Published

2015-07-01

A Bimodal Extension of the Generalized Gamma Distribution

Una extensión bimodal de la distribución gamma generalizada

DOI:

https://doi.org/10.15446/rce.v38n2.51666

Keywords:

Bimodality, Exponential Power Distribution, Generalized Gamma, Skewness (en)
Bimodalidad, Distribución potencia exponencial, Gamma generalizada, Sesgo. (es)

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Authors

  • Mehmet Niyazi Çankaya Ankara University, Ankara, Turkey
  • Yakup Murat Bulut Osmangazi University, Eskisehir, Turkey
  • Fatma Zehra Dogru Ankara University, Ankara, Turkey
  • Olcay Arslan Ankara University, Ankara, Turkey
A bimodal extension of the generalized gamma distribution is proposed by using a mixing approach. Some distributional properties of the new distribution are investigated. The maximum likelihood (ML) estimators for the parameters of the new distribution are obtained. Real data examples are given to show the strength of the new distribution for modeling data.

Una extensión bimodal de la distribución gamma generalizada es propuesta a través de un enfoque de mixturas. Algunas propiedades de la nueva distribución son investigadas. Los estimadores máximo verosímiles (ML por sus siglas en inglés) de los parámetros de la nueva distribución son obtenidos. Algunos ejemplos con datos reales son utilizados con el fin de mostrar las fortalezas de la nueva distribución en la modelación de datos.

https://doi.org/10.15446/rce.v38n2.51666

A Bimodal Extension of the Generalized Gamma Distribution

Una extensión bimodal de la distribución gamma generalizada

MEHMET NIYAZI ÇANKAYA1, YAKUP MURAT BULUT2, FATMA ZEHRA DOGRU3, OLCAY ARSLAN4

1Ankara University, Faculty of Science, Department of Statistics, Ankara, Turkey. Ph.D. Student. Email: mncankaya@ankara.edu.tr
2Osmangazi University, Faculty of Science and Letters, Department of Statistics, Eskisehir, Turkey. Ph.D. Student. Email: ymbulut@ogu.edu.tr
3Ankara University, Faculty of Science, Department of Statistics, Ankara, Turkey. Ph.D. Student. Email: fzdogru@ankara.edu.tr
4Ankara University, Faculty of Science, Department of Statistics, Ankara, Turkey. Professor. Email: oarslan@ankara.edu.tr

Abstract

A bimodal extension of the generalized gamma distribution is proposed by using a mixing approach. Some distributional properties of the new distribution are investigated. The maximum likelihood (ML) estimators for the parameters of the new distribution are obtained. Real data examples are given to show the strength of the new distribution for modeling data.

Key words: Bimodality, Exponential Power Distribution, Generalized Gamma, Skewness.

Resumen

Una extensión bimodal de la distribución gamma generalizada es propuesta a través de un enfoque de mixturas. Algunas propiedades de la nueva distribución son investigadas. Los estimadores máximo verosímiles (ML por sus siglas en inglés) de los parámetros de la nueva distribución son obtenidos. Algunos ejemplos con datos reales son utilizados con el fin de mostrar las fortalezas de la nueva distribución en la modelación de datos.

Palabras clave: bimodalidad, distribución potencia exponencial, gamma generalizada, sesgo.

Texto completo disponible en PDF

References

1. Abdulah, E. & Elsalloukh, H. (2013), 'Analyzing skewed data with the epsilon skew Gamma distribution', Journal of Statistics Applications & Probability 2(3), 195-202.

2. Abdulah, E. & Elsalloukh, H. (2014), 'Bimodal class based on the inverted symmetrized Gamma distribution with applications', Journal of Statistics Applications & Probability 3(1), 1-7.

3. Ahmed, S. E., Goria, M. N. & Hussein, A. (2008), 'Gamma mixture: Bimodality, inflexions and L-moments', Communications in Statistics - Theory and Methods 37(8), 1147-1161.

4. Arellano-Valle, R. B., Cortées, M. A. & Góomez, Héector W. (2010), 'An extension of the epsilon-skew-normal distribution', Communications in Statistics - Theory and Methods 39(5), 912-922.

