1Universidad de Atacama, Facultad de Ingeniería, Departamento de Matemática, Copiapó, Chile. Instructor y estudiante de doctorado en estadística. Email: jolivares@mat.uda.cl
2Universidad de Atacama, Facultad de Ingeniería, Departamento de Matemática, Copiapó, Chile. Profesor asociado. Email: delal@mat.uda.cl
3Universidad de Antofagasta, Facultad de Ciencias Básicas, Departamento de Matemáticas, Antofagasta, Chile. Profesor asociado. Email: hgomez@uantof.cl
]]>
En este trabajo se considera un nuevo enfoque para el estudio de la distribución triangular usando el desarrollo teórico detrás de las distribuciones Skew. La distribución triangular aquí entregada se obtiene por reparametrización de la distribución triangular usual. Se estudian las principales propiedades probabilísticas, incluidos los momentos, coeficientes de asimetría y kurtosis; además, se muestra una representación estocástica para el modelo estudiado, que proporciona un método sencillo y eficiente para la generación de variables aleatorias. Así mismo, se implementa la estimación por el método de los momentos y, a través de un estudio de simulación, se ilustra el comportamiento de las estimaciones de los parámetros.
Palabras clave: distribuciones skew, distribución triangular, asimetría, kurtosis.
In this paper a new approach is considered for studying the triangular distribution using the theoretical development behind Skew distributions. Triangular distribution are obtained by a reparametrization of usual triangular distribution. Main probabilistic properties of the distribution are studied, including moments, asymmetry and kurtosis coefficients, and an stochastic representation, which provides a simple and efficient method for generating random variables. Moments estimation is also implemented. Finally, a simulation study is conducted to illustrate the behavior of the estimation approach proposed.
Key words: Skew distribution, Triangular distribution, Skewness, Kurtosis.
Referencias
1. Arellano-Valle, R. B., Gómez, H. W. & Quintana, F. A. (2005), `Statistical Inference for a General Class of Asymmetric Distributions´, Journal of Statistical Planning and Inference 128, 427-443. [ Links ]
2. Ayyangar, A. S. K. (1941), `The Triangular Distribution´, Mathematics Student 9, 85-87. [ Links ]
3. Azzalini, A. (1985), `A Class of Distribution which Include the Normal Ones´, Scandinavian Journal of Statistics 12, 171-178. [ Links ]
4. Clark, C. E. (1962), `The PERT Model for the Distribution of an Activity´, Operations Research 10, 405-406. [ Links ]
5. Elal-Olivero, D., Gómez, H. W. & Quintana, F. A. (2009), `Bayesian Modeling using a Class of Bimodal Skew-Elliptical Distributions´, Journal of Statistical Planning and Inference 139, 1484-1492. [ Links ]
6. Genton, M. G. (2004), Skew-Elliptical Distributions and their Applications: A Journey Beyond Normality, Chapman & Hall/CRC. [ Links ]
7. Grubbs, F. E. (1962), `Attempts to Validate Certan PERT Statistics or a `Picking on PERT'´, Operations Research 10, 912-915. [ Links ]
8. Johnson, D. (1997), `The Triangular Distribution as a Proxy for the Beta Distribution in Risk Analysis´, The Statistician 46, 387-398. [ Links ]
9. Keefer, D. L. & Bodily, S. E. (1983), `Three-point Approximations for Continuous Random Variables´, Management Science 29(5), 595-609. [ Links ]
10. Kozt, S. & Van Drop, J. R. (2004), Beyond Beta, Other Continuous Families of Distributions with Bounded Support and Aplications, World Scientific Press, Republic of Singapore. [ Links ]
11. MacCrimmon, K. R. & Ryavec, C. A. (1964), `An Analytic Study of the PERT Assumptions´, Operations Research 12, 16-38. [ Links ]
12. Moder, J. J. & Rodgers, E. G. (1968), `Judgment Estimate of the Moments of PERT type Distributions´, Management Science 15(2), 76-83. [ Links ]
13. Mudholkar, G. S. & Hutson, A. D. (2000), `The Epsilon-Skew-Normal Distribution for Analysing Near-Normal Data´, Journal of Statistical Planning and Inference 83, 291-309. [ Links ]
14. Sen, P. K. & Singer, M. J. (1993), Large Sample Method in Statistics, Chapman & Hall, New York, United States. [ Links ]
15. V\~aduva, I. (1971), Computer Generation of Random Variables and Vector Related to PERT Problems, `Proceedings of the 4th conference on probability theory´, Brasov, Rumania, p. 381-395. [ Links ]
16. Van Dorp, J. & Kotz, S. (2002), `A Novel Extension of the Triangular Distribution and its Parameter Estimation´, The Statistician 51, 63-79. [ Links ]
17. Williams, T. M. (1992), `Practical use of Distributions in Network Analysis´, Journal of Operations Research Society 43, 265-270. [ Links ]
@ARTICLE{RCEv32n1a08,
AUTHOR = {Olivares-Pacheco, Juan F. and Elal-Olivero, David and Gómez, Héctor W. and Bolfarine, Heleno},
TITLE = {{Una reparametrización de la distribución triangular basada en las distribuciones skew-simétricas}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2009},
volume = {32},
number = {1},
pages = {145-156}
}