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On comparing coherent systems with heterogeneous components

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  24 March 2016

Francisco J. Samaniego  and
Jorge Navarro
Francisco J. Samaniego*
Affiliation:
University of California, Davis
Jorge Navarro*
Affiliation:
Universidad de Murcia
*
* Postal address: Department of Statistics, University of California, Davis, 399 Crocker Lane, 95616 Davis, CA, USA. Email address: fjsamaniego@ucdavis.edu
** Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es
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Abstract

In this paper we investigate different methods that may be used to compare coherent systems having heterogeneous components. We consider both the case of systems with independent components and the case of systems with dependent components. In the first case, the comparisons are based on the new concept of the survival signature due to Coolen and Coolen-Maturi (2012) which extends the well-known concept of system signatures to the case of components with lifetimes that need not be independent and identically distributed. In the second case, the comparisons are based on the concept of distortion functions. A graphical procedure (called an RR-plot) is proposed as an alternative to the analytical methods when there are two types of components.

Keywords

Coherent systemsystem signaturesurvival signaturedistorted distributionstochastic order

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 88 - 111
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1]Al-Nefaiee, A. H. and Coolen, F. P. A. (2013). Nonparametric predictive inference for system failure time based on bounds for the signature. Proc. Inst. Mech. Engineers O 227, 513–522. Google Scholar
[2]Arcones, M. A., Kvam, P. H. and Samaniego, F. J. (2002). Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J. Amer. Statist. Assoc. 97, 170–182. CrossRefGoogle Scholar
[3]Aslett, L. J. M., Coolen, F. P. A. and Wilson, S. P. (2015). Bayesian inference for reliability of systems and networks using the survival signature. Risk Analysis 35, 1640–1651. CrossRefGoogle ScholarPubMed
[4]Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York. Google Scholar
[5]Boland, P. J. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603. CrossRefGoogle Scholar
[6]Coolen, F. P. A. and Al-Nefaiee, A. H. (2012). Nonparametric predictive inference for failure times of systems with exchangeable components. Proc. Inst. Mech. Engineers O 226, 262–273. Google Scholar
[7]Coolen, F. P. A. and Coolen-Maturi, T. (2012). On generalizing the signature to systems with multiple types of components. In Complex Systems and Dependability, eds W. Zamojski et al., Springer, Berlin, pp. 115–130. Google Scholar
[8]Coolen, F. P. A., Coolen-Maturi, T. and Al-Nefaiee, A. H. (2014). Nonparametric predictive inference for system reliability using the survival signature. Proc. Inst. Mech. Engineers O 228, 437–448. Google Scholar
[9]Hollander, M. and Samaniego, F. J. (2008). The use of stochastic precedence in the comparison of engineered systems. In Advances in Mathematical Modeling for Reliability, IOS, Amsterdam, pp. 129–137. Google Scholar
[10]Hürlimann, W. (2004). Distortion risk measures and economic capital. N. Amer. Actuarial J. 8, 86–95. Google Scholar
[11]Kamińska-Zabierowska, M. and Navarro, J. (2015). Erratum to: Mixture representations of residual lifetimes of used systems. J. Appl. Prob. 52, 1183–1186. Google Scholar
[12]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. Google Scholar
[13]Lindqvist, B. H. and Samaniego, F. J. (2014). On the equivalence of systems of different sizes. Submitted. Google Scholar
[14]Marichal, J.-L. and Mathonet, P. (2013). On the extensions of Barlow–Proschan importance index and system signature to dependent lifetimes. J. Multivariate Anal. 115, 48–56. Google Scholar
[15]Navarro, J. and Rubio, R. (2010). Comparisons of coherent systems using stochastic precedence. TEST 19, 469–486. Google Scholar
[16]Navarro, J., Pellerey, F. and Di Crescenzo, A. (2015). Orderings of coherent systems with randomized dependent components. Europ. J. Operat. Res. 240, 127–139. CrossRefGoogle Scholar
[17]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
[18]Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2011). Signature-based representations for the reliability of systems with heterogeneous components. J. Appl. Prob. 48, 856–867. Google Scholar
[19]Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2013). Mixture representations for the joint distribution of lifetimes of two coherent systems with shared components. Adv. Appl. Prob. 45, 1011–1027. CrossRefGoogle Scholar
[20]Navarro, J., del Aguila, Y., Sordo, M. A. and Suárez-Llorens, A. (2013). Stochastic ordering properties for systems with dependent identically distributed components. Appl. Stoch. Models Business Industry 29, 264–278. CrossRefGoogle Scholar
[21]Navarro, J., del Águila, Y., Sordo, M. A. and Suárez-Llorens, A. (2014). Preservation of reliability classes under the formation of coherent systems. Appl. Stoch. Models Business Industry 30, 444–454. CrossRefGoogle Scholar
[22]Navarro, J., del Águila, Y., Sordo, M. A. and Suárez-Llorens, A. (2014). Preservation of stochastic orders under the formation of generalized distorted distributions. Applications to coherent systems and other concepts. Submitted. Google Scholar
[23]Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008). On the application and extension of system signatures in engineering reliability. Naval Res. Logistics 55, 313–327. Google Scholar
[24]Quiggin, J. (1982). A theory of anticipated utility. J. Econom. Behavior Organization 3, 323–343. Google Scholar
[25]Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliability 34, 69–72. CrossRefGoogle Scholar
[26]Samaniego, F. J. (2006). On the comparison of engineered systems of different sizes. In Proc. Army Conf. Appl. Statist. (Durham, NC), NISS. Google Scholar
[27]Samaniego, F. J. (2007). System Signatures and their Applications in Engineering Reliability. Springer, New York. CrossRefGoogle Scholar
[28]Satyanarayana, A. and Prabhakar, A. (1978). New topological formula and rapid algorithm for reliability analysis of complex networks. IEEE Trans. Reliability 27, 82–100. Google Scholar
[29]Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. Google Scholar
[30]Wang, S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bull. 26, 71–92. CrossRefGoogle Scholar
[31]Wang, S. S. and Young, V. R. (1998). Ordering risks: expected utility theory versus Yaari's dual theory of risk. Insurance Math. Econom. 22, 145–161. CrossRefGoogle Scholar
[32]Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica 55, 95–115. Google Scholar