Published

2015-01-01

The Exponentiated Generalized Gumbel Distribution

Distribución Gumbel exponencializada generalizada

DOI:

https://doi.org/10.15446/rce.v38n1.48806

Keywords:

Gumbel Distribution, Maximum Likelihood, Moment, Rényi Entropy (en)
Distribución Gumbel, Entropía de Rényi, Máxima verosimilitud, Momentos. (es)

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Authors

  • Thiago Andrade Universidade Federal de Pernambuco
  • Heloisa Rodrigues Universidade Federal de Pernambuco
  • Marcelo Bourguignon Universidade Federal do Piauí
  • Gauss Cordeiro Universidade Federal de Pernambuco

A class of univariate distributions called the exponentiated generalized class was recently proposed in the literature. A four-parameter model within this class named the exponentiated generalized Gumbel distribution is defined. We discuss the shapes of its density function and obtain explicit expressions for the ordinary moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. The density function of the order statistic is derived. The method of maximum likelihood is used to estimate model parameters. We determine the observed information matrix. We provide a Monte Carlo simulation study to evaluate the maximum likelihood estimates of model parameters and two applications to real data to illustrate the importance of the new model.

Recientemente fue propuesta una clase de distribuciones univariadas conocida como la clase exponencializada generalizada. Dentro de esta clase se define un modelo con cuatro parámetros conocido como distribución Gumbel exponencializada generalizada. En este artículo estudiamos las formas de la función de densidad de este modelo, obtenemos expresiones explicitas para los momentos ordinarios, las funciones generadora de momentos y cuantílica, para los desvíos medios, las curvas de Bonferroni y Lorenz, y, para la entropía de Rényi. Derivamos la función de densidad de la estadística de orden. Usamos el método de máxima verosimilitud para estimar los parámetros del modelo. Determinamos la matriz de información observada. Presentamos una simulación de Monte Carlo que evalúa las estimativas de máxima verosimilitud de los parámetros del modelo y presentamos dos aplicaciones a datos reales que ilustran la importancia del modelo nuevo.

https://doi.org/10.15446/rce.v38n1.48806

The Exponentiated Generalized Gumbel Distribution

Distribución Gumbel exponencializada generalizada

THIAGO ANDRADE1, HELOISA RODRIGUES2, MARCELO BOURGUIGNON3, GAUSS CORDEIRO4

1Universidade Federal de Pernambuco, Centro de Ciências Exatas e da Natureza, Departamento de Estatística, Recife, Brasil. Graduate student. Email: thiagoan.andrade@gmail.com
2Universidade Federal de Pernambuco, Centro de Ciências Exatas e da Natureza, Departamento de Estatística, Recife, Brasil. Graduate student. Email: heloisa.mrodrigues@ufpe.br
3Universidade Federal do Piauí, Centro de Ciências da Natureza, Departamento de Estatística, Teresina, Brasil. Assistent Professor. Email: m.p.bourguignon@gmail.com
4Universidade Federal de Pernambuco, Centro de Ciências Exatas e da Natureza, Departamento de Estatística, Recife, Brasil. Professor. Email: gauss@de.ufpe.br

Abstract

A class of univariate distributions called the exponentiated generalized class was recently proposed in the literature. A four-parameter model within this class named the exponentiated generalized Gumbel distribution is defined. We discuss the shapes of its density function and obtain explicit expressions for the ordinary moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. The density function of the order statistic is derived. The method of maximum likelihood is used to estimate model parameters. We determine the observed information matrix. We provide a Monte Carlo simulation study to evaluate the maximum likelihood estimates of model parameters and two applications to real data to illustrate the importance of the new model.

Key words: Gumbel Distribution, Maximum Likelihood, Moment, Rényi Entropy.

Resumen

Recientemente fue propuesta una clase de distribuciones univariadas conocida como la clase exponencializada generalizada. Dentro de esta clase se define un modelo con cuatro parámetros conocido como distribución Gumbel exponencializada generalizada. En este artículo estudiamos las formas de la función de densidad de este modelo, obtenemos expresiones explicitas para los momentos ordinarios, las funciones generadora de momentos y cuantílica, para los desvíos medios, las curvas de Bonferroni y Lorenz, y, para la entropía de Rényi. Derivamos la función de densidad de la estadística de orden. Usamos el método de máxima verosimilitud para estimar los parámetros del modelo. Determinamos la matriz de información observada. Presentamos una simulación de Monte Carlo que evalúa las estimativas de máxima verosimilitud de los parámetros del modelo y presentamos dos aplicaciones a datos reales que ilustran la importancia del modelo nuevo.

