On Weighted Generalized Cumulative Residual Entropy of Order n

  • Suchandan Kayal


The present paper considers a shift-dependent measure of uncertainty and its dynamic (residual) version. Various properties have been discussed. Two classes of lifetime distributions are proposed. Further, when m independent and identically distributed observations are available, an estimator of the measure under study is presented using empirical approach. In addition, large sample properties of the estimator are studied. Finally, an application of the proposed measure to the problem related to right-tail risk measure is presented.


WGCRE Increasing convex order Decreasing failure rate Combination mean residual waiting time Proportional hazard models Empirical WGCRE Risk measure 

Mathematics Subject Classification (2010)

94A17 62N05 60E15 


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The author would like to thank the Editor, Associate Editor and the two referees for careful reading and for their comments which greatly improved the article.


  1. Abu-Youssef SE (2002) A moment inequality for decreasing (increasing) mean residual life distributions with hypothesis testing application. Statistics and Probability Letters 57:171–177MathSciNetCrossRefzbMATHGoogle Scholar
  2. Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  3. Asadi M, Zohrevand Y (2007) On dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137:1931–1941MathSciNetCrossRefzbMATHGoogle Scholar
  4. Billingsley P (1986) Probability and measure, 2nd Edn. Wiley, New YorkzbMATHGoogle Scholar
  5. Cover TM, Thomas JA (2006) Elements of information theory. Wiley, New YorkzbMATHGoogle Scholar
  6. Cox DR, Oakes D (2001) Analysis of survival data. Chapman and HallGoogle Scholar
  7. Di Crescenzo A, Longobardi M (2006) On weighted residual and past entropies. Scientiae Mathematicae Japonicae 64:255–266MathSciNetzbMATHGoogle Scholar
  8. Di Crescenzo A, Longobardi M (2009) On cumulative entropies. Journal of Statistical Planning and Inference 139:4072–4087MathSciNetCrossRefzbMATHGoogle Scholar
  9. Kass RE, Ventura V, Cai C (2003) Statistical smooting of neural data. Netw Comput Neural Syst 14:5–15CrossRefGoogle Scholar
  10. Kayal S, Moharana R (2017) On weighted cumulative residual entropy. Journal of Statistics and Management Systems 20:153–173CrossRefGoogle Scholar
  11. Kayid M, Izadkhah S, Alhalees H (2016) Combination of mean residual life order with reliability applications. Statistical Methodology 29:51–69MathSciNetCrossRefGoogle Scholar
  12. Mirali M, Baratpour S, Fakoor V (2017) On weighted cumulative residual entropy. Communications in Statistics-Theory and Methods 46:2857–2869MathSciNetCrossRefzbMATHGoogle Scholar
  13. Misagh F, Panahi Y, Yari GH, Shahi R (2011) Weighted Cumulative entropy and its estimation 2011 IEEE International Conference on Quality and Reliability (ICQR). doi: 10.1109/ICQR.2011.6031765 Google Scholar
  14. Navarro J, Psarrakos G (2017) Characterizations based on generalized cumulative residual entropy functions. Communications in Statistics - Theory and Methods 46:1247–1260MathSciNetCrossRefzbMATHGoogle Scholar
  15. Psarrakos G, Navarro J (2013) Generalized cumulative residual entropy and record values. Metrika 27:623–640MathSciNetCrossRefzbMATHGoogle Scholar
  16. Psarrakos G, Toomaj A (2017) On the generalized cumulative residual entropy with applications in actuarial science. J Comput Appl Math 309:186–199MathSciNetCrossRefzbMATHGoogle Scholar
  17. Pyke R (1965) Spacings. J R Stat Soc 27:395–449zbMATHGoogle Scholar
  18. Ramsay CM (1993) Loading gross premiums for risk without using utility theory. Transactions of the Society of Actuaries XLVGoogle Scholar
  19. Rao M, Chen Y, Vernuri BC, Wang F (2004) Cumulative residual entropy: a new measure of information. IEEE Trans Inf Theory 50:1220–1228MathSciNetCrossRefzbMATHGoogle Scholar
  20. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer Verlag, New YorkCrossRefzbMATHGoogle Scholar
  21. Shannon C (1948) The mathematical theory of communication. Bell Syst Tech J 27:379–423MathSciNetCrossRefzbMATHGoogle Scholar
  22. Wang S (1998) An actuarial index of the righ-tail risk. North American Actuarial Journal 2:88–101MathSciNetCrossRefzbMATHGoogle Scholar
  23. Yang V (2012) Study on cumulative residual entropy and variance as risk measure 5th International conference on business intelligence and financial engineering, published in IEEE, p 4Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia

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