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Representations of component importance for coherent systems with exchangeable components

Part of: Special processes Survival analysis and censored data Distribution theory - Probability

Published online by Cambridge University Press:  16 July 2020

S. Pitzen  and
M. Burkschat
S. Pitzen*
Affiliation:
RWTH Aachen University
M. Burkschat*
Affiliation:
RWTH Aachen University
*
*Postal address: Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany.
*Postal address: Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany.
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Abstract

Two definitions of Birnbaum’s importance measure for coherent systems are studied in the case of exchangeable components. Representations of these measures in terms of distribution functions of the ordered component lifetimes are given. As an example, coherent systems with failure-dependent component lifetimes based on the notion of sequential order statistics are considered. Furthermore, it is shown that the two measures are ordered in the case of associated component lifetimes. Finally, the limiting behavior of the measures with respect to time is examined.

Keywords

coherent systemexchangeabilitycomponent importancesignatureorder statisticssequential order statisticsassociation

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 385 - 406
Copyright
© Applied Probability Trust 2020

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References

Aki, S. and Hirano, K. (1997). Lifetime distributions of consecutive-k-out-of-n:F systems. Nonlin. Anal. 30, 555–562.CrossRefGoogle Scholar
Aliee, H., Borgonovo, E., Glaß, M. and Teich, J. (2017). On the boolean extension of the Birnbaum importance to non-coherent systems. Rel. Eng. Syst. Safety 160, 191–200.10.1016/j.ress.2016.12.013CrossRefGoogle Scholar
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. Wiley, New York.Google Scholar
Balakrishnan, N., Beutner, E. and Kamps, U. (2011). Modeling parameters of a load-sharing system through link functions in sequential order statistics models and associated inference. IEEE Trans. Reliab. 60, 605–611.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Importance of system components and fault tree events. Stoch. Process. Appl. 3, 153–172.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing: Probability Models. To Begin With, Silver Spring, MD.Google Scholar
Bedbur, S., Johnen, M. and Kamps, U. (2019). Inference from multiple samples of Weibull sequential order statistics. J. Multivar. Anal. 169, 381–399.CrossRefGoogle Scholar
Bezgina, E. and Burkschat, M. (2019). On total positivity of exchangeable random variables obtained by symmetrization, with applications to failure-dependent lifetimes. J. Multivar. Anal. 169, 95–109.10.1016/j.jmva.2018.08.015CrossRefGoogle Scholar
Birnbaum, Z. W. (1969). On the importance of different components in a multicomponent system. In Multivariate Analysis II, ed. P. R. Krishnaiah. Academic Press, New York, pp. 581–592.Google Scholar
Burkschat, M. (2009). Systems with failure-dependent lifetimes of components. J. Appl. Prob. 46, 1052–1072.CrossRefGoogle Scholar
Burkschat, M. and Lenz, B. (2009). Marginal distributions of the counting process associated with generalized order statistics. Commun. Statist. Theory Meth. 38, 2089–2106.10.1080/03610920902846513CrossRefGoogle Scholar
Burkschat, M. and Navarro, J. (2011). Aging properties of sequential order statistics. Prob. Eng. Inf. Sci. 25, 449–467.CrossRefGoogle Scholar
Burkschat, M. and Navarro, J. (2013). Dynamic signatures of coherent systems based on sequential order statistics. J. Appl. Prob. 50, 272–287.CrossRefGoogle Scholar
Cramer, E. (2016). Sequential order statistics. In Wiley StatsRef: Statistics Reference Online. Wiley, New York.Google Scholar
Cramer, E. and Kamps, U. (2001). Sequential k-out-of-n systems. In Handbook of Statistics, Vol. 20, Advances in Reliability. eds. N. Balakrishnan and C. R. Rao, Elsevier, Amsterdam, pp. 301–372.Google Scholar
Cramer, E. and Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58, 293–310.CrossRefGoogle Scholar
David, H. A. and Joshi, P. C. (1968). Recurrence relations between moments of order statistics for exchangeable variates. Ann. Math. Statist. 39, 272–274.10.1214/aoms/1177698532CrossRefGoogle Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. Wiley, New York.CrossRefGoogle Scholar
Deshpande, J. V., Dewan, I., Lam, K. F. and Naik-Nimbalkar, U. V. (2019). Tests for specific nonparametric relations between two distribution functions with applications. Appl. Stoch. Models Business Industry 35, 247–259.CrossRefGoogle Scholar
Deshpande, J. V., Dewan, I. and Naik-Nimbalkar, U. V. (2010). A family of distributions to model load sharing systems. J. Statist. Planning Infer. 140, 1441–1451.CrossRefGoogle Scholar
Dewan, I. and Naik-Nimbalkar, U. V. (2011). Load-sharing systems. In Wiley Encyclopedia of Operations Research and Management Science, eds. J. J. Cochran, L. A. Cox, P. Keskinocak, J. P. Kharoufeh and J. C. Smith. Wiley, New York.CrossRefGoogle Scholar