Palabras clave: distribución Gumbel, entropía de Rényi, máxima verosimilitud, momentos.

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References

1. Carrasco, J. M. F., Ortega, E. M. M. & Cordeiro, G. M. (2008), 'A generalized modified Weibull distribution for lifetime modeling', Computational Statistics and Data Analysis 53, 450-462.

2. Chen, G. & Balakrishnan, N. (1995), 'A general purpose approximate goodness-of-fit test', Journal of Quality Technology 27, 154-161.

3. Cordeiro, G. M., Nadarajah, S. & Ortega, E. M. M. (2012), 'The Kumaraswamy Gumbel distribution', Statistical Methods and Applications 21, 139-168.

4. Cordeiro, G. M., Ortega, E. M. M. & Cunha, D. C. C. (2013), 'The exponentiated generalized class of distributions', Journal of Data Science 11, 1-27.

5. Cordeiro, G. M., Ortega, E. M. M. & Silva, G. O. (2011), 'The exponentiated generalized Gamma distribution with application to lifetime data', Journal of Statistical Computation and Simulation 81, 827-842.

6. Cordeiro, G. M. & de Castro, M. (2011), 'A new family of generalized distributions', Journal of Statistical Computation and Simulation 81, 883-893.

7. Eugene, N., Lee, C. & Famoye, F. (2002), 'Beta-normal distribution and its applications', Communications in Statistics - Theory and Methods 31, 497-512.

8. Gupta, R. D. & Kundu, D. (2001), 'Exponentiated exponential distribution: An alternative to Gamma and Weibull distributions', Biometrical Journal 43, 117-130.

9. Hinkley, D. (1977), 'On quick choice of power transformations', The American Statistician 26, 67-69.

10. Kenney, J. F. & Keeping, E. S. (1962), Mathematics of Statistics, 3 edn, Chapman & Hall Ltd, New Jersey.

11. Kotz, S. & Nadarajah, S. (2000), Extreme Value Distributions: Theory and Applications, Imperial College Press, London.

12. Marshall, A. N. & Olkin, I. (1997), 'A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families', Biometrika 84, 641-652.

13. Moors, J. J. (1988), 'A quantile alternative for kurtosis', Journal of the Royal Statistical Society: Series D 37, 25-32.

14. Murthy, D. N. P., Xie, M. & Jiang, R. (2004), Weibull Models, Wiley series in probability and statistics, John Wiley & Sons, NJ.

15. Nadarajah, S. (2006), 'The exponentiated Gumbel distribution with climate application', Environmetrics 17, 13-23.

16. Nadarajah, S. & Gupta, A. K. (2007), 'The exponentiated Gamma distribution with application to drought data', Calcutta Statistical Association Bulletin 59, 29-54.

17. Nadarajah, S. & Kotz, S. (2003), 'The exponentiated Fréchet distribution', Statistics on the Internet. *http://interstat.statjournals.net/YEAR/2003/articles/0312002.pdf

18. Nadarajah, S. & Kotz, S. (2004), 'The beta Gumbel distribution', Mathematical Problems in Engineering 4, 323-332.

19. Ramos, M. W., Marinho, P. R., Silva, R. V. & Cordeiro, G. M. (2013), 'The Exponentiated Lomax Poisson Distribution with an Application to lifetime data', Advances and Applications in Statistics 34, 107-135.

20. Ristic, M. M. & Balakrishnan, N. (2011), 'The Gamma exponentiated exponential distribution', Journal of Statistical Computation and Simulation. doi: 10.1080/00949655.2011.574633.

21. Rêgo, L. C., Cintra, R. J. & Cordeiro, G. M. (2012), 'On some properties of the Beta normal distribution', Communications in Statistics - Theory and Methods 41, 3722-3738.

