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Slashed Exponentiated Rayleigh Distribution
Distribución Slash Rayleigh exponenciada
DOI:
https://doi.org/10.15446/rce.v38n2.51673Keywords:
Exponentiated Rayleigh Distribution, Kurtosis, Maximum Likelihood, Rayleigh Distribution, Slash Distribution (en)Curtosis, Distribución Rayleigh, Distribución Rayleigh exponenciada, Distribución Slash, Máxima verosimilitud. (es)
In this paper we introduce a new distribution for modeling positive data with high kurtosis. This distribution can be seen as an extension of the exponentiated Rayleigh distribution. This extension builds on the quotient of two independent random variables, one exponentiated Rayleigh in the numerator and Beta(q,1) in the denominator with q>0. It is called the slashed exponentiated Rayleigh random variable. There is evidence that the distribution of this new variable can be more flexible in terms of modeling the kurtosis regarding the exponentiated Rayleigh distribution. The properties of this distribution are studied and the parameter estimates are calculated using the maximum likelihood method. An application with real data reveals good performance of this new distribution.
En este trabajo presentamos una nueva distribución para modelizar datos positivos con alta curtosis. Esta distribución puede ser vista como una extensión de la distribución Rayleigh exponenciada. Esta extensión se construye en base al cuociente de dos variables aleatorias independientes, una Raileigh exponenciada en el numerador y una Beta(q; 1) en el denominador con q > 0. La llamaremos variable aleatoria slash Rayleigh exponenciada. Hay evidencias que la distribución de esta nueva variable puede ser más flexible en términos de modelizar la curtosis respecto a la distribución Rayleigh exponenciada. Se estudian las propiedades de esta distribución y se calculan las estimaciones de los parámetros utilizando el método de máxima verosimilitud. Una aplicación con datos reales revela el buen rendimiento de esta nueva distribución.
https://doi.org/10.15446/rce.v38n2.51673
1Universidad de Atacama, Facultad de Ingeniería, Departamento de Matemática, Copiapó, Chile. Professor. Email: hugo.salinas@uda.cl
2Universidad de Atacama, Instituto Tecnológico, Copiapó, Chile. Professor. Email: yuri.iriarte@uda.cl
3Universidad de Sao Paulo, IME, Departamento de Estatística, Sao Paulo, Brasil. Professor. Email: hbolfar@ime.usp.br
In this paper we introduce a new distribution for modeling positive data with high kurtosis. This distribution can be seen as an extension of the exponentiated Rayleigh distribution. This extension builds on the quotient of two independent random variables, one exponentiated Rayleigh in the numerator and Beta(q, 1) in the denominator with q > 0. It is called the slashed exponentiated Rayleigh random variable. There is evidence that the distribution of this new variable can be more flexible in terms of modeling the kurtosis regarding the exponentiated Rayleigh distribution. The properties of this distribution are studied and the parameter estimates are calculated using the maximum likelihood method. An application with real data reveals good performance of this new distribution.
Key words: Exponentiated Rayleigh Distribution, Kurtosis, Maximum Likelihood, Rayleigh Distribution, Slash Distribution.
En este trabajo presentamos una nueva distribución para modelizar datos positivos con alta curtosis. Esta distribución puede ser vista como una extensión de la distribución Rayleigh exponenciada. Esta extensión se construye en base al cuociente de dos variables aleatorias independientes, una Raileigh exponenciada en el numerador y una Beta(q, 1) en el denominador con q > 0. La llamaremos variable aleatoria recortada Rayleigh exponenciada. Hay evidencias que la distribución de esta nueva variable puede ser más flexible en términos de modelizar la curtosis respecto a la distribución Rayleigh exponenciada. Se estudian las propiedades de esta distribución y se calculan las estimaciones de los parámetros utilizando el método de máxima verosimilitud. Una aplicación con datos reales revela el buen rendimiento de esta nueva distribución.
Palabras clave: curtosis, distribución Rayleigh, distribución Rayleigh exponenciada, distribución recortada, máxima verosimilitud.
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References
1. Akaike, H. A. (1974), 'A new look at statistical model identification', IEEE Transaction on Automatic Control 19(6), 716-723.
2. Andrews, D. F. & Herzberg, A. M. (1985), Data: A Collection of Problems From Many Fields for the Student and Research Worker, Springer, New York.
3. Arslan, O. (2008), 'An alternative multivariate skew-slash distribution', Statistics & Probability Letters 78(16), 2756-2761.
4. Barlow, R. E., Toland, R. H. & Freeman, T. (1984), A Bayesian analysis of stress-rupture life of Kevlar 49/epoxy spherical pressure vessels, 'Proceedings of the Canadian conference in application statistics', Marcel Dekker, New York.
5. Cancho, V. G., Bolfarine, H. & Achcar, J. A. (1999), 'A Bayesian analysis of the exponentiated-Weibull distribution', Journal of Applied Statistical Science 8, 227-242.
6. Fang, K. T., Kotz, S. & Ng, K. W. (1990), Symmetric Multivariate and Related Distributions, Chapman and Hall, London-New York.
7. Genc, A. (2007), 'A generalization of the univariate slash by a scale-mixtured exponential power distribution', Communications in Statistics-Simulation and Computation 36, 937-947.
8. Gómez, H. W., Olivares-Pacheco, J. F. & Bolfarine, H. (2009), 'An extension of the generalized Birnbaum-Saunders distribution', Statistics & Probability Letters 79(3), 331-338.
9. Gómez, H. W., Quintana, F. A. & Torres, F. J. (2007), 'A new family of slash-distributions with elliptical contours', Statistics & Probability Letters 77(7), 717-725.
