Abstract
Consider an n–component coherent system having random lifetime \(T_{\boldsymbol{X}}\), where \(\boldsymbol{X} = (X_{1},\ldots,X_{n})\) is the vector of the non-independent components’ lifetimes. Stochastic comparisons of the residual life of \(T_{\boldsymbol{X}}\) at a fixed time t ≥ 0, conditioned on \(\{T_{\boldsymbol{X}} > t\}\) or on \(\{X_{i} > t,\forall i = 1,\ldots,n\}\), are investigated. Sufficient conditions on the vector \(\boldsymbol{X}\) that imply this comparison in the usual stochastic order are provided, together with sufficient conditions under which the lifetime \(T_{\boldsymbol{X}}\) satisfies the NBU aging property.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
Barlow, R. E. and Proschan, F.: Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, Maryland (1981)
Bassan, B. and Spizzichino, F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. Journal of Multivariate Analysis, 93, 313–339 (2005)
Deshpande, J. V., Kochar, S. C. and Singh, H.: Aspects of positive ageing. Journal of Applied Probability, 23, 748–758 (1986)
Durante, F., Foschi, R. and Spizzichino, F.: Threshold copulas and positive dependence. Statistics and Probability Letters, 17, 2902–2909 (2008)
Esary, J. D. and Marshall, A. W.: Coherent life functions. SIAM Journal of Applied Mathematics, 18, 810–814 (1970)
Franco, M., Ruiz, J. M. and Ruiz, M. C.: On the closure of the IFR(2) and NBU(2) classes. Journal of Applied Probability, 38, 235–241 (2001)
Hu, T. and Li, Y.: Increasing failure rate and decreasing reversed hazard rate properties of the minimum and the maximum of multivariate distributions with log-concave densities. Metrika, 65, 325–330 (2007)
Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London (1997)
Khaledi, B.-E. and Shaked, M.: Ordering conditional lifetimes of coherent systems. Journal of Statistical Planning and Inference, 137, 1173–1184 (2007)
Lai, C. D. and Xie, M.: Stochastic Ageing and Dependence for Reliability. Springer, New York (2006)
Li, X. and Chen, J.: Aging properties of the residual life of k-out-of-n systems with independent but non-identical components. Applied Stochastic Models in Business and Industry, 20, 143–153 (2004)
Li, X. and Kochar, S. C.: Some new results of NBU(2) class of life distributions. Journal of Applied Probability, 38, 242–237 (2001)
Li, X., Li, Z. and Jing, B-Y.: Some results about the NBUC class of life distributions. Statistics and Probability Letters, 46, 229–237 (2000)
Li, X. and Lu, J.: Stochastic comparisons on residual life and inactivity time of series and parallel systems. Probability in Engineering and Informational Sciences, 17, 267–275 (2003)
Li, X. and Pellerey, F.: Generalized Marshall-Olkin distributions, and related bivariate aging properties. Journal of Multivariate Analysis, 102, 1399–1409 (2011)
Marshall, A. W. and Olkin, I.: Inequalities: theory of majorization and its applications. Academic Press, Inc., New York (1979)
Müller, A. and Scarsini, M.: Archimedean copulae and positive dependence. Journal of Multivariate Analysis, 93, 434–445 (2005)
Navarro, J., Balakrishnan, N. and Samaniego, F. J.: Mixture representations of residual lifetimes of used systems. Journal of Applied Probability, 45, 1097–1112 (2008)
Navarro, J. and Shaked M.: Some properties of the minimum and the maximum of random variables with joint logconcave distributions. Metrika, 71, 313–317 (2010)
Norros, I.: System weakened by failures. Stochastic Processes and their Applications, 20, 181–196 (1985)
Pellerey, F.: On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas. Kybernetica, 44, 795–806 (2008)
Pellerey, F. and Petakos, K.: On closure property of the NBUC class under formation of parallel systems. IEEE Transaction on Reliability, 51, 452–454 (2002)
Samaniego, F. J.: On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability, 34, 69–72 (1985)
Samaniego, F. J., Balakrishnan, N. and Navarro, J.: Dynamic signatures and their use in comparing the reliability of new and used systems. Naval Research Logistics, 56, 577–591 (2009)
Shaked, M. and Shanthikumar, J. G.: Dynamic multivariate aging notions in reliability theory. Stochastic Processes and their Applications, 38, 85–97 (1991)
Shaked, M. and Shanthikumar, J. G.: Stochastic Orders. Springer, New York (2007)
Zhang, Z.: Ordering conditional general coherent systems with exchangeable components. Journal of Statistical Planning and Inference, 140, 454–460 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Li, X., Pellerey, F., You, Y. (2013). On Used Systems and Systems with Used Components. In: Li, H., Li, X. (eds) Stochastic Orders in Reliability and Risk. Lecture Notes in Statistics(), vol 208. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6892-9_11
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6892-9_11
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6891-2
Online ISBN: 978-1-4614-6892-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)