Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Fisher Information and the Kullback-Leibler Distance in Concomitants of Generalized Order Statistics Under Iterated FGM family

  • Received : 2020.06.04
  • Accepted : 2021.07.06
  • Published : 2022.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

We study the Fisher Information (FI) of m-generalized order statistics (m-GOSs) and their concomitants about the shape-parameter vector of the Iterated Farlie-Gumbel-Morgenstern (IFGM) bivariate distribution. We carry out a computational study and show how the FI matrix (FIM) helps in finding information contained in singly or multiply censored bivariate samples from the IFGM. We also run numerical computations about the FIM for the sub-models of order statistics (OSs) and sequential order statistics (SOSs). We evaluate FI about the mean and the shape-parameter of exponential and power distributions, respectively. Finally, we investigate the Kullback-Leibler distance in concomitants of m-GOSs.

Keywords

Acknowledgement

The authors are grateful to the editors, and the four anonymous reviewers for careful and diligent reading which improved the readability and presentation substantially.

References

  1. M. A. Abd Elgawad, E. M. Alawady, H. M. Barakat and Shengwu Xiong, Concomitants of generalized order statistics from Huang-Kotz Farlie-Gumbel-Morgenstern bivariate distribution: Some information measures, Bull. Malays. Math. Sci. Soc., 43(2020), 2627-2645. https://doi.org/10.1007/s40840-019-00822-9
  2. Z. A. Abo-Eleneen and H. N. Nagaraja, Fisher information in an order statistic and its concomitant, Ann. Inst. Statist. Math., 54(2002), 667-680. https://doi.org/10.1023/A:1022479514859
  3. J. Ahmadi and N. R. Arghami, On the Fisher information in record values, Metrika, 53(2001), 195-206. https://doi.org/10.1007/s001840000089
  4. M. Ahsanullah, V. B. Nevzorov and M. Shakil, Ordered statistics: theory and methods, Atlantis Press, Paris(2013).
  5. M. A. Alawady, H. M. Barakat, S. Xiong and M. A. Abd-Elgawad, Concomitants of generalized order statistics from iterated Farlie-Gumbel-Morgenstern type bivariate distribution, Commun. Statist. Theory Meth., to appear(2020).
  6. M. A. Alawady, H. M. Barakat and M. A. Abd Elgawad, Concomitants of generalized order statistics from bivariate Cambanis family of distributions under a general setting, Bull. Malays. Math. Sci. Soc., 44(5)(2021), 3129--3159. https://doi.org/10.1007/s40840-021-01102-1
  7. I. Bairamov and S. Eryilmaz, Spacings, exceedances and concomitants in progressive type II censoring scheme, J. Statist. Plann. Inf., 136(2006) 527-536. https://doi.org/10.1016/j.jspi.2004.09.002
  8. H. M. Barakat, Limit theory of generalized order statistics, J. Statist. Plann. Inf., 137(2007), 1-11. https://doi.org/10.1016/j.jspi.2005.10.003
  9. H. M. Barakat, H. M. and I. A. Husseiny, Some information measures in concomitants of generalized order statistics under iterated FGM bivariate type, Quaestiones Math., 44(2021), 581-598. https://doi.org/10.2989/16073606.2020.1729271
  10. H. M. Barakat and M. E. El-Adll, Asymptotic theory of extreme dual generalized order statistics, Statist. Probab. Lett., 79(2009), 1252-1259. https://doi.org/10.1016/j.spl.2009.01.015
  11. H. M. Barakat and M. E. El-Adll, On the limit distribution of lower extreme generalized order statistics, Proc. Indian Acad. Sci. Math. Sci., 122(2)(2012), 297-311. https://doi.org/10.1007/s12044-012-0064-9
  12. H. M. Barakat, E. M. Nigm and I. A. Husseiny, Measures of information in order statistics and theirconcomitants for the single iterated Farlie-Gumbel-Morgenstern bivariate distribution, Math. Popul. Stud., 28(2020), 154-175.
  13. H. M. Barakat, E. M. Nigm, M. A. Alawady and I. A. Husseiny, Concomitants of order statistics and record values from iterated FGM type bivariate-generalized exponential distribution, REVSTAT, 19(2)(2019), 291-307.
  14. M. I. Beg and M. Ahsanullah, Concomitants of generalized order statistics from Farlie-Gumbel-Morgenstern distributions, Stat. Methodol., 5(1)(2008) 1-20. https://doi.org/10.1016/j.stamet.2007.04.001
  15. S. S. Buhamra and A. Ahsanullah, Fisher information in concomitants of generalized order statistics in Farlie-Gumbel-Morgenestern distributions, J. Statist. Theory Appl., 4(4)(2005), 387-399.
  16. H. A. David and H. N. Nagaraja, Concomitants of order statistics, Handbook of Statistics, North-Holland, Amsterdam(1998).
  17. G. Hofmann, Fisher information in record data and random observations, Statist. Papers, 45(4)(2004), 517-528. https://doi.org/10.1007/bf02760565
  18. G. Hofmann and H. N. Nagaraja, Fisher information in record data, Metrika, 57(2)(2003), 177-193. https://doi.org/10.1007/s001840200208
  19. J. S. Huang and S. Kotz, Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions, Biometrika, 71(3)(1984), 633-636. https://doi.org/10.1093/biomet/71.3.633
  20. U. Kamps, A concept of generalized order statistics, Teubner, Stuttgart (1995).
  21. U. Kamps and E. Cramer, On distribution of generalized order statistics, Statistics, 35(3)(2001), 269-280. https://doi.org/10.1080/02331880108802736
  22. S. Kullback and R. A. Leibler, On information and suffciency, Ann. Math. Statist., 22(1951), 79-86. https://doi.org/10.1214/aoms/1177729694
  23. K. G. Mehrotra, R. A. Johnson and G. K. Bhattacharyya, Exact Fisher information for censored samples and the extended hazard rate functions, Comm. Statist. A-Theory Methods, 8(15)(1979), 1493-1510. https://doi.org/10.1080/03610927908827846
  24. R. B. Nelson, An introduction to copuls, Lecture Notes in Statist, Springer, New York, 139(1999).
  25. S. Park, Fisher information in order statistics, J. Amer. Statist. Assoc., 91(433)(1996), 385-390. https://doi.org/10.1080/01621459.1996.10476699
  26. S. Tahmasebi and A. A. Jafari, Concomitants of order statistics and record values from Morgenstern type bivariate-generalized exponential distribution, Bull. Malays. Math. Sci. Soc., 38(2015), 1411-1423. https://doi.org/10.1007/s40840-014-0087-8
  27. J. W. Tukey, Which part of the sample contains the information? Proc. Nat. Acad. Sci., USA, 53(1965), 127-134. https://doi.org/10.1073/pnas.53.1.127
  28. G. Zheng and J. L. Gastwirth, Where is the Fisher information in an ordered sample? Statist. Sinica, 10(2000), 1267-1280.