5. Celik, N., Senoglu, B. & Arslan, O. (2015), 'Estimation and testing in one-way ANOVA when the errors are skew-normal', Revista Colombiana de Estadística 38(1), 75-91.

6. Cooray, K. (2013), 'Exponentiated sinh Cauchy distribution with applications', Communications in Statistics - Theory and Methods 42(21), 3838-3852.

7. Cruz-Medina, I. R. (2001), Almost nonparametric and nonparametric estimation in mixture models, Ph.D. Thesis, Pennsylvania State University, Pensilvania.

8. Elal-Olivero, D. (2010), 'Alpha-skew-normal distribution', Proyecciones Journal of Mathematics 29(3), 224-240.

9. Elsalloukh, H., Guardiola, J. H. & Young, M. (2005), 'The epsilon-skew exponential power distribution family', Far East Journal of Theoretical Statistics 16, 97-112.

10. Eugene, N., Lee, C. & Famoye, F. (2002), 'Beta-normal distribution and its applications', Communications in Statistics - Theory and methods 31(4), 497-512.

11. Famoye, F., Lee, C. & Eugene, N. (2004), 'Beta-normal distribution: Bimodality properties and application', Journal of Modern Applied Statistical Methods 3(1), 85-103.

12. Gómez, Héctor W., Elal-Olivero, D., Salinas, H. S. & Bolfarine, H. (2011), 'Bimodal extension based on the skew-normal distribution with application to pollen data', Environmetrics 22(1), 50-62.

13. Genc, A. I. (2013), 'A skew extension of the slash distribution via beta-normal distribution', Statistical Papers 54(2), 427-442.

14. Gui, W. (2014), 'A generalization of the slashed distribution via alpha skew normal distribution', Statistical Methods & Applications 23(4), 547-563.

15. Gómez, Y. M., Bolfarine, H. & Gómez, H. W. (2014), 'A new extension of the exponential distribution', Revista Colombiana de Estadística 37(1), 25-34.

16. Hassan, Y. M. & Hijazi, R. H. (2010), 'A bimodal exponential power distribution', Pakistan Journal of Statistics 26(2), 379-396.

17. Iriarte, Y. A., Gómez, H. W., Varela, H. & Bolfarine, H. (2015), 'Slashed Rayleigh distribution', Revista Colombiana de Estadística 38(1), 31-44.

18. Jamalizadeh, A., Arabpour, A. R. & Balakrishnan, N. (2011), 'A generalized skew two-piece skew-normal distribution', Statistical Papers 52(2), 431-446.

19. Martínez-Flórez, G., Vergara-Cardozo, S. & González, L. M. (2013), 'The family of log-skew-normal alpha-power distributions using precipitation data', Revista Colombiana de Estadística 36(1), 43-57.

20. Mudholkar, G. S. & Hutson, A. D. (2000), 'The epsilon-skew-normal distribution for analyzing near-normal data', Journal of Statistical Planning and Inference 83(2), 291-309.

21. Pereira, J. R., Marques, L. A. & da Costa, J. M. (2012), 'An empirical comparison of EM initialization methods and model choice criteria for mixtures of skew-normal distributions', Revista Colombiana de Estadística 35(3), 457-478.

22. R\^ego, L. C., Cintra, R. J. & Cordeiro, G. M. (2012), 'On some properties of the beta normal distribution', Communications in Statistics - Theory and Methods 41(20), 3722-3738.

23. Rocha, G. H. M. A., Loschi, R. H. & Arellano-Valle, R. B. (2013), 'Inference in flexible families of distributions with normal kernel', Statistics 47(6), 1184-1206.

24. Salinas, H. S., Martínez-Flórez, G. & Moreno-Arenas, G. (2013), 'Censored bimodal symmetric-asymmetric alpha-power model', Revista Colombiana de Estadística 36(2), 287-303.

