, Volume 82, Issue 5, pp 607–631 | Cite as

A past inaccuracy measure based on the reversed relevation transform

  • Antonio Di CrescenzoEmail author
  • Suchandan Kayal
  • Abdolsaeed Toomaj


Numerous information indices have been developed in the information theoretic literature and extensively used in various disciplines. One of the relevant developments in this area is the Kerridge inaccuracy measure. Recently, a new measure of inaccuracy was introduced and studied by using the concept of relevation transform, which is related to the upper record values of a sequence of independent and identically distributed random variables. Along this line of research, we introduce an analogue of the inaccuracy measure based on the reversed relevation transform. We discuss some theoretical merits of the proposed measure and provide several results involving equivalent formulas, bounds, monotonicity and stochastic orderings. Our results are also based on the mean inactivity time and the new concept of reversed relevation inaccuracy ratio.


Cumulative (past) entropy Mean inactivity time Reversed relevation transform Stochastic orders Proportional reversed hazard rates model 

Mathematics Subject Classification

60E15 62B10 62N05 94A17 



The first author is a member of the Research group GNCS of INdAM. The third author is partially supported by a grant from Gonbad Kavous University.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. Asadi M, Zohrevand Y (2007) On the dynamic cumulative residual entropy. J Stat Plan Inference 137:1931–1941MathSciNetCrossRefzbMATHGoogle Scholar
  2. Belzunce F, Martínez-Riquelme C, Ruiz JM, Sordo MA (2017) On the comparison of relative spacings with applications. Methodol Comput Appl Probab 19:357–376MathSciNetCrossRefzbMATHGoogle Scholar
  3. Block HW, Savits TH, Singh H (1998) The reversed hazard rate function. Probab Eng Inf Sci 12:69–90MathSciNetCrossRefzbMATHGoogle Scholar
  4. Burnham KP, Anderson DR (2002) Model selection and multimodel inference. A practical information-theoretic approach, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  5. Choe Y (2017) Information criterion for minimum cross-entropy model selection. arxiv:1704.04315
  6. Di Crescenzo A, Longobardi M (2009) On cumulative entropies. J Stat Plan Inference 139:4072–4087MathSciNetCrossRefzbMATHGoogle Scholar
  7. Di Crescenzo A, Longobardi M (2015) Some properties and applications of cumulative Kullback–Leibler information. Appl Stoch Models Bus Ind 31:875–891MathSciNetCrossRefzbMATHGoogle Scholar
  8. Di Crescenzo A, Martinucci B, Zacks S (2015) Compound Poisson process with a Poisson subordinator. J Appl Prob 52:360–374MathSciNetCrossRefzbMATHGoogle Scholar
  9. Di Crescenzo A, Toomaj A (2015) Extension of the past lifetime and its connection to the cumulative entropy. J Appl Prob 52:1156–1174MathSciNetCrossRefzbMATHGoogle Scholar
  10. Di Crescenzo A, Toomaj A (2017) Further results on the generalized cumulative entropy. Kybernetika 53:959–982MathSciNetzbMATHGoogle Scholar
  11. Ebrahimi N, Soofi ES, Soyer R (2010) Information measures in perspective. Int Stat Rev 78(3):383–412CrossRefzbMATHGoogle Scholar
  12. Fraser DAS (1965) On information in statistics. Ann Math Stat 36:890–896MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kayal S (2016) On generalized cumulative entropies. Probab Eng Inf Sci 30:640–662MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kayal S (2018) On weighted generalized cumulative residual entropy of order \(n\). Methodol Comput Appl Probab 20:487–503MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kayal S, Sunoj SM (2017) Generalized Kerridges inaccuracy measure for conditionally specified models. Commun Stat Theory Methods 46:8257–8268MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kayal S, Sunoj SM, Rajesh G (2017) On dynamic generalized measures of inaccuracy. Statistica 77:133–148Google Scholar
