossRefGoogle Scholar
Pledger, G. and Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Optimizing Methods in Statistics, Academic Press, New York, pp. 89–113.Google Scholar
Proschan, F. and Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J. Multivariate Anal. 6, 608–616.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Spizzichino, F. (2001). Subjective Probability Models for Lifetimes. Chapman & Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Xu, M. and Li, X. (2006). Likelihood ratio order of m-spacings for two samples. J. Statist. Planning Infer. 136, 4250–4258.CrossRefGoogle Scholar
Zhao, P. and Balakrishnan, N. (2009). Characterization of MRL order of fail-safe systems with heterogeneous exponential components. J. Statist. Planning Infer. 139, 3027–3037.CrossRefGoogle Scholar
Zhao, P., Li, X. and Balakrishnan, N. (2009). Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables. J. Multivariate Anal. 100, 952–962.CrossRefGoogle Scholar