25. Sanhueza, A., Leiva, V. & Balakrishnan, N. (2008), 'The generalized Birnbaum-Saunders distribution and its theory, methodology, and application', Communications in Statistics - Theory and Methods 37(5), 645-670.

26. Sanhueza, A., Leiva, V. & López-Kleine, L. (2011), 'On the Student-t mixture inverse gaussian model with an application to protein production', Revista Colombiana de Estadística 34(1), 177-195.

27. Shams, H. S. & Alamatsaz, M. H. (2013), 'Alpha-skew-Laplace distribution', Statistics & Probability Letters 83(3), 774-782.

28. Torres-Avilés, F. J., Icaza, G. & Arellano-Valle, R. B. (2012), 'An extension to the scale mixture of normals for bayesian small-area estimation', Revista Colombiana de Estadística 35(2), 185-204.

29. Varadhan, R. & Gilbert, P. D. (2009), 'BB: An R package for solving a large system of nonlinear equations and for optimizing a high-dimensional nonlinear objective function', Journal of Statistical Software 32(4), 1-26.

[Recibido en julio de 2014. Aceptado en enero de 2015]

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

@ARTICLE{RCEv38n2a04,
    AUTHOR  = {\c{C}ankaya, Mehmet Niyazi and Bulut, Yakup Murat and Do\v{g}ru, Fatma Zehra and Arslan, Olcay},
    TITLE   = {{A Bimodal Extension of the Generalized Gamma Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {2},
    pages   = {371-384}
}

References

Abdulah, E. & Elsalloukh, H. (2013), ‘Analyzing skewed data with the épsilon skew Gamma Distribution’, Journal of Statistics Applications & Probability 2(3), 195–202.

Abdulah, E. & Elsalloukh, H. (2014), ‘Bimodal class based on the inverted symmetrized gamma distribution with applications’, Journal of Statistics Applications & Probability 3(1), 1–7.

Ahmed, S. E., Goria, M. N. & Hussein, A. (2008), ‘Gamma mixture: Bimodality, inflexions and L-moments’, Communications in Statistics - Theory and Methods 37(8), 1147–1161.

Arellano-Valle, R. B., Cortés, M. A. & Gómez, H. W. (2010), ‘An extension of the epsilon-skew-normal distribution’, Communications in Statistics - Theory and Methods 39(5), 912–922.

Celik, N., Senoglu, B. & Arslan, O. (2015), ‘Estimation and testing in one-way ANOVA when the errors are skew-normal’, Revista Colombiana de Estadística 38(1), 75–91.

Cooray, K. (2013), ‘Exponentiated sinh Cauchy distribution with applications’, Communications in Statistics - Theory and Methods 42(21), 3838–3852.

Cruz-Medina, I. R. (2001), Almost nonparametric and nonparametric estimation in mixture models, Ph.D. Thesis, Pennsylvania State University, Pensilvania.

Elal-Olivero, D. (2010), ‘Alpha-skew-normal distribution’, Proyecciones Journal of Mathematics 29(3), 224–240.

Elsalloukh, H., Guardiola, J. H. & Young, M. (2005), ‘The epsilon-skew exponential power distribution family’, Far East Journal of Theoretical Statistics 16, 97–112.

Eugene, N., Lee, C. & Famoye, F. (2002), ‘Beta-normal distribution and its applications’, Communications in Statistics - Theory and methods 31(4), 497–512.

Famoye, F., Lee, C. & Eugene, N. (2004), ‘Beta-normal distribution: Bimodality properties and application’, Journal of Modern Applied Statistical Methods 3(1), 85–103.

Genc, A. I. (2013), ‘A skew extension of the slash distribution via beta-normal distribution’, Statistical Papers 54(2), 427–442.

Gómez, Y. M., Bolfarine, H. & Gómez, H. W. (2014), ‘A new extension of the exponential distribution’, Revista Colombiana de Estadística 37(1), 25–34.

Gómez, H. W., Elal-Olivero, D., Salinas, H. S. & Bolfarine, H. (2011), ‘Bimodal extension based on the skew-normal distribution with application to pollen data’, Environmetrics 22(1), 50–62.

