Services on Demand
Journal
Article
Spanish (pdf)
Article in xml format
Article references
Automatic translation
Send this article by e-mailIndicators
-
Cited by SciELO -
Access statistics
Related links
-
Cited by Google -
Similars in
SciELO -
Similars in Google
Share
Permalink
Revista Colombiana de Estadística
Print version ISSN 0120-1751
Rev.Colomb.Estad. vol.32 no.1 Bogotá Jan./June 2009
1Universidad Autónoma Metropolitana Unidad Iztapalapa, Departamento de Matemáticas, Ciudad de México, México. Profesor investigador. Email: ge@xanum.uam.mx
2Universidad Autónoma Metropolitana Unidad Iztapalapa, Departamento de Matemáticas, Ciudad de México, México. Estudiante de doctorado. Email: cbi206280113@xanum.uam.mx
Las cópulas se han convertido en una herramienta útil para el modelado multivariado tanto estocástico como estadístico. En este artículo se revisan propiedades fundamentales de las cópulas que permitan caracterizar la estructura de dependencia de familias de distribución bivariadas definidas por la cópula. También se describen algunas clases de cópulas, enfatizando en la importancia de la cópula Gaussiana y la familia Arquimediana. Se resalta la utilidad de las cópulas para el modelado de parejas de variables aleatorias continuas y el de las discretas. La aplicación de la cópula se ilustra con la construcción de modelos de regresión de Markov de primer orden para respuestas no Gaussianas.
Palabras clave: dependencia, cópula, medida de asociación, estadísticaaplicada, τ de Kendall, ρ de Spearman, correlaciónserial.
Copulas have become a useful tool for the multivariate modelling in both stochastics and statistics. In this article, fundamental properties that allow the characterization of the dependence structure of families of the bivariate distributions defined by the copula are reviewed. Also, the importance of both the Gaussian copula and the Archimedean family is emphasized while some classes of copulas are described. The usefulness for modelling either discrete or continuous random couples is highlighted. The construction of first-order Markov regression models for non-Gaussian responses illustrates the application of the copula.
Key words: Dependence, Copula, Measure of association, Appliedstatistics, Kendall τ, Spearman ρ, Serial correlation.
Texto completo disponible en PDF
Referencias
1. Ané, T. & Kharoubi, C. (2003), `Dependence Structure and Risk Measure´, The Journal of Business 76, 411-438. [ Links ]
2. Barlow, R. E. & Proschan, F. (1975), Statistical Theory of Reliability and Life Testing, Holt and Rinehart and Winston. [ Links ]
3. Carriere, J. F. (1995), `Removing Cancer when it is Correlated with other Causes of Death´, Biometrical Journal 37, 339-350. [ Links ]
4. Charpentier, A. & Juri, A. (2006), `Limiting Dependence Structures for Tail Events, with Applications to Credit Derivatives´, Journal of Applied Probability 43, 563-586. [ Links ]
5. Clemen, R. T. & Reilly, T. (1999), `Correlations and Copulas for Decision and Risk Analysis´, Management Science 45, 208-224. [ Links ]
6. Denuit, M. & Lambert, P. (2005), `Constraints on Concordance Measures in Bivariate Discrete Data´, Journal of Multivariate Analysis 93, 40-57. [ Links ]
7. Dobric, J. & Schmid, F. (2007), `A goodness of Fit Test for Copulas Based on Rosenblatt's Transformation´, Computational Statistics & Data Analysis 51, 4633-4642. [ Links ]
8. Dunn, P. K. & Smyth, G. K. (1996), `Randomized Quantile Residuals´, Journal of Computational and Graphical Statistics 5, 236-244. [ Links ]
9. Dupuis, D. J. (2005), `Ozone Concentrations: A Robust Analysis of Multivariate Extremes´, Technometrics 47, 191-201. [ Links ]
10. Embrechts, P., McNeil, A. J. & Straumann, D. (2002), Correlation and dependence in risk management: properties and pitfalls, `Risk Management: Value at Risk and Beyond´, p. 176-223. [ Links ]
11. Escarela, G. & Carriere, J. F. (2003), `Fitting Competing Risks with an Assumed Copula´, Statistical Methods in Medical Research 12, 333-349. [ Links ]
12. Escarela, G., Mena, R. H. & Castillo-Morales, A. (2006), `A Flexible Class of Parametric Transition Regression Models Based on Copulas: Application to Poliomyelitis Incidence´, Statistical Methods in Medical Research 15, 593-609. [ Links ]
13. Escarela, G., Pérez-Ruiz, L. C. & Bowater, R. (2009), `A Copula-Based Markov Chain Model for the Analysis of Binary Longitudinal Data´, Journal of Applied Statistics. En prensa. [ Links ]
