Copula Modeling Dependence with Continuous Margins with Application by Using the Package R

Luseen Immanuel Kework


 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.

Full Text:



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.

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.

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., 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: //

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. [44] 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.


  • There are currently no refbacks.

Department of Scientific Publication Office:  The Central Library of Salahaddin University-Erbil, Kirkuk Road, Erbil, Kurdistan, Iraq. Phone:+964 (0)66 2580274, email:, 

Copyright and Reprint Permission: It is the policy of ZANCO to own the copyright to the technical contributions it publishes and to facilitate the appropriate reuse of this material by others. Photocopying is permitted with credit to the source for individuals for individual use. All ZANCO Journals are Open Access Journals.

Copyright © 2015 . All Rights Reserved. Salahaddin University - Erbil