Importance of Tail Dependence in Bivariate Frequency Analysis
This article has a reply.
VIEW THE REPLYThis article has a reply.
VIEW THE REPLYPublication: Journal of Hydrologic Engineering
Volume 12, Issue 4
Abstract
This paper highlights the importance of taking into account the tail dependence in the context of bivariate frequency analysis based on copulas. Three nonparametric estimators of the tail-dependence coefficient are compared by simulations with seven families of copulas. We choose the two estimators most adapted to a bivariate frequency analysis of the annual maximum flows and the corresponding flow hydrograph volumes of the Loire River (France). In this example, the bivariate return period and the conditional density of the volume given that the flow exceeds a given threshold are computed. The results show, as can be expected, that out of the seven copula families tested, five overestimate the return periods of correlated extreme events. These results bring to the forefront the importance of taking into account the tail dependence in order to estimate the risk adequately.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writers thank Nicolas Ampen from the Direction Régionale de l’Environnement des Pays de la Loire for kindly providing the data.
References
Abberger, K. (2005). “A simple graphical method to explore tail-dependence in stock-return pairs.” Appl. Financ. Econ., 15(1), 43–51.
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.
Adamson, P. T., Metcalfe, A. V., and Parmentier, B. (1999). “Bivariate extreme value distributions: An application of the Gibbs Sampler to the analysis of floods.” Water Resour. Res., 35(9), 2825–2832.
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.
Chaire en Hydrologie Statistique. (2002). “Hyfran, logiciel pour l’analyse fréquentielle en hydrologie.” Technical Rep., INRS-ETE, Univ. du Québec, Québec (in French).
Chapman, T. (1999). “A comparison of algorithms for stream flow recession and baseflow separation.” Hydrolog. Process., 13(5), 701–714.
Chow, V. T., Maidment, D. R., and Mays, L. R. (1988). Applied hydrology—Series in water resources and environmental engineering, McGraw-Hill, Singapore.
Coles, S., Heffernan, J., and Tawn, J. (1999). “Dependence measures for extreme value analysis.” Extremes, 2(4), 339–365.
De Michele, C., and Salvadori, G. (2002). “A generalized Pareto intensity-duration model of storm rainfall exploiting 2 copulas.” J. Geophys. Res., 108(D2).
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.” Académie Royale de Belgique, Bulletin de la Classe des Sciences. 5e Série, 65, 274–292 (in French).
Demarta, S., and McNeil, A. J. (2005). “The t copula and related copulas.” Int. Statist. Rev., 73(1), 111–129.
Draisma, G., Drees, H., Ferreira, A., and de Haan, L. (2004). “Bivariate tail estimation, dependence in asymptotic independence.” Bernoulli, 10(2), 251–280.
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 University Press, Cambridge, Mass., 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.
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).
Favre, A.-C., Musy, A., and Morgenthaler, S. (2002). “Two-site modeling of rainfall based on the Neyman-Scott process.” Water Resour. Res., 38(12), 43-1–43-7.
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.
Frahm, G., Junker, M., and Schmidt, R. (2005). “Estimating the tail-dependence coefficient: Properties and pitfalls.” Insur. Math. Econ., 37(1), 80–100.
Furey, P. R., and Gupta, V. K. (2001). “A physically based filter for separating base flow from streamflow time series.” Water Resour. Res., 37(11), 2709–2722.
Genest, C., and Favre, A.-C. (2007). “Everything you always wanted to know about copulas but were afraid to ask.” J. Hydrol. Eng., 12(4), 347–368.
Genest, C., Ghoudi, K., and Rivest, L.-P. (1995). “A semiparametric estimation procedure of dependence parameters in multivariate families of distributions.” Biometrika, 82(3), 543–552.
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., and Rivest, L.-P. (1993). “Statistical inference procedures for bivariate Archimedean copulas.” J. Am. Stat. Assoc., 88(423), 1034–1043.
Goel, N. K., Seth, S. M., and Chandra, S. (1998). “Multivariate modeling of flood flows.” J. Hydraul. Eng., 124(2), 146–155.
Grimaldi, S., and Serinaldi, F. (2006). “Asymmetric copula in multivariate flood frequency analysis.” Adv. Water Resour., 29(8), 1155–1167.
Grimaldi, S., Serinaldi, F., Napolitano, F., and Ubertini, L. (2005). “A 3-copula function application for design hyetograph analysis.” Proc., Sustainable Water Management Solutions for Large Cities, 7th IAHS Scientific Assembly, Foz do Iguaçu, Brazil, 203–211.
