Skip to main content
SpringerLink
Log in

Information Generating Function of Record Values

  • Published:
Mathematical Methods of Statistics Aims and scope Submit manuscript

Abstract

In the present work, we study the information generating (IG) function of record values and examine some main properties of it. We establish some comparison results associated with the IG measure of record values. We show that under equality of two given IG measures of upper record values, the corresponding parent distributions can be determined uniquely. We also present some bounds for the IG measure of upper record values based on upper records of a standard exponential distribution. Further, we provide some results associated with characterization of exponential distribution by maximization (minimization) of IG function of record values under some conditions. We also examine the relative information generating (RIG) measure between the distribution of records values and the corresponding underlying distribution and present some results in this regard. To illustrate the results, several examples have been presented through the paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. R. A. Fisher, ‘‘Tests of significance in harmonic analysis,’’ Proceedings of the Royal Society of London, Series A 125, 54–59 (1929). https://doi.org/10.1098/rspa.1929.0151

    Article  MATH  Google Scholar 

  2. C. E. Shannon, ‘‘A mathematical theory of communication,’’ The Bell System Technical Journal, 27 (3), 379–423 (1948). https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Rényi, ‘‘On measures of entropy and information,’’ in: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1: Contributions to the Theory of Statistics (University of California Press, Berkley, California, 1961), pp. 547–562.

  4. C. Tsallis, ‘‘Possible generalization of Boltzmann-Gibbs statistics,’’ J. Stat. Phys. 52 (1), 479–487 (1988). https://doi.org/10.1007/BF01016429

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Park and I. Kim, ‘‘Cumulative ratio information based on general cumulative entropy,’’ J. Stat. Comput. Simulat. 87 (3), 563–576 (2017). http://doi.org/10.1080/00949655.2016.1219911

    Article  MathSciNet  MATH  Google Scholar 

  6. M. R. Ubriaco, ‘‘Entropies based on fractional calculus,’’ Phys. Lett.373 (30), 2516–2519 (2009). https://doi.org/10.1016/j.physleta.2009.05.026

    Article  MathSciNet  MATH  Google Scholar 

  7. M. A Abbasnejad and N. R. Arghami, ‘‘Rényi entropy properties of records,’’ J. Stat. Plann. Infer. 141 (7), 2312–2320 (2011). https://doi.org/10.1016/j.jspi.2011.01.017

    Article  MATH  Google Scholar 

  8. S. Baratpour, J. Ahmadi, and N. R. Arghami, ‘‘Characterizations based on Rényi entropy of order statistics and record values,’’ J. Stat. Plann. Infer. 138 (8), 2544–2551 (2008). https://doi.org/10.1016/j.jspi.2007.10.024

    Article  MATH  Google Scholar 

  9. S. Kayal, ‘‘Characterization based on generalized entropy of order statistics,’’ Comm. Stat. Theor. Meth. 45 (15), 4628–4636 (2016). https://doi.org/10.1080/03610926.2014.927491

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Kumar, ‘‘Some results on Tsallis entropy measure and \(k\)-record values,’’ Phys. Stat. Mech. Appl. 462, 667–673 (2016). https://doi.org/10.1016/j.physa.2016.05.064

  11. N. Balakrishnan, F. Buono, and M. Longobardi, ‘‘On cumulative entropies in terms of moments of order statistics,’’ Meth. Comput. Appl. Probab. 24 (1), 345–359 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  12. B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, A First Course in Order Statistics (John Wiley, New York, 1992).

    MATH  Google Scholar 

  13. S. Golomb, ‘‘The information generating function of a probability distribution (corresp.),’’ IEEE Trans. Inform. Theor. 12 (1), 75–77 (1966). https://doi.org/10.1109/TIT.1966.1053843

    Article  Google Scholar 

  14. N. Balakrishnan, F. Buono and M. Longobardi, ‘‘On Tsallis extropy with an application to pattern recognition,’’ Stat. Probab. Lett. 180, 109241 (2022). https://doi.org/10.1016/j.spl.2021.109241

  15. F. Lad, G. Sanfilippo, and G. Agro, G. (2015). ‘‘Extropy: Complementary dual of entropy. Stat Sic,’’ 30 (1), 40–58 (2015).https://doi.org/10.1214/14-STS430

    Article  MATH  Google Scholar 

  16. D. E. Clark, ‘‘Local entropy statistics for point processes,’’ IEEE Trans. Inform. Theor. 66 (2), 1155–1163 (2019). https://doi.org/10.1109/TIT.2019.2941213

    Article  MathSciNet  MATH  Google Scholar 

  17. O. Kharazmi and N. Balakrishnan, ‘‘Jensen-information generating function and its connections to some well-known information measures,’’ Stat. Probab. Lett. 170, 108995 (2021). https://doi.org/10.1016/j.spl.2020.108995

  18. O. Kharazmi and N. Balakrishnan, ‘‘Information generating function for order statistics and mixed reliability systems,’’ Comm. Stat. Theor. Meth., 1–10 (2021). https://doi.org/10.1080/03610926.2021.1881123

  19. M. Shaked and J.G. Shanthikumar, Stochastic Orders (Springer, New York, 2007).

    Book  MATH  Google Scholar 

  20. C. Goffman and G. Pedrick, First Course in Functional Analysis, (Prentice Hall, London, 1965).

    MATH  Google Scholar 

  21. M. Fashandi and J. Ahmadi, ‘‘Characterizations of symmetric distributions based on Rényi entropy,’’ Stat. Probab. Lett. 82 (4), 798–804 (2012). https://doi.org/10.1016/j.spl.2012.01.004

    Article  MATH  Google Scholar 

  22. H. R. Higgins, Completeness and Basis Properties of Sets of Special Functions (Cambridge University Press, Cambridge, England, 2004).

    Google Scholar 

  23. J. S. Hwang and G. D. Lin, ‘‘On a generalized moment problem II,’’ Proc. Am. Math. Soc. 91 (4), 577–580 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Guiasu and C. Reischer, ‘‘The relative information generating function’’, Inform. Sci. 35 (3), 235–241 (1985). http://doi.org/10.1016/0020-0255(85)90053-2

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Kullback, and R. A. Leibler, ‘‘On information and sufficiency,’’ Ann. Math. Stat. 22 (1), 79–86 (1951). https://doi.org/10.1214/aoms/1177729694

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Shahid Bahonar University of Kerman, Kerman, Iran

    Zohreh Zamani

  2. Department of Statistics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

    Omid Kharazmi

  3. Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

    Narayanaswamy Balakrishnan

Authors
  1. Zohreh Zamani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Omid Kharazmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Narayanaswamy Balakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Omid Kharazmi.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zamani, Z., Kharazmi, O. & Balakrishnan, N. Information Generating Function of Record Values. Math. Meth. Stat. 31, 120–133 (2022). https://doi.org/10.3103/S1066530722030036

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530722030036

Keywords:

Search

Navigation