Bivariate Gumbel-G Family of Distributions: Statistical Properties, Bayesian and Non-Bayesian Estimation with Application

  • M. S. EliwaEmail author
  • M. El-Morshedy


In this paper, a new class of bivariate distributions called the bivariate Gumbel-G family is proposed, whose marginal distributions are Gumbel-G families. Several of its statistical properties are derived. After introducing the general class, a special model of the new family is discussed in-detail. Bayesian and maximum likelihood techniques are used to estimate the model parameters. Simulation study is carried out to examine the bias and mean square error of Bayesian and maximum likelihood estimators. Finally, a real data set is analyzed for illustrative the flexibility of the proposed bivariate family.


Gumbel-G family of distributions Bivariate distributions Bayesian estimation Maximum likelihood estimation Simulation 



  1. 1.
    Al-Aqtash R (2013) On generating a new family of distributions using the log function. Ph.D thesis, Central Michigan University Mount Pleasant, Michigan.
  2. 2.
    Al-Aqtash R, Famoye F, Lee C (2015) On generating a new family of distributions using the log function. J Probab Stat Sci 13(1):135–152Google Scholar
  3. 3.
    Al-Aqtash R, Lee C, Famoye F (2014) Gumbel–Weibull distribution: properties and applications. J Mod Appl Stat Methods 13:201–225Google Scholar
  4. 4.
    Alizadeh M, Cordeiro GM, Nascimento AD, Lima MD, Ortega EM (2016) Odd-Burr generalized family of distributions with some applications. J Stat Comput Simul 83:326–339Google Scholar
  5. 5.
    Alizadeh M, Ramires TG, MirMostafaee SMTK, Samizadeh M, Ortega EM (2018) A new useful four-parameter extension of the Gumbel distribution: properties, regression model and applications using the GAMLSS framework. Commun Stat Simul Comput.
  6. 6.
    Al-Ruzaiza AS, El-Gohary A (2007) A new class of positively quadrant dependent bivariate distributions with Pareto. Int Math Forum 2(26):1259–1273Google Scholar
  7. 7.
    Andrad TA, Heloisa R, Marcelo B, Cordeiro GM (2015) The exponentiated generalized Gumbel distribution. Rev Colomb Estad 38(1):123–143Google Scholar
  8. 8.
    Balakrishnan N, Shiji K (2014) On a class of bivariate exponential distributions. Stat Probab Lett 85:153–160Google Scholar
  9. 9.
    Barlow RE, Proschan F (1981) Statistical theory of reliability and life testing. Probability models, Silver Spring, MarylandGoogle Scholar
  10. 10.
    Basu AP (1971) Bivariate failure rate. J Am Stat Assoc 66:103–104Google Scholar
  11. 11.
    Bismi G (2005) Bivariate Burr distributions. Ph.D. thesis. Cochin University of Science and Technology, IndiaGoogle Scholar
  12. 12.
    Cordeiro GM, Alizadeh M, Ozel G, Hosseini B, Ortega EM, Altun E (2017) The generalized odd log-logistic family of distributions: properties, regression models and applications. J Stat Comput Simul 87(5):908–932Google Scholar
  13. 13.
    Cordeiro GM, Nadarajah S, Ortega EM (2012) The Kumaraswamy Gumbel distribution. Stat Methods Appl 21:139–168Google Scholar
  14. 14.
    Domma F (2009) Some properties of the bivariate Burr type III distribution. Statistics.
  15. 15.
    El-Bassiouny AH, EL-Damcese M, Abdelfattah M, Eliwa MS (2016) Bivariate exponentaited generalized Weibull–Gompertz distribution. J Appl Probab Stat 11(1):25–46Google Scholar
  16. 16.
    El-Gohary A, El-Bassiouny AH, El-Morshedy M (2016) Bivariate exponentiated modified Weibull extension distribution. J Stat Appl Probab 5(1):67–78Google Scholar
  17. 17.