10. Iriarte, Y. A., Gómez, H. W., Varela, H. & Bolfarine, H. (2015), 'Slashed Rayleigh distribution', Revista Colombiana de Estadística 38(1), 31-44.
11. Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994), Continuous Univariate Distributions, Wiley, New York.
12. Leiva, V., Soto, G., Cabrera, E. & Cabrera, G. (2011), 'Nuevas cartas de control basadas en la distribución Birnbaum-Saunders y su implementación', Revista Colombiana de Estadística 34(1), 147-176.
13. Mosteller, F. & Tukey, J. W. (1977), Data Analysis and Regression: A Second Course in Statistics, Addison-Wesley Series in Behavioral Science: Quan-titative Methods. Reading, Massachusetts.
14. Nadarajah, S., Cordeiro, G. M. & Ortega, E. M. M. (2013), 'The exponentiated Weibull distribution: A survey', Statistical Papers 54(3), 839-877.
15. Olivares--Pacheco, J. F., Cornide-Reyes, H. & Monasterio, M. (2010), 'Una extensión de la distribución Weibull de dos parámetros', Revista Colombiana de Estadística 33(2), 219-231.
16. Olmos, N. M., Varela, H., Gómez, H. W. & Bolfarine, H. (2012), 'An extension of the half-normal distribution', Statistical Papers 53(4), 875-886.
17. R Core Team, (2014), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. *{http://www.R-project.org/}
18. Rogers, W. H. & Tukey, J. W. (1972), 'Understanding some long-tailed symmetrical distributions', Statistica Neerlandica 26, 211-226.
19. Schwarz, G. (1978), 'Estimating the dimension of a model', Annals of Statistics 6, 461-464.
20. Wang, J. & Genton, M. G. (2006), 'The multivariate skew-slash distribution', Journal of Statistical Planning and Inference 136, 209-220.
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv38n2a01,
AUTHOR = {Salinas, Hugo S. and Iriarte, Yuri A. and Bolfarine, Heleno},
TITLE = {{Slashed Exponentiated Rayleigh Distribution}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2015},
volume = {38},
number = {2},
pages = {453-466}
}
References
Akaike, H. A. (1974), ‘A new look at statistical model identification’, IEEE Transaction on Automatic Control 19(6), 716–723.
Andrews, D. F. & Herzberg, A. M. (1985), Data: a collection of problems from many fields for the student and research worker, Springer, New York.
Arslan, O. (2008), ‘An alternative multivariate skew-slash distribution’, Statistics & Probability Letters 78(16), 2756–2761.
Barlow, R. E., Toland, R. H. & Freeman, T. (1984), A Bayesian analysis of stressrupture life of Kevlar 49/epoxy spherical pressure vessels, in ‘Proceedings of the Canadian conference in application statistics’, Marcel Dekker, New York.
Cancho, V. G., Bolfarine, H. & Achcar, J. A. (1999), ‘A Bayesian analysis of the exponentiated-Weibull distribution’, Journal of Applied Statistical Science 8, 227–242.
Fang, K. T., Kotz, S. & Ng, K. W. (1990), Symmetric Multivariate and Related Distributions, Chapman and Hall, London-New York.
Genc, A. (2007), ‘A generalization of the univariate slash by a scale-mixtured exponential power distribution’, Communications in Statistics-Simulation and Computation 36, 937–947.
Gómez, H. W., Olivares-Pacheco, J. F. & Bolfarine, H. (2009), ‘An extension of the generalized Birnbaum-Saunders distribution’, Statistics & Probability Letters 79(3), 331–338.
Gómez, H. W., Quintana, F. A. & Torres, F. J. (2007), ‘A new family of slash-distributions with elliptical contours’, Statistics & Probability Letters 77(7), 717–725.
Iriarte, Y. A., Gómez, H. W., Varela, H. & Bolfarine, H. (2015), ‘Slashed Rayleigh distribution’, Revista Colombiana de Estadística 38(1), 31–44.
Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994), Continuous Univariate Distributions, Wiley, New York.
Leiva, V., Soto, G., Cabrera, E. & Cabrera, G. (2011), ‘Nuevas cartas de control basadas en la distribución Birnbaum-Saunders y su implementación’, Revista Colombiana de Estadística 34(1), 147–176.
Mosteller, F. & Tukey, J. W. (1977), Data Analysis and Regression: A Second Course in Statistics, Addison-Wesley Series in Behavioral Science: Quantitative Methods. Reading, Massachusetts.
Nadarajah, S., Cordeiro, G. M. & Ortega, E. M. M. (2013), ‘The exponentiated Weibull distribution: A survey’, Statistical Papers 54(3), 839–877.
Olivares-Pacheco, J. F., Cornide-Reyes, H. & Monasterio, M. (2010), ‘Una extensión de la distribución Weibull de dos parámetros’, Revista Colombiana de Estadística 33(2), 219–231.
Olmos, N. M., Varela, H., Gómez, H. W. & Bolfarine, H. (2012), ‘An extension of the half-normal distribution’, Statistical Papers 53(4), 875–886.
R Core Team (2014), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.*http://www.R-project.org/
Rogers, W. H. & Tukey, J. W. (1972), ‘Understanding some long-tailed symmetrical distributions’, Statistica Neerlandica 26, 211–226.
Schwarz, G. (1978), ‘Estimating the dimension of a model’, Annals of Statistics 6, 461–464.
Wang, J. & Genton, M. G. (2006), ‘The multivariate skew-slash distribution’, Journal of Statistical Planning and Inference 136, 209–220.
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