Copula Modeling Dependence with Continuous Margins with Application by Using the Package R
This paper presents a class of multivariate probability models, known as “copula models”. The aim of this paper is modeling dependence structure between random variables using copula. The copula-based modeling of multivariate distributions with continuous margins is presented as a succession of rank-based tests: a multivariate test of randomness followed by a test of mutual independence and a series of goodness-of-fit tests. All the tests under study are based on the empirical copula, which is a nonparametric rank-based estimator of the true unknown copula. The principles of the tests are recalled and their implementation in the copula package R is briefly described. Their use in the construction of a copula model from data is thoroughly illustrated on real financial and physical structure data. The majority of the results thereby suggest that the Clayton copula and the t – copula provides an adequate representation of the empirical dependence structures for the performance analytics data and the urine analysis data, respectively.
Keywords: copula, multivariate independence, pseudo-observations, rank-based tests, serial independence, goodness of fit.
Nelsen, R.B. (2006): An introduction to copulas, 2nd edition. Springer Series in Statistics. Springer, p 276, New York.
Genest C, Gendron M, Bourdeau-Brien M (2009a): The Advent of Copulas in Finance. European Journal of Finance, 15, 609 – 618.
Dall’Aglio, G., Kotz, S., and Salinetti, G., editors, (1991): Advances in Probability Distributions with Given Marginals. Kluwer Academic Publishers, Dordrecht.
Genest and Mackay (1986): The joy of copulas: Bivariate distributions with uniform marginals. American Statistician 40, 280–283.
Joe H (1997): Multivariate Models and Dependence Concepts. Chapman and Hall, London.
Frees E, Valdez, E. (1998): Understanding Relationships Using Copulas." North American Actuarial Journal, 2(1), 1 – 26.
Genest, C. (1987): Frank's Family of bivariate distributions, Biometrika, 74, 3, pp. 549-555.
Genest C, Remillard B. (2004): Tests of Independence and Randomness Based on the Empirical Copula Process. Sociedad de Estadistica e Investigacion Operativa,Test, 13(2), 335 – 369.
Genest, Favre, Beliveau, and Jacques (2007): Meta elliptical copulas and their use in frequency analysis of multivariate hydrological data, Water Resour. Res., 43, W09401, doi:10.1029/2006WR005275.
Schweizer, B. (1991): Thirty years of copulas. In: G. Dall’Aglio, S. Kotz, and G. Salinetti (eds.): Advances in Probability Distributions with Given Marginals: Beyond the Copulas. The Netherlands: Kluwer Academic Publishers.
Genest, C and Rivest, L (1993): Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association 88(423), 1034–1043.
Genest, C and Rémillard, B. (2008): Validity of the parametric bootstrap for goodness of-fit testing in semiparametric models. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques, Vol. 44(6), 1096–1127.
Kolev, N. and Paiva, D. (2007): Copula-Based Regression Models. Department of Statistics, University of São Paulo. firstname.lastname@example.org.
Härdle, W, Hautsch, N and Overbeck, L (2009): Applied Quantitative Finance, Second Edition, Springer Berlin Heidelberg, Hardle W, Okhrin O & Okhrin Y:Modeling Dependencies with Copulae.
Alsina, C., Frank, M. and Schweizer, B. (2006): Associative Function: Triangular Norms and Copulas, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
Mari, D and Kotz, S (2001): Correlation and Dependence. Imp. Coll. Press, London.
Genest, C and Werker, B (2001): Conditions for the Asymptotic Semiparametric Efficiency of an Omnibus Estimator of Dependence Parameters in Copula Models. Biometrika 98, pp. 103-112.
Owzar, K and Sen, P (2003): Copulas: concepts and novel applications. METRON - International Journal of Statistics, vol. LXI, n. 3, pp. 323-353.
Smith, M (2003): Modeling sample selection using Archimedean copulas, Econometrics Journal, volume 6, pp. 99–123.
Kang, Long (2007): Modeling the Dependence Structure between Bonds and Stocks: A Multidimensional Copula Approach, Department of Economics, and Indiana University Bloomington.
Kim G, Silvapulle M, Silvapulle P (2007): Comparison of Semiparametric and Parametric Methods for Estimating Copulas." Computational Statistics and Data Analysis, 51(6), 2836 – 2850.
Kim, G, Jung, S, Han, P and Sohn (2008): A copula method for modeling directional dependence of genes, licensee BioMed Central Ltd. BMC Bioinformatics, 9:225 doi:10.1186/1471-2105-9-225.