22. Shirke, D. T. & Kakde, C. S. (2006), 'On exponentiated lognormal distribution', International Journal of Agricultural and Statistical Sciences 2, 319-326.

23. Zea, L. M., Silva, R. B., Bourguignon, M., Santos, A. M. & Cordeiro, G. M. (2012), 'The Beta exponentiated Pareto distribution with application to Bladder Cancer susceptibility', International Journal of Statistics and Probability 1, 8-19.

24. Zografos, K. & Balakrishnan, N. (2009), 'On families of Beta- and generalized Gamma-generated distributions and associated inference', Statistical Methodology 6, 344-362.

[Recibido en diciembre de 2013. Aceptado en octubre de 2014]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv38n1a07,
    AUTHOR  = {Andrade, Thiago and Rodrigues, Heloisa and Bourguignon, Marcelo and Cordeiro, Gauss},
    TITLE   = {{The Exponentiated Generalized Gumbel Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {1},
    pages   = {123-143}
}

References

Carrasco, J. M. F., Ortega, E. M. M. & Cordeiro, G. M. (2008), ‘A generalized modified Weibull distribution for lifetime modeling’, Computational Statistics and Data Analysis 53, 450–462.

Chen, G. & Balakrishnan, N. (1995), ‘A general purpose approximate goodness- of- fit test’, Journal of Quality Technology 27, 154–161.

Cordeiro, G. M. & de Castro, M. (2011), ‘A new family of generalized distributions’, Journal of Statistical Computation and Simulation 81, 883–893.

Cordeiro, G. M., Nadarajah, S. & Ortega, E. M. M. (2012), ‘The Kumaraswamy Gumbel distribution’, Statistical Methods and Applications 21, 139–168.

Cordeiro, G. M., Ortega, E. M. M. & Cunha, D. C. C. (2013), ‘The exponentiated generalized class of distributions’, Journal of Data Science 11, 1–27.

Cordeiro, G. M., Ortega, E. M. M. & Silva, G. O. (2011), ‘The exponentiated generalized Gamma distribution with application to lifetime data’, Journal of Statistical Computation and Simulation 81, 827–842.

Eugene, N., Lee, C. & Famoye, F. (2002), ‘Beta-normal distribution and its applications’, Communications in Statistics - Theory and Methods 31, 497–512.

Gupta, R. D. & Kundu, D. (2001), ‘Exponentiated exponential distribution: An alternative to Gamma and Weibull distributions’, Biometrical Journal 43, 117– 130.

Hinkley, D. (1977), ‘On quick choice of power transformations’, The American Statistician 26, 67–69.

Kenney, J. F. & Keeping, E. S. (1962), Mathematics of Statistics, 3 edn, Chapman & Hall Ltd, New Jersey.

Kotz, S. & Nadarajah, S. (2000), Extreme Value Distributions: Theory and Applications, Imperial College Press, London.

Marshall, A. N. & Olkin, I. (1997), ‘A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families’, Biometrika 84, 641–652.

Moors, J. J. (1988), ‘A quantile alternative for kurtosis’, Journal of the Royal Statistical Society: Series D 37, 25–32.

Murthy, D. N. P., Xie, M. & Jiang, R. (2004), Weibull Models, Wiley series in probability and statistics, John Wiley & Sons, NJ.

Nadarajah, S. (2006), ‘The exponentiated Gumbel distribution with climate application’, Environmetrics 17, 13–23.

Nadarajah, S. & Gupta, A. K. (2007), ‘The exponentiated Gamma distribution with application to drought data’, Calcutta Statistical Association Bulletin 59, 29–54.

Nadarajah, S. & Kotz, S. (2003), ‘The exponentiated Fréchet distribution’, Statistics on the Internet.

*http://interstat.statjournals.net/YEAR/2003/articles/0312002.pdf

Nadarajah, S. & Kotz, S. (2004), ‘The beta Gumbel distribution’, Mathematical Problems in Engineering 4, 323–332.

Ramos, M. W., Marinho, P. R., Silva, R. V. & Cordeiro, G. M. (2013), ‘The Exponentiated Lomax Poisson Distribution with an Application to lifetime data’, Advances and Applications in Statistics 34, 107–135.

