Skip to main content Accessibility help

Stochastic comparisons of coherent systems under different random environments

  • Ebrahim Amini-Seresht (a1), Yiying Zhang (a2) and Narayanaswamy Balakrishnan (a3)


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.


Corresponding author

* 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:
*** Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, ON L85 4K1, Canada.


Hide All
[1]Amini-Seresht, E. and Khaledi, B.-E. (2015). Multivariate stochastic comparisons of mixture models. Metrika 78, 10151034.
[2]Badía, F. G., Sangüesa, C. and Cha, J. H. (2014). Stochastic comparison of multivariate conditionally dependent mixtures. J. Multivariate Anal. 129, 8294.
[3]Balakrishnan, N., Barmalzan, G. and Haidari, A. (2016). Multivariate stochastic comparisons of multivariate mixture models and their applications. J. Multivariate Anal. 145, 3743.
[4]Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.
[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, 273293.
[6]Belzunce, F., Mercader, J.-A., Ruiz, J.-M. and Spizzichino, F. (2009). Stochastic comparisons of multivariate mixture models. J. Multivariate Anal. 100, 16571669.
[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. 330.
[8]Cao, J. H. and Wang, Y. D. (1991). The NBUC and NWUC classes of life distributions. J. Appl. Prob. 28, 473479. (Correction: 29 (1992), 753.)
[9]Deshpande, J. V., Kochar, S. C. and Singh, H. (1986). Aspects of positive ageing. J. Appl. Prob. 23, 748758.
[10]Fernández-Ponce, J. M., Pellerey, F. and Rodríguez-Griñolo, M. R. (2016). Some stochastic properties of conditionally dependent frailty models. Statistics 50, 649666.
[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, 279288.
[12]Hürlimann, W. (2004). Distortion risk measures and economic capital. N. Amer. Actuarial J. 8, 8695.
[13]Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press.
[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, 467498.
[15]Kayid, M., Izadkhah, S. and Zuo, M. J. (2017). Some results on the relative ordering of two frailty models. Statist. Papers 58, 287301.
[16]Kenzin, M. and Frostig, E. (2009). M out of n inspected systems subject to shocks in random environment. Reliab. Eng. System Safety 94, 13221330.
[17]Khaledi, B.-E. and Shaked, M. (2010). Stochastic comparisons of multivariate mixtures. J. Multivariate Anal. 101, 24862498.
[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, 507523.
[19]Li, X. and Da, G. (2010). Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications. J. Multivariate Anal. 101, 10161025.
[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, 332348.
[21]Misra, A. K. and Misra, N. (2012). Stochastic properties of conditionally independent mixture models. J. Statist. Planning Inference 142, 15991607.
[22]Misra, N., Gupta, N. and Gupta, R. D. (2009). Stochastic comparisons of multivariate frailty models. J. Statist. Planning Inference 139, 20842090.
[23]Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.
[24]Nakagawa, T. (1979). Further results of replacement problem of a parallel system in random environment. J. Appl. Prob. 16, 923926.
[25]Nanda, A. K., Singh, H., Misra, N. and Paul, P. (2003). Reliability properties of reversed residual lifetime. Commun. Statist. Theory Meth. 32, 20312042.
[26]Navarro, J. and Rubio, R. (2010). Comparisons of coherent systems using stochastic precedence. TEST 19, 469486.
[27]Navarro, J., Pellerey, F. and Di Crescenzo, A. (2015). Orderings of coherent systems with randomized dependent components. Europ. J. Operat. Res. 240, 127139.
[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, 264278.
[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, 444454.
[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, 529545.
[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. 223232.
[32]Nelsen, R. B. (1999). An Introduction to Copulas. Springer, New York.
[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. 145193.
[34]Petakos, K. and Tsapelas, T. (1997). Reliability analysis for systems in a random environment. J. Appl. Prob. 34, 10211031.
[35]Råde, L. (1976). Reliability systems in random environment. J. Appl. Prob. 13, 407410.
[36]Samaniego, F. J. and Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Anal. Appl. Prob. 48, 88111.
[37]Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
[38]Xu, M. and Li, X. (2008). Negative dependence in frailty models. J. Statist. Planning Inference 138, 14331441.
[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, 409421.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed