Published

2015-07-01

Alpha-Skew Generalized t Distribution

Distribución t generalizada alfa sesgada

DOI:

https://doi.org/10.15446/rce.v38n2.51665

Keywords:

Bimodality, Kurtosis, Maximum Likelihood Estimation, Modeling, Skewness (en)
Bimodalidad, Curtosis, Estimación máximo verosímil, Modelamiento, Sesgo. (es)

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Authors

  • Sukru Acitas Anadolu University, Eskisehir, Turkey
  • Birdal Senoglu Ankara University, Ankara, Turkey
  • Olcay Arslan Ankara University, Ankara, Turkey

The alpha-skew normal (ASN) distribution has been proposed recently in the literature by using standard normal distribution and a skewing approach. Although ASN distribution is able to model both skew and bimodal data, it is shortcoming when data has thinner or thicker tails than normal. Therefore, we propose an alpha-skew generalized t (ASGT) by using the generalized t (GT) distribution and a new skewing procedure. From this point of view, ASGT can be seen as an alternative skew version of GT distribution. However, ASGT differs from the previous skew versions of GT distribution since it is able to model bimodal data sest as well as it nests most commonly used density functions. In this paper, moments and maximum likelihood estimation of the parameters of ASGT distribution are given. Skewness and kurtosis measures are derived based on the first four noncentral moments. The cumulative distribution function (cdf) of ASGT distribution is also obtained. In the application part of the study, two real life problems taken from the literature are modeled by using ASGT distribution.

La distribución normal alfa-sesgada (ASN por sus siglas en inglés) ha sido propuesta recientemente en la literatura mediante el uso de una distribución normal estándar y procedimientos de sesgo. Aunque la distribución ASN es capaz de modelar tanto datos sesgados y bimodales, no es recomendada cuando los datos tienen colas más livianas o pesadas que la distribución normal. Por lo tanto, se propone una distribución t alfa-sesgada generalizada (ASGT por sus siglas en inglés) mediante el uso de la distribución t generalizada (GT por sus siglas en inglés) y un nuevo procedimiento de sesgo. Bajo este punto de vista, la distribución ASGT se puede ver como una alternativa sesgada de la distribución GT. Sin embargo, ASGT difiere de previas versiones sesgadas de la distribución GT puesto que es capaz de modelar datos bimodales y agrupa funciones de densidad más comúnmente usadas. En este artículo, los momentos y la estimación máximo verosímil de los parámetros de la distribución ASGT son derivadas. Medidas del sesgo y la curtosis son derivadas con base a los primeros cuatro momentos no centrales. La función de distribución acumulada (cdf por sus siglas en inglés) de la distribución ASGT es también obtenida. En la parte de aplicación del estudio, dos problemas reales tomados de la literatura son modelados usando la distribución ASGT.

https://doi.org/10.15446/rce.v38n2.51665

Alpha-Skew Generalized t Distribution

Distribución t generalizada alfa sesgada

SUKRU ACITAS1, BIRDAL SENOGLU2, OLCAY ARSLAN3

1Anadolu University, Faculty of Science, Department of Statistics, Eskisehir, Turkey. Professor. Email: sacitas@anadolu.edu.tr
2Ankara University, Faculty of Science, Department of Statistics, Ankara, Turkey. Professor. Email: senoglu@science.ankara.edu.tr
3Ankara University, Faculty of Science, Department of Statistics, Ankara, Turkey. Professor. Email: oarslan@ankara.edu.tr

Abstract

The alpha-skew normal (ASN) distribution has been proposed recently in the literature by using standard normal distribution and a skewing approach. Although ASN distribution is able to model both skew and bimodal data, it is shortcoming when data has thinner or thicker tails than normal. Therefore, we propose an alpha-skew generalized t (ASGT) by using the generalized t (GT) distribution and a new skewing procedure. From this point of view, ASGT can be seen as an alternative skew version of GT distribution. However, ASGT differs from the previous skew versions of GT distribution since it is able to model bimodal data sest as well as it nests most commonly used density functions. In this paper, moments and maximum likelihood estimation of the parameters of ASGT distribution are given. Skewness and kurtosis measures are derived based on the first four noncentral moments. The cumulative distribution function (cdf) of ASGT distribution is also obtained. In the application part of the study, two real life problems taken from the literature are modeled by using ASGT distribution.

Key words: Bimodality, Kurtosis, Maximum Likelihood Estimation, Modeling, Skewness.

Resumen

La distribución normal alfa-sesgada (ASN por sus siglas en inglés) ha sido propuesta recientemente en la literatura mediante el uso de una distribución normal estándar y procedimientos de sesgo. Aunque la distribución ASN es capaz de modelar tanto datos sesgados y bimodales, no es recomendada cuando los datos tienen colas más livianas o pesadas que la distribución normal. Por lo tanto, se propone una distribución t alfa-sesgada generalizada (ASGT por sus siglas en inglés) mediante el uso de la distribución t generalizada (GT por sus siglas en inglés) y un nuevo procedimiento de sesgo. Bajo este punto de vista, la distribución ASGT se puede ver como una alternativa sesgada de la distribución GT. Sin embargo, ASGT difiere de previas versiones sesgadas de la distribución GT puesto que es capaz de modelar datos bimodales y agrupa funciones de densidad más comúnmente usadas. En este artículo, los momentos y la estimación máximo verosímil de los parámetros de la distribución ASGT son derivadas. Medidas del sesgo y la curtosis son derivadas con base a los primeros cuatro momentos no centrales. La función de distribución acumulada (cdf por sus siglas en inglés) de la distribución ASGT es también obtenida. En la parte de aplicación del estudio, dos problemas reales tomados de la literatura son modelados usando la distribución ASGT.

Palabras clave: bimodalidad, curtosis, estimación máximo verosímil, \linebreak modelamiento, sesgo.

Texto completo disponible en PDF

References

1. Acitas, S., Senoglu, B. & Arslan, O. (2013), Alpha-skew generalized t distribution, International Conference Applied Statistics, Slovenia.

2. Arellano-Valle, R. B., Cortes, M. A. & Gomez, H. W. (2010), 'An extension of the epsilon-skew-normal distribution', Communications in Statistics - Theory and Methods 39, 912-922.

3. Arslan, O. (2004), 'Family of multivariate generalized t distributions', Journal of Multivariate Analysis 89, 329-337.

4. Arslan, O. & Genc, A. I. (2003), 'Robust location and scale estimation based on the univariate generalized t distribution', Communications in Statistics - Theory and Methods 32, 1505-1525.

5. Butler, R. J., McDonald, J. B., Nelson, R. D. & White, S. (1990), 'Robust and partially adaptive estimation of regression models', The Review of Economics and Statistics 72, 321-327.

6. Castillo, N. O., Gomez, H. W. & Bolfarine, H. (2011), 'Epsilon Birnbaum-Saunders distribution family: Properties and inference', Statistical Papers 52, 871-883.

7. 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.

8. Choy, S. T. B. & Chan, J. S. K. (2008), 'Scale mixtures distributions in statistical modelling', Australian & New Zealand Journal of Statistics 50(2), 135-146.

9. Diaz-Garcia, J. A. & Leiva-Sanchez, V. (2005), 'A new family of life distributions based on the contoured elliptically distributions', Journal of Statistical Planning and Inference 128(2), 445-457.

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

11. Fung, T. & Seneta, E. (2010), 'Extending the multivariate generalised t and generalised VG distributions', Proyecciones Journal of Mathematics 101, 154-164.

12. Genc, A. I. (2013), 'The generalized t Birnbaum-Saunders Family', Statistics 47, 613-625.

13. Genton, G. G. (2004), Skew-Elliptical Distributions and their Applications: A Journey beyond Normality, Chapman & Hall/CRC, Boca Raton, Florida.

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

15. Gomez, H. W., Olivares-Pacheco, J. F. & Bolfarine, H. (2009), 'An extension of the generalized Birnbaum-Saunders distribution', Statistics & Probability Letters 79, 331-338.

16. Johnson, N. L., Kotz, S. & Balakrishnan, N. (2004), Continuous Univariate Distributions, Vol. 2, 2 edn, John Wiley, New York.

17. Kantar, Y. M., Usta, I. & Acitas, S. (2011), 'A Monte Carlo simulation study on partially adaptive estimators of linear regression models', Journal of Applied Statistics 38, 1681-1699.

18. Kasap, P., Senoglu, B., Arslan, O. & Acitas, S. (2011), 'Estimating the location and scale parameters of the generalized t distribution', E-journal of New World Science Academy Physical Sciences 6, 103-111.

19. Lye, J. N. & Martin, V. L. (1993), 'Robust estimation, nonnormalities and generalized exponential distributions', Journal of the American Statistical Association 88, 261-267.

20. Ma, Y. & Genton, M. G. (2004), 'Flexible class of skew-symmetric distributions', Scandinavian Journal of Statistics 31, 459-468.

21. 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.

22. McDonald, J. B. & Nelson, R. D. (1989), 'Alternative beta estimation for the market model using partially adaptive techniques', Communications in Statistics - Theory and Methods 18(11), 4039-4058.

23. McDonald, J. B. & Nelson, R. D. (1993), 'Beta estimation in the market model: Skewness and leptokurtosis', Communications in Statistics - Theory and Methods 22(10), 2843-2862.

24. McDonald, J. B. & Newey, W. K. (1988), 'Partially adaptive estimation of regression models via the generalized t distribution', Econometric Theory 4, 428-457.

25. Nadarajah, S. (2008), 'On the generalized t (GT) distribution', Statistics 42, 467-473.

26. 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.

27. Theodossiou, P. (1998), 'Financial data and the skewed generalized t distribution', Management Science Part 1 12, 1650-1661.

28. Tiku, M. L., Islam, M. Q. & Selcuk, S. A. (2001), 'Nonnormal regression. II. Symmetric distributions', Communications in Statistics - Theory and Methods 30(6), 1021-1045.

29. Venegas, O., Rodríguez, F., Gomez, H. W., Olivares-Pacheco, J. F. & Bolfarine, H. (2012), 'Robust modeling using the generalized epsilon-skew-t distribution', Journal of Applied Statistics 39(12), 2685-2698.

30. Vilca-Labra, F. & Leiva-Sánchez, V. (2006), 'A new fatigue life model based on the family of skew-elliptical distributions', Communications in Statistics - Theory and Methods 35(2), 229-244.

31. Wang, D. & Romagnoli, J. A. (2005), 'Generalized t distribution and its applications to process data reconciliation and process monitoring', Transactions of the Institute of Measurement and Control 27(5), 367-390.

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

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

@ARTICLE{RCEv38n2a03,
    AUTHOR  = {Acitas, Sukru and Senoglu, Birdal and Arslan, Olcay},
    TITLE   = {{Alpha-Skew Generalized t Distribution}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {2},
    pages   = {353-370}
}

References

Acitas, S., Senoglu, B. & Arslan, O. (2013), Alpha-skew generalized t distribution, International Conference Applied Statistics, Slovenia.

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

Arslan, O. (2004), ‘Family of multivariate generalized t distributions’, Journal of Multivariate Analysis 89, 329–337.

Arslan, O. & Genc, A. I. (2003), ‘Robust location and scale estimation based on the univariate generalized t distribution’, Communications in Statistics – Theory and Methods 32, 1505–1525.

Butler, R. J., McDonald, J. B., Nelson, R. D. & White, S. (1990), ‘Robust and partially adaptive estimation of regression models’, The Review of Economics and Statistics 72, 321–327.

Castillo, N. O., Gomez, H. W. & Bolfarine, H. (2011), ‘Epsilon Birnbaum-Saunders distribution family: Properties and inference’, Statistical Papers 52, 871–883.

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.

Choy, S. T. B. & Chan, J. S. K. (2008), ‘Scale mixtures distributions in statistical modelling’, Australian & New Zealand Journal of Statistics 50(2), 135–146.

Diaz-Garcia, J. A. & Leiva-Sanchez, V. (2005), ‘A new family of life distributions based on the contoured elliptically distributions’, Journal of Statistical Planning and Inference 128(2), 445–457.

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

Fung, T. & Seneta, E. (2010), ‘Extending the multivariate generalised t and generalised VG distributions’, Proyecciones Journal of Mathematics 101, 154–164.

Genc, A. I. (2013), ‘The generalized t Birnbaum-Saunders Family’, Statistics 47, 613–625.

Genton, G. G. (2004), Skew-Elliptical Distributions and their Applications: A Journey Beyond Normality, Chapman & Hall/CRC, Boca Raton, Florida.

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

Gomez, H. W., Olivares-Pacheco, J. F. & Bolfarine, H. (2009), ‘An extension of the generalized Birnbaum-Saunders distribution’, Statistics & Probability Letters 79, 331–338.

Johnson, N. L., Kotz, S. & Balakrishnan, N. (2004), Continuous Univariate Distributions, Vol. 2, 2 edn, John Wiley, New York.

Kantar, Y. M., Usta, I. & Acitas, S. (2011), ‘A monte carlo simulation study on partially adaptive estimators of linear regression models’, Journal of Applied Statistics 38, 1681–1699.

Kasap, P., Senoglu, B., Arslan, O. & Acitas, S. (2011), ‘Estimating the location and scale parameters of the generalized t distribution’, E-journal of New World Science Academy Physical Sciences 6, 103–111.

Lye, J. N. & Martin, V. L. (1993), ‘Robust estimation, nonnormalities and generalized exponential distributions’, Journal of the American Statistical Association 88, 261–267.