  17. Karlin S (1968) Total positivity. Stanford University Press, Stanford, CAzbMATHGoogle Scholar
  18. Kayid M, Ahmad IA (2004) On the mean inactivity time ordering with reliability applications. Probab Eng Inf Sci 18:395–409MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kerridge DF (1961) Inaccuracy and inference. J R Stat Soc B 23:184–194MathSciNetzbMATHGoogle Scholar
  20. Krakowski M (1973) The relevation transform and a generalization of the Gamma distribution function. Reve Francaise d’Automatiqe, Informatigue et Recherche Operationnelle 7:107–120MathSciNetzbMATHGoogle Scholar
  21. Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22:79–86MathSciNetCrossRefzbMATHGoogle Scholar
  22. Kumar V, Taneja HC (2015) Dynamic cumulative residual and past inaccuracy measures. J Stat Theory Appl 14:399–412MathSciNetGoogle Scholar
  23. Kumar V, Taneja HC, Srivastava R (2011) A dynamic measure of inaccuracy between two past lifetime distributions. Metrika 74:1–10MathSciNetCrossRefzbMATHGoogle Scholar
  24. Kundu C, Di Crescenzo A, Longobardi M (2016) On cumulative residual (past) inaccuracy for truncated random variables. Metrika 79:335–356MathSciNetCrossRefzbMATHGoogle Scholar
  25. Nanda AK, Singh H, Misra N, Paul P (2003) Reliability properties of reversed residual lifetime. Commun Stat Theory Methods 32:2031–2042MathSciNetCrossRefzbMATHGoogle Scholar
  26. Nath P (1968) Inaccuracy and coding theory. Metrika 13:123–135MathSciNetCrossRefzbMATHGoogle Scholar
  27. Navarro J, del Aguila Y, Asadi M (2010) Some new results on the cumulative residual entropy. J Stat Plan Inference Infer:310–322MathSciNetCrossRefzbMATHGoogle Scholar
  28. Navarro J, Psarrakos G (2017) Characterizations based on generalized cumulative residual entropy functions. Commun Stat Theory Methods 46:1247–1260MathSciNetCrossRefzbMATHGoogle Scholar
  29. Orsingher E, Polito F (2010) Composition of poissonprocesses. In: Oleg V (ed) Proceedings of XIV international conference on eventological mathematics and related fields. State Trade and Economic Institute, Siberian Federal University, Krasn, Krasnoyarsk, pp 13–18Google Scholar
  30. Orsingher E, Polito F (2012) Compositions, random sums and continued random fractions of Poisson and fractional Poisson processes. J Stat Phys 148:233–249MathSciNetCrossRefzbMATHGoogle Scholar
  31. Park S, Rao M, Shin DW (2012) On cumulative residual Kullback–Leibler information. Stat Prob Lett 82:2025–2032MathSciNetCrossRefzbMATHGoogle Scholar
  32. Psarrakos G, Di Crescenzo A (2018) A residual inaccuracy measure based on the relevation transform. Metrika 81:37–59MathSciNetCrossRefzbMATHGoogle Scholar
  33. Psarrakos G, Navarro J (2013) Generalized cumulative residual entropy and record values. Metrika 76:623–640MathSciNetCrossRefzbMATHGoogle Scholar
  34. Rao M, Chen Y, Vemuri B, Fei W (2004) Cumulative residual entropy: a new measure of information. IEEE Trans Inf Theory 50(6):1220–1228MathSciNetCrossRefzbMATHGoogle Scholar
  35. Rezaei M, Gholizadeh B, Izadkhah S (2015) On relative reversed hazard rate order. Commun Stat Theory Methods 44:300–308MathSciNetCrossRefzbMATHGoogle Scholar
  36. Shaked M, Shanthikumar JG (2007) Stochastic orders and their applications. Academic Press, San DiegozbMATHGoogle Scholar
  37. Taneja HC, Kumar V, Srivastava R (2009) A dynamic measure of inaccuracy between two residual lifetime distributions. Int Math Forum 25:1213–1220MathSciNetzbMATHGoogle Scholar
  38. Toomaj A, Sunoj S, Navarro J (2017) Some properties of the cumulative residual entropy of coherent and mixed systems. J Appl Probab 54:379–393MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di SalernoFiscianoItaly
  2. 2.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia
  3. 3.Faculty of Basic Sciences and Engineering, Department of Mathematics and StatisticsGonbad Kavous UniversityGonbad KavousIran

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