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Effective procedure of verifying stochastic ordering of system lifetimes

  • Tomasz Rychlik (a1), Jorge Navarro (a2) and Rafael Rubio (a2)

Abstract

The Samaniego signature is a relevant tool for studying the performance of a system whose component lifetimes are exchangeable. It is well known that the stochastic ordering of the signatures of two systems implies the same for the respective system lifetimes. We prove that the reverse claim is not true when the component lifetimes are independent and identically distributed. There exist small proportions of systems with stochastically ordered lifetimes whose signatures are not ordered. We present a simple procedure in order to check whether the system lifetimes are stochastically ordered even if their signatures are not comparable.

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Corresponding author

* Postal address: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00656 Warsaw, Poland. Email address: trychlik@impan.pl
** Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain.
*** Email address: jorgenav@um.es
**** Email address: rafael.rubio.baeza@gmail.com

References

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[1]Boland, P. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597603.
[2]Boland, P. and Samaniego, F. J. (2004). The signature of a coherent system and its applications in reliability. In Mathematical Reliability: An Expository Perspective, eds R. Soyer, T. A. Mazzuchi, and N. D. Singpurwalla (Internat. Ser. Operat. Res. Managament Sci. 67), Kluwer, Boston, pp. 129.
[3]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.
[4]Navarro, J. (2016). Stochastic comparisons of generalized mixtures and coherent systems. Test 25, 150169.
[5]Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008). On the application and extension of system signatures to problems in engineering reliability. Naval Res. Logistics 55, 313327.
[6]Navarro, J. and Rubio, R. (2009). Computations of signatures of coherent systems with five components. Commun. Statist. Simul. Comput. 39, 6884.
[7]Navarro, J. and Rubio, R. (2011). A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components. Naval Res. Logistics 58, 478489.
[8]Navarro, J. and Samaniego, F. J. (2017). An elementary proof of the “no internal zeros” property of system signatures. Preprint. Available a https://www.researchgate.net/publication/314208606.
[9]Ross, S. M., Shahshahani, M.and Weiss, G. (1980). On the number of component failures in systems whose component lives are exchangeable. Math. Operat. Res. 5, 358365.
[10]Rychlik, T. (2001). Projecting Statistical Functionals (Lecture Notes Statist. 160), Springer, New York.
[11]Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. R-34, 6972.

Keywords

MSC classification

Effective procedure of verifying stochastic ordering of system lifetimes

  • Tomasz Rychlik (a1), Jorge Navarro (a2) and Rafael Rubio (a2)

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