Gui, W. (2014), ‘A generalization of the slashed distribution via alpha skew normal distribution’, Statistical Methods & Applications 23(4), 547–563.

Hassan, Y. M. & Hijazi, R. H. (2010), ‘A bimodal exponential power distribution’, Pakistan Journal of Statistics 26(2), 379–396.

Iriarte, Y. A., Gómez, H. W., Varela, H. & Bolfarine, H. (2015), ‘Slashed Rayleigh distribution’, Revista Colombiana de Estadística 38(1), 31–44.

Jamalizadeh, A., Arabpour, A. R. & Balakrishnan, N. (2011), ‘A generalized skew two-piece skew-normal distribution’, Statistical Papers 52(2), 431–446.

Martínez-Flórez, G., Vergara-Cardozo, S. & González, L. M. (2013), ‘The family of log-skew-normal alpha-power distributions using precipitation data’, Revista Colombiana de Estadística 36(1), 43–57.

Mudholkar, G. S. & Hutson, A. D. (2000), ‘The epsilon-skew-normal distribution for analyzing near-normal data’, Journal of Statistical Planning and Inference 83(2), 291–309.

Pereira, J. R., Marques, L. A. & da Costa, J. M. (2012), ‘An empirical comparison of EM initialization methods and model choice criteria for mixtures of skewnormal distributions’, Revista Colombiana de Estadística 35(3), 457–478.

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(20), 3722–3738.

Rocha, G. H. M. A., Loschi, R. H. & Arellano-Valle, R. B. (2013), ‘Inference in flexible families of distributions with normal kernel’, Statistics 47(6), 1184– 1206.

Salinas, H. S., Martínez-Flórez, G. & Moreno-Arenas, G. (2013), ‘Censored bimodal symmetric-asymmetric alpha-power model’, Revista Colombiana de Estadística 36(2), 287–303.

Sanhueza, A., Leiva, V. & Balakrishnan, N. (2008), ‘The generalized Birnbaum- Saunders distribution and its theory, methodology, and application’, Communications in Statistics - Theory and Methods 37(5), 645–670

Sanhueza, A., Leiva, V. & López-Kleine, L. (2011), ‘On the Student-t mixture inverse gaussian model with an application to protein production’, Revista Colombiana de Estadística 34(1), 177–195.

Shams, H. S. & Alamatsaz, M. H. (2013), ‘Alpha-skew-Laplace distribution’, Statistics & Probability Letters 83(3), 774–782.

Torres-Avilés, F. J., Icaza, G. & Arellano-Valle, R. B. (2012), ‘An extension to the scale mixture of normals for bayesian small-area estimation’, Revista Colombiana de Estadística 35(2), 185–204.

Varadhan, R. & Gilbert, P. D. (2009), ‘BB: An R package for solving a large system of nonlinear equations and for optimizing a high-dimensional nonlinear objective function’, Journal of Statistical Software 32(4), 1–26.

How to Cite

APA

Çankaya, M. N., Bulut, Y. M., Dogru, F. Z. and Arslan, O. (2015). A Bimodal Extension of the Generalized Gamma Distribution. Revista Colombiana de Estadística, 38(2), 353–370. https://doi.org/10.15446/rce.v38n2.51666

ACM

[1]
Çankaya, M.N., Bulut, Y.M., Dogru, F.Z. and Arslan, O. 2015. A Bimodal Extension of the Generalized Gamma Distribution. Revista Colombiana de Estadística. 38, 2 (Jul. 2015), 353–370. DOI:https://doi.org/10.15446/rce.v38n2.51666.

ACS

(1)
Çankaya, M. N.; Bulut, Y. M.; Dogru, F. Z.; Arslan, O. A Bimodal Extension of the Generalized Gamma Distribution. Rev. colomb. estad. 2015, 38, 353-370.

ABNT

ÇANKAYA, M. N.; BULUT, Y. M.; DOGRU, F. Z.; ARSLAN, O. A Bimodal Extension of the Generalized Gamma Distribution. Revista Colombiana de Estadística, [S. l.], v. 38, n. 2, p. 353–370, 2015. DOI: 10.15446/rce.v38n2.51666. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/51666. Acesso em: 3 dec. 2024.

