1Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Medellín, Colombia. Profesor Asociado. Email: mcjarami@unal.edu.co
2Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Medellín, Colombia. Profesor Asistente. Email: cmlopera@unal.edu.co
3Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Medellín, Colombia. Profesor Asistente. Email: ecmanota@unal.edu.co ]]>
La distribución Weibull bivariada es muy importante en confiabilidad y en análisis de supervivencia. La dependencia para este tipo de problemas ha venido cobrando gran importancia en años recientes. En la literatura, se conocen algoritmos para generar una distribución Weibull univariada y distribuciones bivariadas con marginales independientes. En este artículo, se presenta un algoritmo para generar tiempos de falla Weibull bivariados dependientes, usando una representación cópula para la función de confiabilidad Weibull bivariada. Tal representación se obtiene utilizando modelos cópula arquimedianos. En particular, se utilizó la familia Gumbel. Se realizó una aplicación del algoritmo cópula, cuyos resultados fueron validados exitosamente.
Palabras clave: distribución bivariada, datos dependientes, cópula.
The bivariate Weibull distribution is very important in both reliability and survival analysis. The dependence for these kind of problems has been gaining great importance in recent years. In the literature, there are algorithms to generate univariate Weibull distributions and bivariate Weibull distributions with independent marginal distributions. In this paper, we present an algorithm to generate dependent bivariate Weibull failure times using a copula representation for the bivariate Weibull reliability function. Such representation is obtained using archimedean copula models. In particular, we used the Gumbels family. An application of the copula algorithm was done and the results were successfully validated.
Key words: Bivariate distribution, Dependent data, Copula.
Referencias
1. Bedford, T. (2005), Competing Risk Modeling in Reliability, `Modern Statistical and Mathematical Methods in Reliability´, World Scientific Publishing Co, Singapore. [ Links ]
2. Bouyè, E., Durrleman, V., Bikeghbali, A., Riboulet, G. & Rconcalli, T. (2000), `Copulas for Finance - A Reading Guide and Some Applications´, Manuscript, Financial Econometrics Research Center. [ Links ]
3. Canty, A. & Ripley, B. D. (2007), Boot: Bootstrap R (S-Plus) Functions (Canty). S original by Canty, A. R port by Ripley, B. D. Package version 1.2-28. [ Links ]
4. Chambers, J. M., Mallows, C. L. & Stuck, B. W. (1976), `A Method for Simulating Stable Random Variables´, Journal of the American Statistical Association 71(354), 340-344. [ Links ]
5. Cherubini, U., Luciano, E. & Vecchiato, W. (2004), Copula Methods in Finance, John Wiley & Sons, New York. [ Links ]
6. Clayton, D. G. (1978), `A Model for Association in Bivariate Life Tables and its Application in Epidemiological Studies of Familial Tendency in Chronic Disease Incidence´, Biometrika 65(1), 141-152. [ Links ]
7. Conover, W. J. (1999), Practical Nonparametric Statistics, third edn, John Wiley & Sons, New York, United States. [ Links ]
8. Crowder, M. (2001), Classical Competing Risks, Chapman & Hall, London, United Kingdom. [ Links ]
9. Denuit, M., Dhaene, J., Goovaerts, M. & Kaas, R. (2005), Actuarial Theory for Dependent Risks: Measures, Orders and Models, John Wiley & Sons, Great Britain. [ Links ]
10. Embrechts, P., Lindskog, F. & McNeil, A. (2003), Modelling Dependence with Copulas and Applications to Risk Management, `Handbook of Heavy Tailed Distribution in Finance´, Elsevier, North-Holland, Netherlands. [ Links ]
11. Escarela, G. & Carrière, J. F. (2003), `Fitting Competing Risks with an Assumed Copula´, Statistical Methods in Medical Research 12(4), 333-349. [ Links ]
12. Frank, M. J. (1979), `On the Simultaneous Associativity of f(x,y) and x + y - f(x,y)´, Aequationes Mathematicae 19(1), 194-226. [ Links ]
13. Frees, E. W., Carrière, J. F. & Valdez, E. A. (1996), `Annuity Valuation with Dependent Mortality´, Journal of Risk and Insurance 63(2), 229-261. [ Links ]
14. Frees, E. W. & Valdez, E. A. (1998), `Understanding Relationships Using Copulas´, North American Actuarial Journal 2(1), 1-25. [ Links ]
15. Frees, E. W. & Wang, P. (2005), `Credibility Using Copulas´, North American Actuarial Journal 9(2), 31-48. [ Links ]
16. Genest, C. & Favre, A. C. (2007), `Everything You Always Wanted to Know About Copula Modeling but were Afraid to Ask´, Journal of Hydrologic Engineering 12(4), 347-368. [ Links ]
17. Gumbel, E. J. (1960), `Bivariate Exponential Distributions´, Journal of the American Statistical Association 55(292), 698-707. [ Links ]
18. Joe, H. (1997), Multivariate Models and Dependence Concepts, Chapman & Hall, New York, United States. [ Links ]
19. Lu, J. & Bhattacharyya, G. K. (1990), `Some New Constructions of Bivariate Weibull Models´, Annals of the Institute of Statistical Mathematics 42(3), 543-559. [ Links ]
20. Lund, U. & Agostinelli, C. (2007), CircStats: Circular Statistics, from ``Topics in Circular Statistics'' (2001). S-plus original by Lund, U. R port by Agostinelli, C. R package version 0.2-3. [ Links ]
21. Nelsen, R. B. (2006), An Introduction to Copulas, second edn, Springer, New York, United States. [ Links ]
22. R Development Core Team, (2007), 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 [ Links ]
23. Samorodnitsky, G. & Taqqu, M. S. (1994), Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York, United States. [ Links ]
24. Venables, W. N. & Ripley, B. D. (2002), Modern Applied Statistics with S, fourth edn, Springer, New York, United States. ISBN 0-387-95457-0. [ Links ]
25. Wang, W. & Wells, M. T. (2000), `Model Selection and Semiparametric Inference for Bivariate Failure-Time Data´, Journal of the American Statistical Association 95(449), 62-72. [ Links ]
26. Yan, J. (2006), Multivariate Modeling with Copulas and Engineering Applications, `Handbook in Engineering Statistics´, Springer, New York, United States. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv31n2a03, ]]>
AUTHOR = {Jaramillo, Mario César and Lopera, Carlos Mario and Manotas, Eva Cristina and Yáñez, Sergio},
TITLE = {{Generación de tiempos de falla dependientes Weibull bivariados usando cópulas}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2008},
volume = {31},
number = {2},
pages = {169-181}
}