Eryilmaz, S. (2013). Component importance for linear consecutive-k-out-of-n and m-consecutive-k-out-of-n systems with exchangeable components. Naval Res. Logistics 60, 313–320.CrossRefGoogle Scholar
Eryilmaz, S. (2013). Joint reliability importance in linear m-consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 62, 862–869.CrossRefGoogle Scholar
Eryilmaz, S. (2015). Component importance in coherent systems with exchangeable components. J. Appl. Prob. 52, 851–863.CrossRefGoogle Scholar
Eryilmaz, S., Oruc, O. E. and Oger, V. (2016). Joint reliability importance in coherent systems with exchangeable dependent components. IEEE Trans. Reliab. 65, 1562–1570.10.1109/TR.2016.2570543CrossRefGoogle Scholar
Eryilmaz, S., Coolen, F. P. A. and Coolen-Maturi, T. (2018). Marginal and joint reliability importance based on survival signature. Reliab. Eng. Syst. Safety 172, 118–128.CrossRefGoogle Scholar
Esary, J. D. and Marshall, A. W. (1970). Coherent life functions. SIAM J. Appl. Math. 18, 810–814.CrossRefGoogle Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.10.1214/aoms/1177698701CrossRefGoogle Scholar
Hollander, M. and, E. A. (1995). Dynamic reliability models with conditional proportional hazards. Lifetime Data Anal. 1, 377–401.CrossRefGoogle ScholarPubMed
Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables with applications. Ann. Statist. 11, 286–295.10.1214/aos/1176346079CrossRefGoogle Scholar
Kamps, U. (1995). A concept of generalized order statistics. J. Statist. Planning Infer. 48, 1–23.CrossRefGoogle Scholar
Kamps, U. (2014). Generalized order statistics. In Wiley StatsRef: Statistics Reference Online. Wiley, New York.Google Scholar
Kamps, U. and Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35, 269–280.CrossRefGoogle Scholar
Kong, Y. and Ye, Z.-S. (2016). A cumulative-exposure-based algorithm for failure data from a load-sharing system. IEEE Trans. Reliab/ 65, 1001–1013.CrossRefGoogle Scholar
Kuo, W. and Zhu, X. (2012). Importance Measures in Reliability, Risk, and Optimization: Principles and Applications. Wiley, New York.10.1002/9781118314593CrossRefGoogle Scholar
Kuo, W. and Zhu, X. (2012). Relations and generalizations of importance measures in reliability. IEEE Trans. Reliab. 61, 659–674.CrossRefGoogle Scholar
Kuo, W. and Zhu, X. (2012). Some recent advances on importance measures in reliability. IEEE Trans. Reliab. 61, 344–360.10.1109/TR.2012.2194196CrossRefGoogle Scholar
Kvam, P. H. and Pena, E. A. (2005). Estimating load-sharing properties in a dynamic reliability system. J. Am. Statist. Assoc. 100, 262–272.CrossRefGoogle Scholar
Mahmoud, B. and Eryilmaz, S. (2014). Joint reliability importance in a binary k-out-of-n:G system with exchangeable dependent components. Quality Tech. Quant. Manage. 11, 453–460.CrossRefGoogle Scholar
Marichal, J.-L. (2012). Subsignatures of systems. J. Multivar. Anal. 124, 226–236.10.1016/j.jmva.2013.11.002CrossRefGoogle Scholar
Marichal, J.-L. (2014). Computing subsignatures of systems with exchangeable component lifetimes. Statist. Prob. Lett. 94, 128–134.CrossRefGoogle Scholar
Marichal, J.-L. and Mathonet, P. (2013). On the extensions of Barlow–Proschan importance index and system signature to dependent lifetimes. J. Multivar. Anal. 115, 48–56.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1967). A multivariate exponential distribution. J. Am. Statist. Assoc. 62, 30–44.CrossRefGoogle Scholar
Miziuła, P. and Navarro, J. (2019). Birnbaum importance measure for reliability systems with dependent components. IEEE Trans. Reliab. 68, 439–450.CrossRefGoogle Scholar
Navarro, J. and Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Res. Logistics 58, 123–135.10.1002/nav.20445CrossRefGoogle Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2007). Properties of coherent systems with dependent components. Commun. Statist. Theory Meth. 36, 175–191.CrossRefGoogle Scholar
Ramamurthy, K. G. (1990). Coherent Structures and Simple Games. Kluwer, Dordrecht.10.1007/978-94-009-2099-6CrossRefGoogle Scholar
Samaniego, F. J. (2007). System Signatures and their Applications in Engineering Reliability. International Series in OperationsResearch & Management Science. Springer, New York.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1987). Multivariate hazard rates and stochastic ordering. Adv. Appl. Prob. 19, 123–137.CrossRefGoogle Scholar
Sutar, S. S. and Naik-Nimbalkar, U. V. (2014). Accelerated failure time models for load sharing systems. IEEE Trans. Reliab. 63, 706–714.CrossRefGoogle Scholar
Vaurio, J. K. (2016). Importances of components and events in non-coherent systems and risk models. Reliab. Eng. Syst. Safety 147, 117–122.CrossRefGoogle Scholar
Zhang, X. and Wilson, A. (2017). System reliability and component importance under dependence: A copula approach. Technometrics 59, 215–224.CrossRefGoogle Scholar
Zhu, X. and Kuo, W. (2014). Importance measures in reliability and mathematical programming. Ann. Operat. Res. 212, 241–267.CrossRefGoogle Scholar