Concomitants of Ordered Variables from Huang–Kotz FGM Type Bivariate Generalized Exponential Distribution

  • H. M. BarakatEmail author
  • E. M. Nigm
  • A. H. Syam


We introduce the Huang–Kotz Morgenstern type bivariate generalized exponential distribution. Some distributional properties of concomitants of order statistics as well as record values for this family are studied. Recurrence relations between single and product moments of concomitants are obtained. Moreover, the rank and the asymptotic behavior of concomitants of order statistics are investigated.


Concomitants Order statistics Record values Generalized exponential distribution Huang–Kotz FGM family 

Mathematical Subject Classification

62B10 62G30 



The authors are grateful to the Editor in Chief, Professor Rosihan M. Ali, and the anonymous referees for suggestions and comments that improved the presentation substantially.


  1. 1.
    Ahsanullah, M.: Records and concomitants. Bull. Malays. Math. Sci. Soc. 32(2), 101–117 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ahsanullah, M., Shakil, M.: Characterizations of Rayleigh distribution based on order statistics and record values. Bull. Malays. Math. Sci. Soc. 36(3), 625–635 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ahsanullah, M., Shakil, M., Kibria, G.M.: A note on a characterization of Gompeertz–Verhulst distribution. J. Stat. Theory Appl. 13(1), 17–26 (2013)Google Scholar
  4. 4.
    Ahuja, J.C.: On certain properties of the generalized Gompertz distribution. Indian J. Stat. Ser. B 31, 541–544 (1969)Google Scholar
  5. 5.
    Ahuja, J.C., Nash, S.W.: The generalized Gompertz–Verhulst family of distributions. Sankhya Ser. A 29, 141–156 (1967)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Amblard, C., Girard, S.: Symmetry and dependence properties within a semiparametric family of bivariate copulas. J. Nonparametric Stat. 14(6), 715–727 (2002)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: Records, Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1998)Google Scholar
  8. 8.
    Bairamov, I., Kotz, S.: Dependence structure and symmetry of Huang–Kotz FGM distributions and their extensions. Metrika 56(1), 55–72 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Barakat, H.M., El-Shandidy, M.A.: Computing the distribution and expected value of the concomitant rank order statistics. Commun. Stat. Theory Meth. 33(11), 2575–2594 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Barakat, H.M., Nigm, E.M., Harpy, M.H.: Limit theorems of order statistics and record values from the gamma and Kumaraswamy-generated-distributions. Bull. Malays. Math. Sci. (2016). doi: 10.1007/s40840-016-0356-9 zbMATHGoogle Scholar
  11. 11.
    Bdair, O.M., Raqab, M.Z.: Mean residual life of kth records under double monitoring. Bull. Malays. Math. Sci. Soc. 37(2), 457–464 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Beg, M.I., Ahsanullah, M.: Concomitants of generalized order statistics from Farlie–Gumbel–Morgenstern distributions. Stat. Methodol. 5, 1–20 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Bhattacharya, P.K.: Convergence of sample paths of normalized sums of induced order statistics. Ann. Stat. 2, 1034–1039 (1974)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chen, Z., Bai, Z., Sinha, B.K.: Ranked Set Sampling: Theory and Applications. Springer, New York (2004)zbMATHGoogle Scholar
  15. 15.
    David, H.A.: Concomitants of order statistics. Bull. Int. Stat. Inst. 45, 295–300 (1973)MathSciNetGoogle Scholar
  16. 16.
    David, H.A., Nagaraja, H.N.: Concomitants of order statistics. In: Balakrishnan, N., Rao, C.R. (eds.) Order Statistics: Theory & Methods, pp. 487–513. Elsevier, Amsterdam (1998)Google Scholar
  17. 17.
    David, H.A., Nagaraja, H.N.: Order Statistics, 3rd edn. Wiley, New York (2003)zbMATHGoogle Scholar
  18. 18.
    David, H.A., O’Connell, M.J., Yang, S.S.: Distribution and expected value of the rank of a concomitant and an order statistic. Ann. Stat. 5, 216–223 (1977)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fischer, M., Klein, I.: Constructing generalized FGM copulas by means of certain univariate distributions. Metrika 65(243), 260 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Galambos, J.: The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Krieger, Malabar (1987)zbMATHGoogle Scholar
  21. 21.
    Gompertz, B.: On the nature of the function expressive of the human mortality and on a new mode of determining the value of life contingencies. Philos. Transduct. R. Soc. Lond. 115, 513–585 (1825)Google Scholar
  22. 22.
    Gupta, R.D., Kundu, D.: Generalized exponential distributions. Aust. N. Z. J. Stat. 41, 173–188 (1999)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Houchens, R.L.: Record Value Theory and Inference. ProQuest LLC, Ann Arbor (1984)Google Scholar
  24. 24.
    Huang, J.S., Kotz, S.: Modifications of the Farlie–Gumbel–Morgenstern distributions. A tough hill to climb. Metrika 49, 135–145 (1999)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mokhlis, N.A., Khames, S.K.: Concomitants of record values from a general Farlie–Gumbel–Morgenstern distributoion. J. Adv. Math. 7(3), 1328–1340 (2014a)Google Scholar
  26. 26.
    Mokhlis, N.A., Khames, S.K.: On concomitants of record values from generalized Farlie–Gumbel–Morgenstern distributoion. J. Stat. Sci. Appl. 2, 175–192 (2014b)Google Scholar
  27. 27.
    Tahmasebi, S., Behboodian, J.: Shannon information for concomitants of generalized order statistics in Farlie–Gumbel–Morgenstern (FGM) family. Bull. Malays. Math. Sci. Soc. 35(4), 975–981 (2012)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Tahmasebi, S., Jafari, A.A.: Estimators for the parameter mean of Morgenstern type bivariate generalized exponential distribution using ranked set sampling. SORT 38(2), 161–180 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Tahmasebi, S., Jafari, A.A.: Concomitants of order statistics and record values from Morgenstern type bivariate-generalized exponential distribution. Bull. Malays. Math. Sci. Soc. 38, 1411–1423 (2015)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Tahmasebi, S., Jafari, A.A., Afshari, M.: Concomitants of dual generalized order statistics from Morgenstern type bivariate generalized exponential distribution. J. Stat. Theory Appl. 14(1), 1–12 (2015)MathSciNetGoogle Scholar
  31. 31.
    Tahmasebi, S., Jafari, A.A., Ahsanullah, M.: Properties on concomitants of generalized order statistics from a bivariate Rayleigh distribution. Bull. Malays. Math. Sci. Soc (2016). doi: 10.1007/s40840-015-0297-8 zbMATHGoogle Scholar
  32. 32.
    Verhulstt, P.F.: Notice sur la loi la population suit dans son accroissement. Correspondence mathematique et physique publiee L. A. J. Quetelet 10, 112–121 (1838)Google Scholar
  33. 33.
    Verhulst, P.F.: Recherches matheatiques sur la loi d’accorissement de la population. Nouvelles Mem. Ser. (2J) 18, 38 (1845)Google Scholar
  34. 34.
    Verhulst, P.F.: Deuxieme memoire sur la loi d’accorissement de la population. Memoires de L’Academie Royale des Sciences des Lettres et des beaux-Arts de Belgique Ser. 2 20, 32 (1847)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceZagazig UniversityZagazigEgypt
  2. 2.Higher Technological Institute10th of Ramadan cityEgypt

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