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Stochastic comparisons of coherent systems under different random environments

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

Published online by Cambridge University Press:  26 July 2018

Ebrahim Amini-Seresht ,
Yiying Zhang  and
Narayanaswamy Balakrishnan
Ebrahim Amini-Seresht*
Affiliation:
Bu-Ali Sina University
Yiying Zhang*
Affiliation:
The University of Hong Kong
Narayanaswamy Balakrishnan*
Affiliation:
McMaster University
*
* Postal address: Department of Statistics, Bu-Ali Sina University, Hamedan, Iran.
** Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: zhangyiying@outlook.com
*** Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, ON L85 4K1, Canada.
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Abstract

For many practical situations in reliability engineering, components in the system are usually dependent since they generally work in a collaborative environment. In this paper we build sufficient conditions for comparing two coherent systems under different random environments in the sense of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orders. Applications and numerical examples are provided to illustrate all the theoretical results established here.

Keywords

Coherent systemrandom environmentstochastic orderdistortion functiondependent component

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Secondary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 459 - 472
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Amini-Seresht, E. and Khaledi, B.-E. (2015). Multivariate stochastic comparisons of mixture models. Metrika 78, 1015–1034. 10.1007/s00184-015-0538-8Google Scholar
[2]Badía, F. G., Sangüesa, C. and Cha, J. H. (2014). Stochastic comparison of multivariate conditionally dependent mixtures. J. Multivariate Anal. 129, 82–94. Google Scholar
[3]Balakrishnan, N., Barmalzan, G. and Haidari, A. (2016). Multivariate stochastic comparisons of multivariate mixture models and their applications. J. Multivariate Anal. 145, 37–43. Google Scholar
[4]Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York. Google Scholar
[5]Belzunce, F., Franco, M., Ruiz, J.-M. and Ruiz, M. C. (2001). On partial orderings between coherent systems with different structures. Prob. Eng. Inf. Sci. 15, 273–293. 10.1017/S0269964801152095Google Scholar
[6]Belzunce, F., Mercader, J.-A., Ruiz, J.-M. and Spizzichino, F. (2009). Stochastic comparisons of multivariate mixture models. J. Multivariate Anal. 100, 1657–1669. Google Scholar
[7]Boland, P. J. and Samaniego, F. J. (2004). The signature of a coherent system and its applications in reliability. In Mathematical Reliability: An Expository Perspective, Kluwer, Boston, MA, pp. 3–30. Google Scholar
[8]Cao, J. H. and Wang, Y. D. (1991). The NBUC and NWUC classes of life distributions. J. Appl. Prob. 28, 473–479. (Correction: 29 (1992), 753.) Google Scholar
[9]Deshpande, J. V., Kochar, S. C. and Singh, H. (1986). Aspects of positive ageing. J. Appl. Prob. 23, 748–758. 10.2307/3214012Google Scholar
[10]Fernández-Ponce, J. M., Pellerey, F. and Rodríguez-Griñolo, M. R. (2016). Some stochastic properties of conditionally dependent frailty models. Statistics 50, 649–666. Google Scholar
[11]Franco, M., Ruiz, M. C. and Ruiz, J. M. (2003). A note on closure of the ILR and DLR classes under formation of coherent systems. Statist. Papers 44, 279–288. Google Scholar
[12]Hürlimann, W. (2004). Distortion risk measures and economic capital. N. Amer. Actuarial J. 8, 86–95. 10.1080/10920277.2004.10596130Google Scholar
[13]Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press. Google Scholar
[14]Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467–498. Google Scholar
[15]Kayid, M., Izadkhah, S. and Zuo, M. J. (2017). Some results on the relative ordering of two frailty models. Statist. Papers 58, 287–301. Google Scholar
[16]Kenzin, M. and Frostig, E. (2009). M out of n inspected systems subject to shocks in random environment. Reliab. Eng. System Safety 94, 1322–1330. 10.1016/j.ress.2009.02.005Google Scholar
[17]Khaledi, B.-E. and Shaked, M. (2010). Stochastic comparisons of multivariate mixtures. J. Multivariate Anal. 101, 2486–2498. Google Scholar
[18]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
[19]Li, X. and Da, G. (2010). Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications. J. Multivariate Anal. 101, 1016–1025. Google Scholar
[20]Lindqvist, B. H., Samaniego, F. J. and Huseby, A. B. (2016). On the equivalence of systems of different sizes, with applications to system comparisons. Anal. Appl. Prob. 48, 332–348. Google Scholar
[21]Misra, A. K. and Misra, N. (2012). Stochastic properties of conditionally independent mixture models. J. Statist. Planning Inference 142, 1599–1607. 10.1016/j.jspi.2012.01.012Google Scholar
[22]Misra, N., Gupta, N. and Gupta, R. D. (2009). Stochastic comparisons of multivariate frailty models. J. Statist. Planning Inference 139, 2084–2090. Google Scholar
[23]Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester. Google Scholar
[24]Nakagawa, T. (1979). Further results of replacement problem of a parallel system in random environment. J. Appl. Prob. 16, 923–926. 10.2307/3213159Google Scholar
[25]Nanda, A. K., Singh, H., Misra, N. and Paul, P. (2003). Reliability properties of reversed residual lifetime. Commun. Statist. Theory Meth. 32, 2031–2042. Google Scholar
[26]Navarro, J. and Rubio, R. (2010). Comparisons of coherent systems using stochastic precedence. TEST 19, 469–486. Google Scholar
[27]Navarro, J., Pellerey, F. and Di Crescenzo, A. (2015). Orderings of coherent systems with randomized dependent components. Europ. J. Operat. Res. 240, 127–139. Google Scholar
[28]Navarro, J., del Águila, 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. 10.1002/asmb.1917Google Scholar
[29]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. Google Scholar
[30]Navarro, J., del Águila, Y., Sordo, M. A. and Suárez-Llorens, A. (2016). Preservation of stochastic orders under the formation of generalized distorted distributions: applications to coherent systems. Methodol. Comput. Appl. Prob. 18, 529–545. Google Scholar
[31]Nelsen, R. B. (1996). Nonparametric measures of multivariate association. In Distributions with Fixed Marginals and Related Topics (IMS Lecture Notes Monogr. Ser. 28), Institute of Mathematical Statistics, Hayward, CA, pp. 223–232. 10.1214/lnms/1215452621Google Scholar
[32]Nelsen, R. B. (1999). An Introduction to Copulas. Springer, New York. Google Scholar
[33]Persona, A., Sgarbossa, F. and Pham, H. (2016). Systemability: a new reliability function for different environments. In Quality and Reliability Management and Its Applications, Springer, London, pp. 145–193. 10.1007/978-1-4471-6778-5_6Google Scholar
[34]Petakos, K. and Tsapelas, T. (1997). Reliability analysis for systems in a random environment. J. Appl. Prob. 34, 1021–1031. Google Scholar
[35]Råde, L. (1976). Reliability systems in random environment. J. Appl. Prob. 13, 407–410. Google Scholar
[36]Samaniego, F. J. and Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Anal. Appl. Prob. 48, 88–111. Google Scholar
[37]Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. Google Scholar
[38]Xu, M. and Li, X. (2008). Negative dependence in frailty models. J. Statist. Planning Inference 138, 1433–1441. Google Scholar
[39]Zhang, Y., Amini-Seresht, E. and Ding, W. (2017). Component and system active redundancies for coherent systems with dependent components. Appl. Stoch. Models Business Industry 33, 409–421. Google Scholar