    Gumbel EJ (1958) Statistics of extremes. Columbia University Press, New YorkGoogle Scholar
  18. 18.
    Hiba ZM (2016) Bivariate inverse Weibull distribution. J Stat Comput Simul 86(12):2335–2345Google Scholar
  19. 19.
    Klara P, Jesper R (2010) Exponentiated Gumbel distribution for estimation of return levels of significant wave height. J Environ Stat 1(3).
  20. 20.
    Kotz S, Nadarajah S (2000) Extreme value distributions: theory and applications. Imperial College Press, LondonGoogle Scholar
  21. 21.
    Kundu D, Gupta K (2013) Bayes estimation for the Marshall–Olkin bivariate Weibull distribution. J Comput Stat Data Anal 57(1):271–281Google Scholar
  22. 22.
    Kundu D, Gupta RD (2009) Bivariate generalized exponential distribution. J Multivar Anal 100:581–593Google Scholar
  23. 23.
    Lai CD, Xie M (2000) A new family of positive quadrant dependent bivarite distributions. Stat Probab Lett 46:359–364Google Scholar
  24. 24.
    Lehmann EL (1966) Some concepts of dependence. Ann Math Stat 37:1137–1153Google Scholar
  25. 25.
    Marshall AW, Olkin I (1967) A multivariate exponential model. J Am Stat Assoc 62:30–44Google Scholar
  26. 26.
    Marshall AW, Olkin I (1997) A new methods for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652Google Scholar
  27. 27.
    Meintanis SG (2007) Test of fit for Marshall–Olkin distributions with applications. J Stat Plan Inference 137:3954–3963Google Scholar
  28. 28.
    Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1091Google Scholar
  29. 29.
    Mohamed I, Eliwa MS, El-Morshedy M (2017) Bivariate exponentiated generalized linear exponential distribution with applications in reliability analysis. arXiv:1710.00502
  30. 30.
    Nadarajah S (2006) The exponentiated Gumbel distribution with climate application. Environmetrics 17:13–23Google Scholar
  31. 31.
    Nadarajah S, Kotz S (2004) The beta Gumbel distribution. Math Probl Eng 4:323–332Google Scholar
  32. 32.
    Nadarajah S, Kotz S (2006) The exponentiated type distributions. Acta Appl Math 92:97–111Google Scholar
  33. 33.
    Nelsen RB (1999) An introduction to copulas, 2nd edn. Springer, New YorkGoogle Scholar
  34. 34.
    Roozegar R, Jafari A (2016) On bivariate exponentiated extended Weibull family of distributions. Ciênc nat St Maria 38(2):564–576Google Scholar
  35. 35.
    Sarhan A, Balakrishnan N (2007) A new class of bivariate distributions and its mixture. J Multivar Anal 98:1508–1527Google Scholar
  36. 36.
    Sarhan A, Hamilton DC, Smith B, Kundu D (2011) The bivariate generalized linear failure rate distribution and its multivariate extension. Comput Stat Data Anal 55(1):644–654Google Scholar
  37. 37.
    Scarf P (1992) Estimation for a four parameter generalized extreme value distribution. Commun Stat Theory Methods 21:2185–2201Google Scholar
  38. 38.
    Silva FG, Percontini A, de Brito E, Ramos MW, Venancio R, Cordeiro GM (2017) The odd Lindley-G family of distributions. Aust J Stat 46:65–87Google Scholar
  39. 39.
    Tahir MH, Adnan HM, Cordeiro GM, Hamedani GG, Mansoor M, Zubair M (2016) The Gumbel–Lomax distribution: properties and applications. J Stat Theory Appl 15(1):61–79Google Scholar
  40. 40.
    Tahir MH, Zubair M, Mansoor M, Cordeiro GM, Alizadeh M, Hamedani GG (2016) A new Weibull-G family of distributions. Hacet J Math Stat 45(2):629–647Google Scholar
  41. 41.
    Wagner BS, Artur JL (2013) Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes. Statistics 47(6):1321–1342Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

Personalised recommendations