On Weighted Extended Cumulative Residual Entropy of k-th Upper Record

  • Rajesh Moharana
  • Suchandan KayalEmail author
Part of the Lecture Notes on Data Engineering and Communications Technologies book series (LNDECT, volume 21)


Generalized cumulative residual entropy has been shown an alternative uncertainty measure to cumulative residual entropy in the literature. We find its applications in the theory of communication, actuarial and computer science. In this chapter, we consider a shift-dependent version of the generalized cumulative residual entropy for the k-th upper record and its dynamic version. The advantage of the proposed measure over the existing measure is discussed. Various results including some useful properties, effect of affine transformations, bounds, stochastic ordering and aging properties are obtained. In addition, inequalities based on the proportional hazard rate model are derived. Further, we obtain some characterization results for various probability models based on the proposed information measure. A nonparametric estimator of the proposed measure is provided and then, its asymptotic normality is established.


Weighted extended cumulative residual entropy k-th upper record value Affine transformation Characterization Empirical function 


  1. 1.
    Asadi M, Zohrevand Y (2007) On dynamic cumulative residual entropy. J Stat Plan Inference 137(6):1931–1941MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baratpour S (2010) Characterizations based on cumulative residual entropy of first-order statistics. Commun Stat-Theory Methods 39(20):3645–3651MathSciNetCrossRefGoogle Scholar
  3. 3.
    Di Crescenzo A, Longobardi M (2006) On weighted residual and past entropies. Scientiae Mathematicae Japonicae 64:255–266MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dziubdziela W, Kopocinski B (1976) Limiting properties of the kth record values. Appl Math 2(15):187–190zbMATHGoogle Scholar
  5. 5.
    Ebrahimi N (1996) How to measure uncertainty in the residual life time distribution. Sankhya Indian J Stat Ser A 58:48–56MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hwang JS, Lin GD (1984) On a generalized moment problem II. Proc Am Math Soc 91(4):577–580MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kamps U (1998) Characterizations of distributions by recurrence relations and identities for moments of order statistics. In: Handbook of Statistics, vol 16, pp 291–311zbMATHGoogle Scholar
  8. 8.
    Kapodistria S, Psarrakos G (2012) Some extensions of the residual lifetime and its connection to the cumulative residual entropy. Probab Eng Inf Sci 26(1):129–146MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kayal S (2016) On generalized cumulative entropies. Probab Eng Inf Sci 30(4):640–662MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kayal S (2018) On weighted generalized cumulative residual entropy of order \(n\). Method Comput Appl Probabi 20(2):487–503Google Scholar
  11. 11.
    Kayal S, Moharana R (2017a) On weighted cumulative residual entropy. J Stat Manag Syst 20(2):153–173MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kayal S, Moharana R (2017b) On weighted measures of cumulative entropy. Int J Math Stat 18(3):26–46MathSciNetGoogle Scholar
  13. 13.
    Mirali M, Baratpour S, Fakoor V (2017) On weighted cumulative residual entropy. Commun Stat-Theory Methods 46(6):2857–2869MathSciNetCrossRefGoogle Scholar
  14. 14.
    Moharana R, Kayal S (2017) On weighted Kullback-Leibler divergence for doubly truncated random variables. RevStat (to appear)Google Scholar
  15. 15.
    Navarro J, Aguila YD, Asadi M (2010) Some new results on the cumulative residual entropy. J Stat Plan Inference 140(1):310–322MathSciNetCrossRefGoogle Scholar
  16. 16.
    Navarro J, Psarrakos G (2017) Characterizations based on generalized cumulative residual entropy functions. Commun Stat-Theory Methods 46(3):1247–1260MathSciNetCrossRefGoogle Scholar
  17. 17.
    Psarrakos G, Economou P (2017) On the generalized cumulative residual entropy weighted distributions. Commun Stat-Theory Methods 46(22):10914–10925MathSciNetCrossRefGoogle Scholar
  18. 18.
    Psarrakos G, Navarro J (2013) Generalized cumulative residual entropy and record values. Metrika 76(5):623–640MathSciNetCrossRefGoogle Scholar
  19. 19.
    Psarrakos G, Toomaj A (2017) On the generalized cumulative residual entropy with applications in actuarial science. J Comput Appl Math 309:186–199MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pyke R (1965) Spacings. J R Stat Soc Ser B (Methodol) 27:395–449zbMATHGoogle Scholar
  21. 21.
    Rao M (2005) More on a new concept of entropy and information. J Theor Probab 18(4):967–981MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rao M, Chen Y, Vemuri BC, Wang F (2004) Cumulative residual entropy: a new measure of information. IEEE Trans Inf Theory 50(6):1220–1228MathSciNetCrossRefGoogle Scholar
  23. 23.
    Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New YorkCrossRefGoogle Scholar
  24. 24.
    Shannon CE (1948) The mathematical theory of communication. Bell Syst Tech J 27:379–423MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tahmasebi S, Eskandarzadeh M, Jafari AA (2017) An extension of generalized cumulative residual entropy. J Stat Theory Appl 16(2):165–177MathSciNetGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia
  2. 2.Faculty of Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia

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