Ma, Y. & Genton, M. G. (2004), ‘Flexible class of skew-symmetric distributions’, Scandinavian Journal of Statistics 31, 459–468.

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.

McDonald, J. B. & Nelson, R. D. (1989), ‘Alternative beta estimation for the market model using partially adaptive techniques’, Communications in Statistics – Theory and Methods 18(11), 4039–4058.

McDonald, J. B. & Nelson, R. D. (1993), ‘Beta estimation in the market model: Skewness and leptokurtosis’, Communications in Statistics – Theory and Methods 22(10), 2843–2862.

McDonald, J. B. & Newey, W. K. (1988), ‘Partially adaptive estimation of regression models via the generalized t distribution’, Econometric Theory 4, 428– 457.

Nadarajah, S. (2008), ‘On the generalized t (GT) distribution’, Statistics 42, 467– 473.

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.

Theodossiou, P. (1998), ‘Financial data and the skewed generalized t distribution’, Management Science Part 1 12, 1650–1661.

Tiku, M. L., Islam, M. Q. & Selcuk, S. A. (2001), ‘Nonnormal regression. II. Symmetric distributions’, Communications in Statistics – Theory and Methods 30(6), 1021–1045.

Venegas, O., Rodríguez, F., Gomez, H. W., Olivares-Pacheco, J. F. & Bolfarine, H. (2012), ‘Robust modeling using the generalized epsilon- skew- t distribution’, Journal of Applied Statistics 39(12), 2685–2698.

Vilca-Labra, F. & Leiva-Sánchez, V. (2006), ‘A new fatigue life model based on the family of skew-elliptical distributions’, Communications in Statistics – Theory and Methods 35(2), 229–244.

Wang, D. & Romagnoli, J. A. (2005), ‘Generalized t distribution and its applications to process data reconciliation and process monitoring’, Transactions of the Institute of Measurement and Control 27(5), 367–390.

How to Cite

APA

Acitas, S., Senoglu, B. and Arslan, O. (2015). Alpha-Skew Generalized t Distribution. Revista Colombiana de Estadística, 38(2), 371–384. https://doi.org/10.15446/rce.v38n2.51665

ACM

[1]
Acitas, S., Senoglu, B. and Arslan, O. 2015. Alpha-Skew Generalized t Distribution. Revista Colombiana de Estadística. 38, 2 (Jul. 2015), 371–384. DOI:https://doi.org/10.15446/rce.v38n2.51665.

ACS

(1)
Acitas, S.; Senoglu, B.; Arslan, O. Alpha-Skew Generalized t Distribution. Rev. colomb. estad. 2015, 38, 371-384.

ABNT

ACITAS, S.; SENOGLU, B.; ARSLAN, O. Alpha-Skew Generalized t Distribution. Revista Colombiana de Estadística, [S. l.], v. 38, n. 2, p. 371–384, 2015. DOI: 10.15446/rce.v38n2.51665. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/51665. Acesso em: 22 jan. 2025.

Chicago

Acitas, Sukru, Birdal Senoglu, and Olcay Arslan. 2015. “Alpha-Skew Generalized t Distribution”. Revista Colombiana De Estadística 38 (2):371-84. https://doi.org/10.15446/rce.v38n2.51665.

Harvard

Acitas, S., Senoglu, B. and Arslan, O. (2015) “Alpha-Skew Generalized t Distribution”, Revista Colombiana de Estadística, 38(2), pp. 371–384. doi: 10.15446/rce.v38n2.51665.

IEEE

[1]
S. Acitas, B. Senoglu, and O. Arslan, “Alpha-Skew Generalized t Distribution”, Rev. colomb. estad., vol. 38, no. 2, pp. 371–384, Jul. 2015.

MLA

Acitas, S., B. Senoglu, and O. Arslan. “Alpha-Skew Generalized t Distribution”. Revista Colombiana de Estadística, vol. 38, no. 2, July 2015, pp. 371-84, doi:10.15446/rce.v38n2.51665.

Turabian

Acitas, Sukru, Birdal Senoglu, and Olcay Arslan. “Alpha-Skew Generalized t Distribution”. Revista Colombiana de Estadística 38, no. 2 (July 1, 2015): 371–384. Accessed January 22, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/51665.

Vancouver

1.
Acitas S, Senoglu B, Arslan O. Alpha-Skew Generalized t Distribution. Rev. colomb. estad. [Internet]. 2015 Jul. 1 [cited 2025 Jan. 22];38(2):371-84. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/51665

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