Joe, H. (1997). Multivariate models and dependence concepts—Monographs on statistics and applied probability, Chapman and Hall, London.
Joe, H., Smith, R. L., and Weissman, I. (1992). “Bivariate threshold models for extremes.” J. R. Stat. Soc. Ser. B (Methodol.), 54(1), 171–183.
Joerin, C., Beven, K. J., Iorgulescu, I., and Musy, A. (2002). “Uncertainty in hydrograph separations based on geochemical mixing models.” J. Hydrol., 255(1–4), 90–106.
Juri, A., and Wüthrich, M. V. (2002). “Copula convergence theorems for tail events.” Insur. Math. Econ., 24(1), 139–148.
Juri, A., and Wüthrich, M. V. (2004). “Tail dependence from a distributional point of view.” Extremes, 6(3), 213–246.
Nathan, R. J., and McMahon, T. A. (1990). “Evaluation of automated techniques for base flow and recession analyses.” Water Resour. Res., 26(7), 1465–1473.
Nelsen, R. B. (1999). An introduction to copulas, Lecture Notes in Statistics, Springer, New York.
Oakes, D. (1994). “Multivariate survival distributions.” J. Nonparam. Stat., 3(3–4), 343–354.
Pardé, M. (1933). Fleuves et rivières, Armand Colin, Paris.
Raynal-Villasenor, J. A., and Salas, J. D. (1987). “Multivariate extreme value distributions in hydrological analyses.” Water for the future: Hydrology in perspective, Proc., Rome Symp., IAHS, 111–119.
Salvadori, G. (2004). “Bivariate return periods via 2-copulas.” Stat. Methodol., 1(2), 129–144.
Salvadori, G., and De Michele, C. (2004). “Frequency analysis via copulas: Theoretical aspects and applications to hydrological events.” Water Resour. Res., 40(W12511).
Schmidt, R. (2002). “Tail dependence for elliptically contoured distributions.” Math. Methods Oper. Res., 55(2), 301–327.
Schmidt, R. (2003). “Dependencies of extreme events in finance. Modelling, statistics, and data analysis.” Ph.D. thesis, Univ. Ulm, Fakultät für Mathematik und Wirtschaftswissenschaften, Germany.
Schmidt, R. (2005). Statistical tools in finance and insurance, Chap. 3, Springer, New York, 65–92.
Schmidt, R., and Stadtmüller, U. (2006). “Nonparametric estimation of tail dependence.” Scand. J. Stat., 32(2), 307–335.
Serinaldi, F., Grimaldi, S., Napolitano, F., and Ubertini, L. (2004). “A 3-copula function application to flood frequency analysis.” Proc., IASTED Int. Conf. on Environmental Modelling and Simulation, St. Thomas, Virgin Islands, 202–206.
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.” Ann. Inst. Stat. Math., 11, 195–210.
Singh, V. P., and Zhang, L. (2004a). “Bivariate rainfall frequency analysis using the copula method.” Proc., 6th Int. Conf. on Hydroscience and Engineering, Brisbane, Australia.
Singh, V. P., and Zhang, L. (2004b). “Stochastic dependence modeling in environmental hydrology.” Proc., Int. Conf. on Hydraulic Engineering, Research and Practice, Indian Institute of Technology, Roorkee, India, 46–59.
Sklar, A. (1959). “Fonctions de répartition à dimensions et leurs marges.” Publ. Inst. Stat. Univ. Paris, 8, 229–231.
Stedinger, J. R., Vogel, R. M., and Foufoula-Georgiou, E. (1993). Handbook of hydrology, McGraw-Hill, New York, 18.1–18.66.
Szilagyi, J. (2004). “Heuristic continuous base flow separation.” J. Hydrol. Eng., 9(4), 311–318.
Tallaksen, L. M. (1995). “A review of baseflow recession analysis.” J. Hydrol., 165(1–4), 349–370.
Yue, S. (2000). “The bivariate lognormal distribution to model a multivariate flood episode.” Hydrolog. Process., 14(14), 2575–2588.
Yue, S., Ouarda, T. B. M. J., and Bobée, B. (2001). “A review of bivariate gamma distributions for hydrological application.” J. Hydrol., 246(1–4), 1–18.
Yue, S., and Rasmussen, P. (2002). “Bivariate frequency analysis: discussion of some useful concepts in hydrological application.” Hydrolog. Process., 16(14), 2881–2898.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 12 • Issue 4 • July 2007
Pages: 394 - 403
Copyright
© 2007 ASCE.
History
Received: Aug 29, 2006
Accepted: Aug 29, 2006
Published online: Jul 1, 2007
Published in print: Jul 2007
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.