Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T22:59:26.966Z Has data issue: false hasContentIssue false
  • English
  • Français

Effective procedure of verifying stochastic ordering of system lifetimes

Part of: Survival analysis and censored data Nonparametric inference Distribution theory - Probability

Published online by Cambridge University Press:  16 January 2019

Tomasz Rychlik ,
Jorge Navarro  and
Rafael Rubio
Tomasz Rychlik*
Affiliation:
Polish Academy of Sciences
Jorge Navarro*
Affiliation:
Universidad de Murcia
Rafael Rubio*
Affiliation:
Universidad de Murcia
*
* Postal address: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00656 Warsaw, Poland. Email address: [email protected]
** Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain.
** Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Coherent systemmixed systemSamaniego signaturestochastic order

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 62N05: Reliability and life testing
Secondary: 62G30: Order statistics; empirical distribution functions
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1261 - 1271
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Boland, P. (2001). Signatures of indirect majority systems. J. Appl. Prob. 38, 597–603.Google Scholar
[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. 1–29.Google Scholar
[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, 507–523.Google Scholar
[4]Navarro, J. (2016). Stochastic comparisons of generalized mixtures and coherent systems. Test 25, 150–169.Google Scholar
[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, 313–327.Google Scholar
[6]Navarro, J. and Rubio, R. (2009). Computations of signatures of coherent systems with five components. Commun. Statist. Simul. Comput. 39, 68–84.Google Scholar
[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, 478–489.Google Scholar
[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.Google Scholar
[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, 358–365.Google Scholar
[10]Rychlik, T. (2001). Projecting Statistical Functionals (Lecture Notes Statist. 160), Springer, New York.Google Scholar
[11]Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. R-34, 69–72.Google Scholar