Journal of Applied Probability

A note on the mixture representation of the conditional residual lifetime of a coherent system

Xiuying Feng, Shuhong Zhang, and Xiaohu Li

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

Article information

J. Appl. Probab., Volume 50, Number 2 (2013), 475-485.

First available in Project Euclid: 19 June 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E15: Exact distribution theory
Secondary: 60k10

Coherent system order statistics residual lifetime signature stochastic order


Feng, Xiuying; Zhang, Shuhong; Li, Xiaohu. A note on the mixture representation of the conditional residual lifetime of a coherent system. J. Appl. Probab. 50 (2013), no. 2, 475--485. doi:10.1239/jap/1371648955.

Export citation


  • Balakrishnan, N., Belzunce, F., Hami, N. and Khaledi, B. E. (2010). Univariate and multivariate likelihood ratio ordering of generalized order statistics and associated conditional variables. Prob. Eng. Inf. Sci. 24, 441–455.
  • Belzunce, F., Gurler, S. and Ruiz, J. M. (2011). Revisiting multivariate likelihood ratio ordering results for order statistics. Prob. Eng. Inf. Sci. 25, 355–368.
  • Ery\ilmaz, S. (2013). On residual lifetime of coherent systems after the $r$th failure. Statist. Papers 54, 243–250.
  • Gertsbakh, I., Shpungin, Y. and Spizzichino, F. (2012). Two-dimensional signatures. J. Appl. Prob. 49, 416–429.
  • Goliforushani, S., Asadi, M. and Balakrishnan, N. (2012). On the residual and inactivity times of the components of used coherent systems. J. Appl. Prob. 49, 385–404.
  • Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press.
  • Khaledi, B.-E. and Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Planning Infer. 137, 1173–1184.
  • Kochar, S. and Xu, M. (2010). On residual lifetimes of $k$-out-of-$n$ systems with nonidentical components. Prob. Eng. Inf. Sci. 24, 109–127.
  • 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.
  • Li, X. and Zhang, Z. (2008). Some stochastic comparisons of conditional coherent systems. Appl. Stoch. Models Business Industry 24, 541–549.
  • Mahmoudi, M. and Asadi, M. (2011). The dynamic signature of coherent systems. IEEE Trans. Reliab. 60, 817–822.
  • Misra, N. and van der Meulen, E. C. (2003). On stochastic properties of $m$-spacings. J. Statist. Planning Infer. 115, 683–697.
  • Navarro, J., Balakrishnan, N. and Samaniego, F. J. (2008). Mixture representations of residual lifetimes of used systems. J. Appl. Prob. 45, 1097–1112.
  • Navarro, J., Samaniego, F. J. and Balakrishnan, N. (2010). The joint signature of coherent systems with shared components. J. Appl. Prob. 47, 235–253.
  • 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.
  • Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. R-34, 69–72.
  • Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York.
  • Samaniego, F. J., Balakrishnan, N. and Navarro, J. (2009). Dynamic signatures and their use in comparing the reliability of new and used systems. Naval Res. Logistics 56, 577–591.
  • Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
  • Zhang, Z. (2010). Mixture representations of inactivity times of conditional coherent systems and their applications. J. Appl. Prob. 47, 876–885.
  • Zhang, Z. (2011). Ordering new conditional residual lifetimes of $k$-out-of-$n$ systems. Commun. Statist. Theory Meth. 40, 1591–1600.
  • Zhang, Z. and Li, X. (2010). Some new result on stochastic orders and aging properties of coherent systems. IEEE Trans. Reliab. 59, 718–724.