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Some new results on stochastic comparisons of coherent systems using signatures

Part of: Sampling theory, sample surveys Nonparametric inference Survival analysis and censored data Distribution theory - Probability

Published online by Cambridge University Press:  04 May 2020

Ebrahim Amini-Seresht ,
Baha-Eldin Khaledi  and
Subhash Kochar
Ebrahim Amini-Seresht*
Affiliation:
Bu-Ali Sina University
Baha-Eldin Khaledi*
Affiliation:
Florida International University
Subhash Kochar*
Affiliation:
Portland State University
*
*Postal address: Department of Statistics, Bu-Ali Sina University, Hamedan, Iran. Email address: [email protected]
**Postal address: Department of Mathematics and Statistics, Florida International University, Miami, USA. Email address: [email protected]
***Postal address: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, USA. Email address: [email protected]
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Abstract

We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed component lifetimes. Some numerical examples are also provided to illustrate the theoretical results.

Keywords

Likelihood ratio orderhazard rate orderreversed hazard rate ordercoherent systems

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62N05: Reliability and life testing 62G30: Order statistics; empirical distribution functions 62D05: Sampling theory, sample surveys
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 156 - 173
Copyright
© Applied Probability Trust 2020

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References

Belzunce, F., Franco, M., Ruiz, J. M. and Ruiz, M. C. (2001). On partial orderings between coherent systems with different structure. Prob. Eng. Inf. Sci. 15, 273–293.CrossRefGoogle Scholar
Gupta, R. C. (2002). Reliability of a k out of n system of components sharing a common environment. Appl. Math. Lett. 15, 837–844.CrossRefGoogle Scholar
Khaledi, B. and Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Planning Infer. 137, 1173–1184.CrossRefGoogle Scholar
Kochar, S., Mukerjee, H. and Samaniego, F. (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
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, New York.Google Scholar
Nanda, A. K., Jain, K. and Singh, H. (1998). Preservation of some partial orderings under the formation of coherent systems. Statist. Prob. Lett. 39, 123–131.CrossRefGoogle Scholar
Navarro, J. (2016). Stochastic comparisons of generalized mixtures and coherent systems. Test 25, 150–169.CrossRefGoogle Scholar
Navarro, J., Aguila, Y. A., Sordo, M. and Suarez-Llorens, A. (2013). Stochastic ordering properties for systems with dependent identically distributed components. Appl. Stoch. Models Business Industry 29, 264–278.CrossRefGoogle Scholar
Navarro, J. and Samaniego, F. J. (2017). An elementary proof of the no “internal zeros” property of system signatures. In Proc. 10th Int. Conf. on Mathematical Methods in Reliability (MMR).Google Scholar
Navarro, J, Samaniego, F. J. and Balakrishnan, N. (2010). The joint signature of coherent systems with shared components. J. Appl. Prob 47, 235–253.CrossRefGoogle Scholar
Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008). Applications and extensions of system signatures in engineering reliability. Naval Res. Logistics 55, 313–327.CrossRefGoogle Scholar
Ross, S., Shahshahani, M. and Weiss, G. (1980). On the number of component failures in systems whose component lives are exchangeable. Math. Operat. Res. 5, 358–365.CrossRefGoogle Scholar
Rychlik, T., Navarro, J. and Rubio, R. (2018). Effective procedure of verifying stochastic ordering of system lifetimes. J. Appl. Prob 55, 1261–1271.CrossRefGoogle Scholar
Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72.CrossRefGoogle Scholar
Samaniego, F. J. (2007). System Signatures and their Applications in Engineering Reliability. Springer, New York.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.10.1007/978-0-387-34675-5CrossRefGoogle Scholar
Zhang, Z. (2010). Ordering conditional general coherent systems with exchangeable components.. J. Statist. Planning Infer. 140, 454–460.CrossRefGoogle Scholar
Zhang, Z. and Meeker, W. Q. (2013). Mixture representations of reliability in coherent systems and preservation results under double monitoring. Commun. Statist. Theory Meth. 42, 385–397.CrossRefGoogle Scholar