Alejandro Quiroz Flores (2008): Copula Functions and Bivariate Distributions for Survival Analysis: An Application to Political Survival, Wilf Family Department of Politics, New York University, USA.
Kuethe, Hubbs and Waldorf (2009): Copula Models for Spatial Point Patterns and Processes. email@example.com.
Genest C, Remillard B, Beaudoin D (2009b): Goodness-of-Fit Tests for Copulas: A Review and a Power Study." Insurance: Mathematics and Economics, 44, 199 – 213.
Fischbach, Pascal (2010): Copula-Models in the Electric Power Industry, Master Thesis, Graduate School of Business Administration, Economics, University of St.Gallen.
Aschke, S, Siburg, K & Stoimenov,P (2011): Modeling dependence of extreme events in energy markets using tail copulas. tu-dortmund.de/MathPreprints, Fakultat fur Mathematik, Technische Universitat Dortmund, Vogelpothsweg 87, 44227 Dortmund.
Trivedi, P. and Zimmer, D (2007): Copula Modeling: An Introduction for Practitioners. Published, sold and distributed by: now Publishers Inc. p. 126.
Cherubini G, Vecchiato W, Luciano E (2004): Copula Models in Finance. John Wiley & Sons Ltd, p. 308, England.
Denuit, Dhaene, Goovaerts and Kaas (2005): Actuarial Theory for Dependent Risks, John Wiley & Sons Ltd, p, 460, England.
Dowd, Kevin (2005): Measuring Market Risk. Second Edition, John Wiley & Sons Ltd., England.
McNeil A, Frey R, Embrechts P (2005): Quantitative Risk Management. Princeton University Press, New Jersey.
Balakrishnan, N, Lai, Chin-Diew (2009): Continuous Bivariate Distributions. Springer Science+Business Media, LLC, p. 712.
Sklar, A. (1996): Random variables, distribution functions, and copulas – a personal look backward and forward. IMS Lecture Notes Monogr. Series., vol. 28, pp. 1–14. Inst. Math. Statist., Hayward, CA.
Genest C, Favre AC (2007): Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask. Journal of Hydrological Engineering, 12, 347 –368.
R Development Core Team (2009): R: A Language andEnvironment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL. http: //www.R-project.org/.
Yan J, Kojadinovic I (2010): copula: Multivariate Dependence with Copulas. R package version 0.9-5.
Gregoire V, Genest C, Gendron M (2008): Using Copulas to Model Price Dependence in Energy Markets. Energy Risk, 5(5), 58 –64.
Yan J (2007): Enjoy the Joy of Copulas: With a Package copula. Journal of Statistical Software, 21(4), 1 –21.
Genest C, Ghoudi K, Rivest LP (1995): A Semiparametric Estimation Procedure of Dependence Parameters in Multivariate Families of Distributions. Biometrika, 82, 543 –552.
Genest C, Quessy JF, Remillard B (2007): Goodness-of-Fit Procedures for Copulas Models Based on the Probability Integral Transformation. Scandinavian Journal of Statistics, 33, 337 –366.
Kojadinovic I and Yan J (2010): Modeling Multivariate Distributions with Continuous Margins Using the copula R Package, Journal of Statistical Software ,Volume 34, Issue 9.
Embrechts, P, Lindskog F and McNeil A (2003): Modelling Dependence with Copulas and Applications to Risk Management. In Handbook of Heavy Tailed Distributions in Finance, Elsevier, 329-384.  Genest C, Quessy JF, Remillard B (2002): Tests of serial independence based on Kendall’s process. The Canadian Journal of Statistics, Vol. 30, No. 3, Pages 1–21.
Berg D (2009): Copula Goodness-of-Fit Testing: An Overview and Power Comparison. The European Journal of Finance, 15, 675 – 70.
Chen X, Fan Y (2005): Pseudo-Likelihood Ratio Tests for Semiparametric Multivariate Copula Model Selection. Canadian Journal of Statistics, 33, 389 – 414.
Genest C, Remillard B, Beaudoin D (2009b): Goodness-of-Fit Tests for Copulas: A Review and a Power Study. Insurance: Mathematics and Economics, 44, 199 –213.
Nelsen, R. (2002): Properties and Applications of Copulas: A Brief Survey, Lewis and Clark College / Mount Holyoke College.
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