14. Frees, E. W. & Valdez, E. A. (1998), `Understanding Relationships Using Copulas´, North American Actuarial Journal 2, 1-25. [ Links ]
15. Fréchet, M. (1951), `Sur les Tableaux de Corrélation dont les Marges sont Donnés´, Annales de l'Université de Lyon(14), 53-77 %key Series \bf3. [ 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, 347-368. [ Links ]
17. Genest, C. & MacKay, R. J. (1986a), `The Joy of Copulas: Bivariate Distributions with Uniform Marginals´, The American Statistician 40, 280-283. [ Links ]
18. Genest, C. & MacKay, R. J. (1986b), `Copules Archimédiennes et Familles de Lois Bidimensionnelles dont les Marges sont Donnés´, The Canadian Journal of Statistics 14, 145-159. [ Links ]
19. Genest, C. & Rivest, L. (1993), `Statistical Inference Procedures for Bivariate Archimedian Copulas.´, Journal of the American Statistical Association 88, 1034-1043. [ Links ]
20. Hougaard, P. (1986), `A Class of Multivariate Failure Time Distributions´, Biometrika 73, 671-678. [ Links ]
21. Huard, D., Évin, G. & Favre, A. C. (2006), `Bayesian Copula Selection´, Computational Statistics & Data Analysis 51, 809-822. [ Links ]
22. Ihaka, R. & Gentleman, R. (1996), `R: A Language for Data Analysis and Graphics´, Journal of Computational and Graphical Statistics 5, 299-314. [ Links ]
23. Jan, Y. (2007), `Enjoy the Joy of Copulas with a Package Copula´, Journal of Statistical Software 21, 1-21. [ Links ]
24. Joe, H. (1997), Multivariate Models and Dependence Concepts, Chapman & Hall, New York, United States. [ Links ]
25. Klugman, S. & Parsa, R. (1999), `Fitting Bivariate loss Distributions with Copulas´, Insurance: Mathematics and Economics 24, 139-148. [ Links ]
26. Lehmann, E. L. (1966), `Some Concepts of Dependence´, Annals of Mathematical Statistics 37, 1137-1153. [ Links ]
27. Nelsen, R. B. (1999), An Introduction to Copulas, Springer, New York, United States. [ Links ]
28. Nelsen, R. (1991), Copulas and association, `Advances in Probability Distributions with Given Marginals: Beyond the Copulas´, Kluwer, Dordrecht, Netherlands, p. 51-74. [ Links ]
29. Oakes, D. (1989), `Bivariate Survival Models Induced by Frailties´, Journal of the American Statistical Association 84, 487-493. [ Links ]
30. Rodríguez-Lallena, J. A. & Úbeda-Flores, M. (2004), `A New Class of Bivariate Copulas´, Statistics & Probability Letters 66, 315-325. [ Links ]
31. Rényi, A. (1959), `On Measures of Dependence´, Acta Mathematica Academiae Scientiarum Hungaricae 10, 441-451. [ Links ]
32. Scarsini, M. (1984), `On Measures of Concordance´, Stochastica 8, 201-218. [ Links ]
33. Schweizer, B. & Sklar, A. (1983), Probabilistic Metric Spaces, Dover Publications, New York, United States. [ Links ]
34. Schweizer, B. & Wolff, E. F. (1981), `On Nonparametric Measures of Dependence for Random Variables´, The Annals of Statistics 9, 879-885. [ Links ]
35. Schäbe, H. (1997), `Parameter Estimation for a Special Class of Markov Chains´, Statistical Papers 38, 303-327. [ Links ]
36. Sklar, A. (1959), `Fonctions de Répartition a n Dimensions et Leurs Marges´, Publications de l'Institut Statistique de l'Université de Paris 8, 229-231. [ Links ]
37. Smith, R. L. (1989), `Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone´, Statistical Science 4, 367-377. [ Links ]
38. Song, P. X. K. (2000), `Multivariate Dispersion Models Generated from Gaussian Copula´, Scandinavian Journal of Statistics 27, 305-320. [ Links ]
39. Stephenson, A. (2002), `Evd: Extreme Value Distributions´, R News 2(2), 31-32. [ Links ]
40. Wang, W. & Wells, M. (2000), `Model Selection and Semiparametric Inference for Bivariate Failure-Time Data´, Journal of the American Statistical Association 95, 62-76. [ Links ]
41. Whelan, N. (2004), `Sampling from Archimedian Copulas´, Quantitative Finance 4, 339-352. [ Links ]
42. Yanagimoto, T. & Okamoto, M. (1969), `Partial Orderings of Permutations and Monoticity of a Rank Correlation Statistic´, Annals of the Institute of Statistical Mathematics 21, 489-506. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv32n1a03,
AUTHOR = {Escarela, Gabriel and Hernández, Angélica},
TITLE = {{Modelado de parejas aleatorias usando cópulas}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2009},
volume = {32},
number = {1},
pages = {33-58}
}
