Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask
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VIEW THE REPLYPublication: Journal of Hydrologic Engineering
Volume 12, Issue 4
Abstract
This paper presents an introduction to inference for copula models, based on rank methods. By working out in detail a small, fictitious numerical example, the writers exhibit the various steps involved in investigating the dependence between two random variables and in modeling it using copulas. Simple graphical tools and numerical techniques are presented for selecting an appropriate model, estimating its parameters, and checking its goodness-of-fit. A larger, realistic application of the methodology to hydrological data is then presented.
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Acknowledgments
Partial funding in support of this work was provided by the Natural Sciences and Engineering Research Council of Canada, the fonds québécois de la recherche sur la nature et les technologies, the Institut de finance mathématique de Montréal, and Hydro-Québec.
References
Abdous, B., Genest, C., and Rémillard, B. (2005). “Dependence properties of meta-elliptical distributions.” Statistical modeling and analysis for complex data problems, Springer, New York, 1–15.
Abdous, B., and Ghoudi, K. (2005). “Non-parametric estimators of multivariate extreme dependence functions.” J. Nonparam. Stat., 17(8), 915–935.
Ali, M. M., Mikhail, N. N., and Haq, M. S. (1978). “A class of bivariate distributions including the bivariate logistic.” J. Multivariate Anal., 8(3), 405–412.
Bâ, K. M., Díaz-Delgado, C., and Cârsteanu, A. (2001). “Confidence intervals of quantile in hydrology computed by an analytical method.” Natural Hazards, 24(1), 1–12.
Barbe, P., Genest, C., Ghoudi, K., and Rémillard, B. (1996). “On Kendall’s process.” J. Multivariate Anal., 58(2), 197–229.
Bertino, S. (1977). “Sulla dissomiglianza tra mutabili cicliche.” Metron, 35(1–2), 53–88.
Biau, G., and Wegkamp, M. H. (2005). “A note on minimum distance estimation of copula densities.” Stat. Probab. Lett., 73(2), 105–114.
Bobée, B., and Ashkar, F. (1991). The gamma family and derived distributions applied in hydrology, Water Resources, Littleton, Colo.
Borkowf, C. B. (2002). “Computing the nonnull asymptotic variance and the asymptotic relative efficiency of Spearman’s rank correlation.” Comput. Stat. Data Anal., 39(3), 271–286.
Capéraà, P., Fougères, A.-L., and Genest, C. (1997). “A nonparametric estimation procedure for bivariate extreme value copulas.” Biometrika, 84(3), 567–577.
Capéraà, P., Fougères, A.-L., and Genest, C. (2000). “Bivariate distributions with given extreme value attractor.” J. Multivariate Anal., 72(1), 30–49.
Chen, X., and Fan, Y. (2005). “Pseudo-likelihood ratio tests for semiparametric multivariate copula model selection.” Can. J. Stat., 33(3), 389–414.
Cherubini, U., Luciano, E., and Vecchiato, W. (2004). Copula methods in finance, Wiley, New York.
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), 141–151.
De Michele, C., and Salvadori, G. (2002). “A generalized Pareto intensity–duration model of storm rainfall exploiting 2-copulas.” J. Geophys. Res., 108(D2), 1–11.
De Michele, C., Salvadori, G., Canossi, M., Petaccia, A., and Rosso, R. (2005). “Bivariate statistical approach to check adequacy of dam spillway.” J. Hydrol. Eng., 10(1), 50–57.
Deheuvels, P. (1979). “La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d’indépendance.” Bull. Cl. Sci., Acad. R. Belg., 65(6), 274–292.
Deheuvels, P. (1981). “A Kolmogorov–Smirnov type test for independence and multivariate samples.” Rev. Roum. Math. Pures Appl., 26(2), 213–226.
Devroye, L. (1986). Nonuniform random variate generation, Springer, New York.
Dhaene, J., and Goovaerts, M. J. (1996). “Dependency of risks and stop-loss order.” Astin Bull., 26(2), 201–212.
Embrechts, P., McNeil, A. J., and Straumann, D. (2002). “Correlation and dependence in risk management: Properties and pitfalls.” Risk management: Value at risk and beyond (Cambridge, 1998), Cambridge University Press, Cambridge, U.K., 176–223.
Fang, H.-B., Fang, K.-T., and Kotz, S. (2002). “The meta-elliptical distributions with given marginals.” J. Multivariate Anal., 82(1), 1–16 [corr. J. Multivariate Anal., 94(1), 222–223].