Rêgo, L. C., Cintra, R. J. & Cordeiro, G. M. (2012), ‘On some properties of the Beta normal distribution’, Communications in Statistics - Theory and Methods 41, 3722–3738.

Ristic, M. M. & Balakrishnan, N. (2011), ‘The Gamma exponentiated exponential distribution’, Journal of Statistical Computation and Simulation. doi: 10.1080/00949655.2011.574633.

Shirke, D. T. & Kakde, C. S. (2006), ‘On exponentiated lognormal distribution’, International Journal of Agricultural and Statistical Sciences 2, 319–326.

Tahir, M. & Nadarajah, S. (2013), ‘Parameter induction in continuous univariate distributions - Part I: Well-established G-classes’, Communications in Statistics - Theory and Methods.

Zea, L. M., Silva, R. B., Bourguignon, M., Santos, A. M. & Cordeiro, G. M. (2012), ‘The Beta exponentiated Pareto distribution with application to Bladder Cancer susceptibility’, International Journal of Statistics and Probability 1, 8–19.

Zografos, K. & Balakrishnan, N. (2009), ‘On families of Beta- and generalized Gamma-generated distributions and associated inference’, Statistical Methodology 6, 344–362.

How to Cite

APA

Andrade, T., Rodrigues, H., Bourguignon, M. and Cordeiro, G. (2015). The Exponentiated Generalized Gumbel Distribution. Revista Colombiana de Estadística, 38(1), 123–143. https://doi.org/10.15446/rce.v38n1.48806

ACM

[1]
Andrade, T., Rodrigues, H., Bourguignon, M. and Cordeiro, G. 2015. The Exponentiated Generalized Gumbel Distribution. Revista Colombiana de Estadística. 38, 1 (Jan. 2015), 123–143. DOI:https://doi.org/10.15446/rce.v38n1.48806.

ACS

(1)
Andrade, T.; Rodrigues, H.; Bourguignon, M.; Cordeiro, G. The Exponentiated Generalized Gumbel Distribution. Rev. colomb. estad. 2015, 38, 123-143.

ABNT

ANDRADE, T.; RODRIGUES, H.; BOURGUIGNON, M.; CORDEIRO, G. The Exponentiated Generalized Gumbel Distribution. Revista Colombiana de Estadística, [S. l.], v. 38, n. 1, p. 123–143, 2015. DOI: 10.15446/rce.v38n1.48806. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/48806. Acesso em: 29 jan. 2025.

Chicago

Andrade, Thiago, Heloisa Rodrigues, Marcelo Bourguignon, and Gauss Cordeiro. 2015. “The Exponentiated Generalized Gumbel Distribution”. Revista Colombiana De Estadística 38 (1):123-43. https://doi.org/10.15446/rce.v38n1.48806.

Harvard

Andrade, T., Rodrigues, H., Bourguignon, M. and Cordeiro, G. (2015) “The Exponentiated Generalized Gumbel Distribution”, Revista Colombiana de Estadística, 38(1), pp. 123–143. doi: 10.15446/rce.v38n1.48806.

IEEE

[1]
T. Andrade, H. Rodrigues, M. Bourguignon, and G. Cordeiro, “The Exponentiated Generalized Gumbel Distribution”, Rev. colomb. estad., vol. 38, no. 1, pp. 123–143, Jan. 2015.

MLA

Andrade, T., H. Rodrigues, M. Bourguignon, and G. Cordeiro. “The Exponentiated Generalized Gumbel Distribution”. Revista Colombiana de Estadística, vol. 38, no. 1, Jan. 2015, pp. 123-4, doi:10.15446/rce.v38n1.48806.

Turabian

Andrade, Thiago, Heloisa Rodrigues, Marcelo Bourguignon, and Gauss Cordeiro. “The Exponentiated Generalized Gumbel Distribution”. Revista Colombiana de Estadística 38, no. 1 (January 1, 2015): 123–143. Accessed January 29, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/48806.

Vancouver

1.
Andrade T, Rodrigues H, Bourguignon M, Cordeiro G. The Exponentiated Generalized Gumbel Distribution. Rev. colomb. estad. [Internet]. 2015 Jan. 1 [cited 2025 Jan. 29];38(1):123-4. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/48806

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