Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-29T11:14:19.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Some properties of the cumulative residual entropy of coherent and mixed systems

Part of: Communication, information Survival analysis and censored data

Published online by Cambridge University Press:  22 June 2017

A. Toomaj ,
S. M. Sunoj  and
J. Navarro
A. Toomaj*
Affiliation:
Gonbad Kavous University
S. M. Sunoj*
Affiliation:
Cochin University of Science and Technology
J. Navarro*
Affiliation:
Universidad de Murcia
*
* Postal address: Department of Statistics, Gonbad Kavous University, Gonbad Kavous, Iran. Email address: ab.toomaj@gonbad.ac.ir
** Postal address: Department of Statistics, Cochin University of Science and Technology, Cochin 682 022, Kerala, India. Email address: smsunoj@cusat.ac.in
*** Postal address: Facultad de Matematicas, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es
Get access Rights & Permissions [Opens in a new window]

Abstract

Recently, Rao et al. (2004) introduced an alternative measure of uncertainty known as the cumulative residual entropy (CRE). It is based on the survival (reliability) function F̅ instead of the probability density function f used in classical Shannon entropy. In reliability based system design, the performance characteristics of the coherent systems are of great importance. Accordingly, in this paper, we study the CRE for coherent and mixed systems when the component lifetimes are identically distributed. Bounds for the CRE of the system lifetime are obtained. We use these results to propose a measure to study if a system is close to series and parallel systems of the same size. Our results suggest that the CRE can be viewed as an alternative entropy (dispersion) measure to classical Shannon entropy.

Keywords

Coherent systemcumulative residual entropystochastic ordersystem signature

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 94A17: Measures of information, entropy
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 379 - 393
Copyright
Copyright © Applied Probability Trust 2017 

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] Asadi, M. and Zohrevand, Y. (2007). On the dynamic cumulative residual entropy. J. Statist. Planning Infer. 137, 1931–1941. CrossRefGoogle Scholar
[2] Asadi, M., Ebrahimi, N., Soofi, E. S. and Zarezadeh, S. (2014). New maximum entropy methods for modeling lifetime distribution. Naval Res. Logistics 61, 427–434. CrossRefGoogle Scholar
[3] Baratpour, S. (2010). Characterizations based on cumulative residual entropy of first-order statistics. Commun. Statist. Theory Meth. 39, 3645–3651. CrossRefGoogle Scholar
[4] Baratpour, S. and Rad, A. H. (2012). Testing goodness-of-fit for exponential distribution based on cumulative residual entropy. Commun. Statist. Theory Meth. 41, 1387–1396. CrossRefGoogle Scholar
[5] Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. Silver Spring, MD. Google Scholar
[6] Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory, 2nd edn. Wiley-Interscience, Hoboken, NJ. Google Scholar
[7] D'Andrea, A. and De Sanctis, L. (2015). The Kruskal-Katona theorem and a characterization of system signatures. J. Appl. Prob. 52, 508–518. CrossRefGoogle Scholar
[8] Ebrahimi, N., Jalali, N. Y and Soofi, E. S. (2014). Comparison, utility, and partition of dependence under absolutely continuous and singular distributions. J. Multivariate Anal. 131, 32–50. CrossRefGoogle Scholar
[9] Ebrahimi, N., Kirmani, S. N. U. A. and Soofi, E. S. (2007). Multivariate dynamic information. J. Multivariate Anal. 98, 328–349. CrossRefGoogle Scholar
[10] Ebrahimi, N., Maasoumi, E. and Soofi, E. S. (1999). Ordering univariate distributions by entropy and variance. J. Econometrics 90, 317–336. CrossRefGoogle Scholar
[11] Klein, I., Mangold, B. and Doll, M. (2016). Cumulative paired ϕ-entropy. Entropy 18, 248, 45pp. CrossRefGoogle Scholar
[12] 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. 3.0.CO;2-D>CrossRefGoogle Scholar
[13] Nadarajah, S. and Zografos, K. (2005). Expressions for Rényi and Shannon entropies for bivariate distributions. Inf. Sci. 170, 173–189. CrossRefGoogle Scholar
[14] Navarro, J. (2016). Stochastic comparisons of generalized mixtures and coherent systems. TEST 25, 150–169. CrossRefGoogle Scholar
[15] Navarro, J. and Rubio, R. (2010). Computations of signatures of coherent systems with five components. Commun. Statist. Simul. Comput. 39, 68–84. CrossRefGoogle Scholar
[16] Navarro, J., del Aguila, Y. and Asadi, M. (2010). Some new results on the cumulative residual entropy. J. Statist. Planning Infer. 140, 310–322. CrossRefGoogle Scholar
[17] Navarro, J., del Aguila, 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. CrossRefGoogle Scholar
[18] Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008). On the application and extension of system signatures in engineering reliability. Naval Res. Logistics 55, 313–327. CrossRefGoogle Scholar
[19] Park, S. and Kim, I. (2014). On cumulative residual entropy of order statistics. Statist. Prob. Lett. 94, 170–175. CrossRefGoogle Scholar
[20] Psarrakos, G. and Navarro, J. (2013). Generalized cumulative residual entropy and record values. Metrika 76, 623–640. CrossRefGoogle Scholar
[21] Rao, M. (2005). More on a new concept of entropy and information. J. Theoret. Prob. 18, 967–981. CrossRefGoogle Scholar
[22] Rao, M., Chen, Y., Vemuri, B. C. and Wang, F. (2004). Cumulative residual entropy: a new measure of information. IEEE Trans. Inf. Theory 50, 1220–1228. CrossRefGoogle Scholar
[23] Samaniego, F. J. (2007). System Signatures and Their Applications in Engineering Reliability. Springer, New York. CrossRefGoogle Scholar
[24] Samaniego, F. J. and Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Adv. Appl. Prob. 48, 88–111. CrossRefGoogle Scholar
[25] Shaked, M. and Shanthikumar, J. G. (2006). Stochastic Orders and Their Applications. Academic Press, San Diego. Google Scholar
[26] Taneja, H. and Kumar, V. (2012). On dynamic cumulative residual inaccuracy measure. In Proc. World Congress on Engineering, Vol. I, pp. 153–156. Google Scholar
[27] Toomaj, A. and Doostparast, M. (2014). A note on signature-based expressions for the entropy of mixed r-out-of-n systems. Naval Res. Logistics 61, 202–206. CrossRefGoogle Scholar
[28] Toomaj, A. and Doostparast, M. (2016). On the Kullback Leibler information for mixed systems. Internat. J. Systems Sci. 47, 2458–2465. CrossRefGoogle Scholar
[29] Wang, F. E., Vemuri, B. C., Rao, M. and Chen, Y. (2003). A new & robust information theoretic measure and its application to image alignment. In Internat. Conf. on Information Processing in Medical Imaging, Springer, Berlin, pp. 3880–4000. Google Scholar
[30] Yang, L. (2012). Study on cumulative residual entropy and variance as risk measure. In 5th Internat. Conf. on Business Intelligence and Financial Engineering, IEEE, pp. 210–213. Google Scholar