Favre, A.-C., El Adlouni, S., Perreault, L., Thiémonge, N., and Bobée, B. (2004). “Multivariate hydrological frequency analysis using copulas.” Water Resour. Res., 40(W01101), 1–12.
Fermanian, J.-D. (2005). “Goodness-of-fit tests for copulas.” J. Multivariate Anal., 95(1), 119–152.
Fermanian, J.-D., and Scaillet, O. (2003). “Nonparametric estimation of copulas for time series.” J. Risk, 5(4), 25–54.
Fisher, N. I., and Switzer, P. (1985). “Chi-plots for assessing dependence.” Biometrika, 72(2), 253–265.
Fisher, N. I., and Switzer, P. (2001). “Graphical assessment of dependence: Is a picture worth 100 tests?” Am. Stat., 55(3), 233–239.
Frank, M. J. (1979). “On the simultaneous associativity of and .” Aequ. Math., 19(2–3), 194–226.
Fréchet, M. (1951). “Sur les tableaux de corrélation dont les marges sont données.” Ann. Univ. Lyon Sect. A. (3), 14(3), 53–77.
Fredricks, G. A., and Nelsen, R. B. (2002). “The Bertino family of copulas.” Distributions with given marginals and statistical modelling, Kluwer, Dordrecht, The Netherlands, 81–91.
Frees, E. W., and Valdez, E. A. (1998). “Understanding relationships using copulas.” North Am. Act. J., 2(1), 1–25.
Gaenssler, P., and Stute, W. (1987). Seminar on empirical processes, Vol. 9, Birkhäuser, Basel, Switzerland.
Galambos, J. (1975). “Order statistics of samples from multivariate distributions.” J. Am. Stat. Assoc., 70(3), 674–680.
Genest, C. (1987). “Frank’s family of bivariate distributions.” Biometrika, 74(3), 549–555.
Genest, C., and Boies, J.-C. (2003). “Detecting dependence with Kendall plots.” Am. Stat., 57(4), 275–284.
Genest, C., Ghoudi, K., and Rivest, L.-P. (1995). “A semiparametric estimation procedure of dependence parameters in multivariate families of distribution.” Biometrika, 82(3), 543–552.
Genest, C., Ghoudi, K., and Rivest, L.-P. (1998). “Discussion of ‘Understanding relationships using copulas’ by E. W. Frees and E. A. Valdez.” North Am. Act. J., 2(3), 143–149.
Genest, C., and MacKay, R. J. (1986). “Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données.” Can. J. Stat., 14(2), 145–159.
Genest, C., Quessy, J.-F., and Rémillard, B. (2006). “Goodness-of-fit procedures for copula models based on the probability integral transformation.” Scand. J. Stat., 33(2), 337–366.
Genest, C., and Rémillard, B. (2004). “Tests of independence and randomness based on the empirical copula process.” Test, 13(2), 335–369.
Genest, C., and Rémillard, B. (2005). “Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models.” Technical Rep. G-2005-51, Groupe d’Études et de Recherche en Analyse des Décisions, Montréal.
Genest, C., and Rivest, L.-P. (1993). “Statistical inference procedures for bivariate Archimedean copulas.” J. Am. Stat. Assoc., 88(3), 1034–1043.
Genest, C., and Rivest, L.-P. (2001). “On the multivariate probability integral transformation.” Stat. Probab. Lett., 53(4), 391–399.
Genest, C., and Verret, F. (2005). “Locally most powerful rank tests of independence for copula models.” J. Nonparam. Stat., 17(5), 521–539.
Genest, C., and Werker, B. J. M. (2002). “Conditions for the asymptotic semiparametric efficiency of an omnibus estimator of dependence parameters in copula models.” Distributions with given marginals and statistical modeling, Kluwer, Dordrecht, The Netherlands, 103–112.
Geoffroy, J. (1958). “Contribution à la théorie des valeurs extrêmes.” Publ. Inst. Stat. Univ. Paris, 7, 37–185.
Ghoudi, K., Khoudraji, A., and Rivest, L.-P. (1998). “Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles.” Can. J. Stat., 26(1), 187–197.
Gijbels, I., and Mielniczuk J. (1990). “Estimating the density of a copula function.” Commun. Stat: Theory Meth., 19(2), 445–464.
Gumbel, É J. (1960). “Distributions des valeurs extrêmes en plusieurs dimensions.” Publ. Inst. Stat. Univ. Paris, 9, 171–173.
