Skip to main content
Log in

Information Generating Function of \(\boldsymbol{k}\)-Record Values and Its Applications

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

Abstract

In this paper, we study the information-generating (IG) measure of \(k\)-record values and examine some of its main properties. We establish some bounds for the IG measure of \(k\)-record values. In addition, we present some results related to the characterization of an exponential distribution by maximization (minimization) of the IG measure of record values under certain conditions. We also examine the relative information generating (RIG) measure between the distribution of record values and the corresponding underlying distribution and present some results in this regard. Several examples have been provided throughout the study to illustrate the results. We also consider the problem of estimation of the IG measure for a two-parameter Weibull distribution based on the upper \(k\)-record values.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

REFERENCES

  1. M. Abbasnejad and N. R. Arghami, ‘‘Renyi entropy properties of records,’’ Journal of Statistical Planning and Inference 141 (7), 2312–2320 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Ahsanullah, ‘‘Record values-theory and applications. University Press of America, Maryland, United States,’’ Journal of Statistical Planning and Inference 141 (7), 2312–2320 (2004).

    Google Scholar 

  3. L. Al-Labadi and S. Berry, ‘‘Bayesian estimation of extropy and goodness of fit tests,’’ Journal of Applied Statistics 49 (2), 357–370 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  4. B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, A first course in order statistics, vol. 54 (SIAM, Philadelphia, 1992).

    MATH  Google Scholar 

  5. N. Balakrishnan, F. Buono, and M. Longobardi, ‘‘On cumulative entropies in terms of moments of order statistics,’’ Methodology and Computing in Applied Probability 24 (1), 345–359 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Bansal and N. Gupta, ‘‘Weighted extropies and past extropy of order statistics and \(k\)-record values,’’ Communications in Statistics-Theory and Methods 51 (17), 6091–6108 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Baratpour, J. Ahmadi, and N. R. Arghami, ‘‘Characterizations based on Renyi entropy of order statistics and record values,’’ Journal of Statistical Planning and Inference 138 (8), 2544–2551 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  8. P. J. Bickel and E. L. Lehmann, Descriptive Statistics for Non-Parametric Models, III. Dispersion. In Selected Works of E.L. Lehmann (Springer, 2012), p. 499–518.

    Google Scholar 

  9. M. Chacko and P. Asha, ‘‘Estimation of entropy for generalized exponential distribution based on record values,’’ Journal of the Indian Society for Probability and Statistics 19, 79–96 (2018).

    Article  Google Scholar 

  10. M. Chacko and P. Asha, ‘‘Estimation of entropy for Weibull distribution based on record values,’’ Journal of Statistical Theory and Applications 20 (2), 279–288 (2021).

    Article  Google Scholar 

  11. M. Chacko and L. Muraleedharan, ‘‘Inference based on \(k\)-record values from generalized exponential distribution,’’ Statistica 78 (1), 37–56 (2018).

    MATH  Google Scholar 

  12. M. Chacko and M. Shy Mary, ‘‘Concomitants of \(k\)-record values arising from Morgenstern family of distributions and their applications in parameter estimation,’’ Statistical Papers 54 (1), 21–46 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Chandler, ‘‘The distribution and frequency of record values,’’ Journal of the Royal Statistical Society: Series B (Methodological) 14 (2), 220–228 (1952).

    MathSciNet  MATH  Google Scholar 

  14. S. Chib and E. Greenberg, ‘‘Understanding the Metropolis-Hastings algorithm,’’ The American Statistician 49 (4), 327–335 (1995).

    Google Scholar 

  15. Y. Cho, H. Sun, and K. Lee, ‘‘Estimating the entropy of a Weibull distribution under generalized progressive hybrid censoring,’’ Entropy 17 (1), 102–122 (2015).

    Article  Google Scholar 

  16. D. E. Clark, ‘‘Local entropy statistics for point processes,’’ IEEE Transactions on Information Theory 66 (2), 1155–1163 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  17. W. Dziubdziela and B. Kopociński, ‘‘Limiting properties of the \(k\)-th record values,’’ Applicationes Mathematicae 2 (15), 187–190 (1976).

    Article  MATH  Google Scholar 

  18. R. A. Fisher, ‘‘Tests of significance in harmonic analysis,’’ Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 125 (796), 54–59 (1929).

    MATH  Google Scholar 

  19. S. Golomb, ‘‘The information generating function of a probability distribution (corresp.),’’ IEEE Transactions on Information Theory 12 (1), 75–77 (1966).

    Article  MathSciNet  Google Scholar 

  20. S. Guiasu and C. Reischer, ‘‘The relative information generating function,’’ Information sciences 35 (3), 235–241 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  21. C. B. Guure, N. A. Ibrahim, and A. O. M. Ahmed, ‘‘Bayesian estimation of two-parameter Weibull distribution using extension of Jeffreys’ prior information with three loss functions,’’ Mathematical Problems in Engineering (2012).

  22. A. S. Hassan and A. N. Zaky, ‘‘Entropy Bayesian estimation for Lomax distribution based on record,’’ Thailand Statistician 19 (1), 95–114 (2021).

    MATH  Google Scholar 

  23. J. M. Joyce, Kullback-Leibler Divergence, in: International Encyclopedia of Statistical Science (Springer, New York, 2011), p. 720–722.

    Google Scholar 

  24. S. Kayal, ‘‘Characterization based on generalized entropy of order statistics,’’ Communications in Statistics-Theory and Methods 45 (15), 4628–4636 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  25. O. Kharazmi and N. Balakrishnan, ‘‘Information generating function for order statistics and mixed reliability systems,’’ Communications in Statistics-Theory and Methods, 1–10 (2021).

  26. O. Kharazmi and N. Balakrishnan, ‘‘Jensen-information generating function and its connections to some well-known information measures,’’ Statistics & Probability Letters 170, 108995 (2021).

  27. V. Kumar, ‘‘Some results on Tsallis entropy measure and \(k\)-record values,’’ Physica A: Statistical Mechanics and its Applications 462, 667–673 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  28. F. Lad, G. Sanfilippo, and G. Agro, ‘‘Extropy: Complementary dual of entropy,’’ Statistical Science 30 (1), 40–58 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  29. V. B. Nevzorov, ‘‘Records: Mathematical Theory, Translation of mathematical monographs,’’ American Mathematical Society, Providence, Rhode Island, USA 194 (2001).

    Google Scholar 

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

    Book  MATH  Google Scholar 

  31. C. E. Shannon, ‘‘A mathematical theory of communication,’’ The Bell system technical journal 27 (3), 379–423 (1948).

    Article  MathSciNet  MATH  Google Scholar 

  32. Z. Zamani, O. Kharazmi, and N. Balakrishnan, ‘‘Information Generating Function of Record Values,’’ Mathematical Methods of Statistics 31 (3), 120–133 (2022).

    Article  MathSciNet  MATH  Google Scholar 

Download references

ACKNOWLEDGEMENTS

The authors would like to thank the editor and the reviewers for their constructive comments, which helped to improve the quality of the paper.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Manoj Chacko or Annie Grace.

Ethics declarations

The authors declare that they have no conflicts of interest.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chacko, M., Grace, A. Information Generating Function of \(\boldsymbol{k}\)-Record Values and Its Applications. Math. Meth. Stat. 32, 176–196 (2023). https://doi.org/10.3103/S106653072303002X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S106653072303002X

Keywords:

Navigation