American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: Editor-in-chief: Mohamed Seddeek
Open Access
Journal Browser
American Journal of Applied Mathematics and Statistics. 2020, 8(1), 9-20
DOI: 10.12691/ajams-8-1-2
Open AccessArticle

A New Gumbel Generated Family of Distributions: Properties, Bivariate Distribution and Application

Elebe E. Nwezza1, , Chinonyerem V. Ogbuehi2, Uchenna U. Uwadi1 and C.O. Omekara2

1Department of Mathematics/Computer Science/Statistics/Informatics, Alex Ekwueme Federal University Ndufu alike, Ikwo, Nigeria

2Department of Statistics, Micheal Okpara University of Agriculture, Umudike, Nigeria

Pub. Date: January 19, 2020

Cite this paper:
Elebe E. Nwezza, Chinonyerem V. Ogbuehi, Uchenna U. Uwadi and C.O. Omekara. A New Gumbel Generated Family of Distributions: Properties, Bivariate Distribution and Application. American Journal of Applied Mathematics and Statistics. 2020; 8(1):9-20. doi: 10.12691/ajams-8-1-2


In this paper, we propose a new class of Gumbel generated distributions called Gumbel-Marshall-Olkin family of distributions. The new family of distributions is represented as linear mixture of exponentiated-G distribution. Some of the sub-models are presented. We derived some characterizations such as the quantile, moments, moment generating function, entropy and order statistics of the proposed family of distributions. The estimation of the unknown parameters of the new class of distribution is through the maximum likelihood. The consistency of the MLEs of the sub-model is assessed by means of simulation. Furthermore, we derive the bivariate density function of the new class of distributions. Two real life data sets are used to illustrate the potential usefulness of the sub-models of the proposed class of distributions. The results of the applications clearly indicate that the sub-models of the proposed class of distribution provided better fit among the other competing models.

Gumbel distribution Marshall-Olkin distribution Bivariate distribution Moment Maximum Likelihood

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit


[1]  Marshall A.W. and Olkin I. (1997). A new method for adding a parameter to a family of distribution with application to exponential and Weibull families. Biometrika, 84(3): 641-652.
[2]  Mudholkar G.S., Srivastava D.K. and Freimer M. (1995). The exponentiated Weibull family: A reanalysis of the bus Motor-failure data, Technometrics, 37(4) 436-445.
[3]  Eugene N., Lee C. and Famoye F., (2002). Beta-Normal distribution and its applications. Commun. Statis-theory meth., 31(4):497-512.
[4]  Shaw, W. T. and Buckley, I. R. C. (2007). The alchemy of probability distributions: beyond Gram-Charlier expansions and a skew-kurtotic-normal distribution from a rank transmutation map. Research report.
[5]  Zografos, K. and Balakrishnan, N. (2009). On families of beta and generalized gamma-generated distributions and associated inference, Statistical Methodology 6:344-362.
[6]  Cordeiro G. M and De Castro M. (2011). A new family of generalized distributions. Journal of statistical computation and simulation, 81(7): 883-898.
[7]  Alexander C., Cordeiro G.M., Ortega E.M.M and Sarabia J.M. (2012). Generalized beta generated distributions. Computational statistics and data analysis, 56:1880-1897.
[8]  Alzaatreh, A., Lee C. Famoye F. (2013). A new method for generating families of continuous distributions. Metron, 71: 63-79.
[9]  Al-Aqtash R., Lee C. and Famoye F. (2014). Gumbel-Weibull distribution: Properties and application. Journal of Modern applied statistical methods, 13(2)201-225.
[10]  Gupta R. D. and Kundu D. (2001). Exponentiated exponential family: An alternative to gamma and Weibull distributions. Biometrika Journal, 43(1):117-130.
[11]  Cordeiro G. M., Ortega E.M.M., and Da Cunha D.C.C. (2013). The Exponentiated generalized class of distributions. Journal of data science, 11: 1-27.
[12]  Nadarajah S. and Kotz S. (2006). The exponentiated type distributions. Acta Appl. Math., 92:97-111.
[13]  Cordeiro G.M., Alizadeh M., Ozel G., Hosseini B., Ortega E.M.M., and Altun E. (2016). The generalized odd log-logistic family of distributions:properties, regression models and applications, Journal of statistical computation and simulation.
[14]  Butler R. J. and McDonald J.B. (1989). Using Incomplete moments to measure inequality. Journal of Econometrics, 42: 109-119.
[15]  Rényi, A. (1961). On Measures of Entropy and Information. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, 547-561, University of California Press, Berkeley, Calif.
[16]  Nadarajah S., Cordeiro G.M., and Ortega E.M.M. (2015). The Zografos-Balakrishnan-G family of distributions: Mathematical properties and applications. Comm. Stat. Theory Methods, 44: 186-215.
[17]  Chen, G. and Balakrishnan, N., (1995). A General Purpose Approximate Goodness-of-Fit Test. Journal of Quality Technology, 27:2, 154-161.
[18]  Feigl P, Zelen M (1965) .Estimation of exponential probabilities with concomitant information. Biometrics, 21:826-838.
[19]  Lee C.., Famoye, F., and Olumolade, O. (2007). Beta-Weibull Distribution: Some Properties and Applications to Censored Data. Journal of Modern Applied Statistical Methods, 6(1): 173-186.
[20]  Meeker W. Q. and Escobar L.A., (1998). Statistical methods for reliability data. Wiley New York.
[21]  Tahir ,M. H., Alizadeh, M., Mansoor, M., Cordeiro, G. M., and Zubair, M. (2016). The Weibull-power function distribution with applications. Hacettepe Journal of Mathematics and Statistics, 45(1): 245-265.
[22]  Rajab, M., Aleem, M., Nawaz, T. and Daniyal M. (2013). On Five Parameter Beta Lomax Distribution. Journal of Statistics, 20: 102-118.
[23]  Tahir, M.H., Hussain, A.M., Cordeiro, G.M., Hamedani, G.G., Mansoor, M. and Zubair, M. (2015). The Gumbel-Lomax Distribution: Properties and Applications. Journal of Statistical Theory and Applications, 15 (1): 61-79.
[24]  El-Bassiouny, A.H., Abdo, N. F., and Shahen, H.S., (2015). Exponential Lomax Distribution. International Journal of Computer Applications, 121(13): 24-29.
[25]  Zubair, M., Cordeiro, G. M., Tahir, M. H., Mahmood, M., Mansoor, M. (2017). A Study of Logistic-Lomax Distribution and Its Applications. Journal of Probability and Statistical Science, 15(1): 29-46.