Hoeffding, W. (1940). “Maszstabinvariante Korrelationstheorie.” Schriftenr. Math. Inst. Inst. Angew. Math. Univ. Berlin, 5, 181–233.
Hoeffding, W. (1948). “A class of statistics with asymptotically normal distribution.” Ann. Math. Stat., 19(3), 293–325.
Hüsler, J., and Reiss, R.-D. (1989). “Maxima of normal random vectors: Between independence and complete dependence.” Stat. Probab. Lett., 7(4), 283–286.
Joe, H. (1993). “Parametric families of multivariate distributions with given margins.” J. Multivariate Anal., 46(2), 262–282.
Joe, H. (1997). Multivariate models and dependence concepts, Chapman and Hall, London.
Joe, H. (2005). “Asymptotic efficiency of the two-stage estimation method for copula-based models.” J. Multivariate Anal., 94(2), 401–419.
Kim, G., Silvapulle, M. J., and Silvapulle, P. (2007). “Comparison of semiparametric and parametric methods for estimating copulas.” Comp. Stat. Data Anal., 51(6), 2836–2850.
Kimeldorf, G., and Sampson, A. R. (1975). “Uniform representations of bivariate distributions.” Commun. Stat: Theory Meth., 4(7), 617–627.
Klaassen, C. A. J., and Wellner, J. A. (1997). “Efficient estimation in the bivariate normal copula model: Normal margins are least favorable.” Bernoulli, 3(1), 55–77.
Kruskal, W. H. (1958). “Ordinal measures of association.” J. Am. Stat. Assoc., 53(4), 814–861.
Lehmann, E. L. (1966). “Some concepts of dependence.” Ann. Math. Stat., 37(5), 1137–1153.
Nelsen, R. B. (1986). “Properties of a one-parameter family of bivariate distributions with specified marginals.” Commun. Stat: Theory Meth., 15(11), 3277–3285.
Nelsen, R. B. (1999). An introduction to copulas, Springer, New York.
Nelsen, R. B., Quesada-Molina, J. J., Rodríguez-Lallena, J. A., and Úbeda-Flores, M. (2003). “Kendall distribution functions.” Stat. Probab. Lett., 65(3), 263–268.
Oakes, D. (1982). “A model for association in bivariate survival data.” J. R. Stat. Soc. Ser. B (Stat. Methodol.), 44(3), 414–422.
Oakes, D. (1994). “Multivariate survival distributions,” J. Nonparam. Stat., 3(3-4), 343–354.
Plackett, R. L. (1965). “A class of bivariate distributions.” J. Am. Stat. Assoc., 60(2), 516–522.
Quesada-Molina, J. J. (1992). “A generalization of an identity of Hoeffding and some applications.” J. Ital. Stat. Soc., 3, 405–411.
R Development Core Team. (2004), “R: A language and environment for statistical computing.” R Foundation for Statistical Computing, Vienna.
Salvadori, G., and De Michele, C. (2004). “Frequency analysis via copulas: Theoretical aspects and applications to hydrological events.” Water Resour. Res., 40(W12511), 1–17.
Samara, B., and Randles, R. H. (1988). “A test for correlation based on Kendall’s tau.” Commun. Stat: Theory Meth., 17(9), 3191–3205.
Schucany, W. R., Parr, W. C., and Boyer, J. E. (1978). “Correlation structure in Farlie–Gumbel–Morgenstern distributions.” Biometrika, 65(3), 650–653.
Shih, J. H., and Louis, T. A. (1995). “Inferences on the association parameter in copula models for bivariate survival data.” Biometrics, 51(4), 1384–1399.
Sibuya, M. (1960). “Bivariate extreme statistics. I.” Ann. Inst. Stat. Math. Tokyo, 11, 195–210.
Sklar, A. (1959). “Fonctions de répartition à dimensions et leurs marges.” Publ. Inst. Stat. Univ. Paris, 8, 229–231.
Tawn, J. A. (1988). “Bivariate extreme value theory: Models and estimation.” Biometrika, 75(3), 397–415.
Tsukahara, H. (2005). “Semiparametric estimation in copula models.” Can. J. Stat., 33(3), 357–375.
Wang, W., and Wells, M. T. (2000). “Model selection and semiparametric inference for bivariate failure-time data (with discussion).” J. Am. Stat. Assoc., 95(1), 62–76.
Whelan, N. (2004). “Sampling from Archimedean copulas.” Quant. Finance, 4(3), 339–352.
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Received: Aug 29, 2006
Accepted: Aug 29, 2006
Published online: Jul 1, 2007
Published in print: Jul 2007
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