Chicago

Çankaya, Mehmet Niyazi, Yakup Murat Bulut, Fatma Zehra Dogru, and Olcay Arslan. 2015. “A Bimodal Extension of the Generalized Gamma Distribution”. Revista Colombiana De Estadística 38 (2):353-70. https://doi.org/10.15446/rce.v38n2.51666.

Harvard

Çankaya, M. N., Bulut, Y. M., Dogru, F. Z. and Arslan, O. (2015) “A Bimodal Extension of the Generalized Gamma Distribution”, Revista Colombiana de Estadística, 38(2), pp. 353–370. doi: 10.15446/rce.v38n2.51666.

IEEE

[1]
M. N. Çankaya, Y. M. Bulut, F. Z. Dogru, and O. Arslan, “A Bimodal Extension of the Generalized Gamma Distribution”, Rev. colomb. estad., vol. 38, no. 2, pp. 353–370, Jul. 2015.

MLA

Çankaya, M. N., Y. M. Bulut, F. Z. Dogru, and O. Arslan. “A Bimodal Extension of the Generalized Gamma Distribution”. Revista Colombiana de Estadística, vol. 38, no. 2, July 2015, pp. 353-70, doi:10.15446/rce.v38n2.51666.

Turabian

Çankaya, Mehmet Niyazi, Yakup Murat Bulut, Fatma Zehra Dogru, and Olcay Arslan. “A Bimodal Extension of the Generalized Gamma Distribution”. Revista Colombiana de Estadística 38, no. 2 (July 1, 2015): 353–370. Accessed December 3, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/51666.

Vancouver

1.
Çankaya MN, Bulut YM, Dogru FZ, Arslan O. A Bimodal Extension of the Generalized Gamma Distribution. Rev. colomb. estad. [Internet]. 2015 Jul. 1 [cited 2024 Dec. 3];38(2):353-70. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/51666

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CrossRef citations7

1. Arturo Tozzi, James F. Peters, Mehmet Niyazi Çankaya. (2018). The informational entropy endowed in cortical oscillations. Cognitive Neurodynamics, 12(5), p.501. https://doi.org/10.1007/s11571-018-9491-3.

2. Najme Sharifipanah, Rahim Chinipardaz, Gholam Ali Parham. (2020). A new class of weighted bimodal distribution with application to gamma-ray burst duration data. Journal of Applied Statistics, 47(13-15), p.2785. https://doi.org/10.1080/02664763.2020.1815669.

3. Anderson Ara, Francisco Louzada. (2022). Alpha Skew Gaussian Naïve Bayes Classifier. International Journal of Information Technology & Decision Making, 21(01), p.441. https://doi.org/10.1142/S0219622021500644.

4. Roberto Vila, Victor Serra, Mehmet Niyazi Çankaya, Felipe Quintino. (2024). A general class of trimodal distributions: properties and inference. Journal of Applied Statistics, 51(8), p.1446. https://doi.org/10.1080/02664763.2023.2207785.

5. Mehmet Niyazi Çankaya, Jan Korbel. (2017). On statistical properties of Jizba–Arimitsu hybrid entropy. Physica A: Statistical Mechanics and its Applications, 475, p.1. https://doi.org/10.1016/j.physa.2017.02.009.

6. Emrah Altun, Hüseyin Tatlıdil, Gamze Özel. (2019). Conditional ASGT-GARCH Approach to Value-at-Risk. Iranian Journal of Science and Technology, Transactions A: Science, 43(1), p.239. https://doi.org/10.1007/s40995-018-0484-1.

7. Mehmet Niyazi Çankaya, Abdullah Yalçınkaya, Ömer Altındaǧ, Olcay Arslan. (2019). On the robustness of an epsilon skew extension for Burr III distribution on the real line. Computational Statistics, 34(3), p.1247. https://doi.org/10.1007/s00180-018-0859-y.

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