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Bayesian copula selection. (English) Zbl 1157.62359
Summary: In recent years, the use of copulas has grown extremely fast and with it, the need for a simple and reliable method to choose the right copula family. Existing methods pose numerous difficulties and none is entirely satisfactory. We propose a Bayesian method to select the most probable copula family among a given set. The copula parameters are treated as nuisance variables, and hence do not have to be estimated. Furthermore, by a parameterization of the copula density in terms of Kendall’s $$\tau$$, the prior on the parameter is replaced by a prior on $$\tau$$, conceptually more meaningful. The prior on $$\tau$$, common to all families in the set of tested copulas, serves as a basis for their comparison. Using simulated data sets, we study the reliability of the method and observe the following: (1) the frequency of successful identification approaches 100% as the sample size increases, (2) for weakly correlated variables, larger samples are necessary for reliable identification.

##### MSC:
 62F15 Bayesian inference 62H20 Measures of association (correlation, canonical correlation, etc.) 62H05 Characterization and structure theory for multivariate probability distributions; copulas
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##### References:
 [1] Bagdonavicius, V.; Malov, S.; Nikulin, M., Characterizations and semiparametric regression estimation in Archimedean copulas, J. appl. statist. sci., 8, 3, 137-153, (1999) · Zbl 0923.62052 [2] Bouyé, E., Durrleman, A., Nikeghbali, A., Riboulet, G., Roncalli, T., 2000. Copulas for finance—a reading guide and some applications. Technical Report, Groupe de Recherche Opérationnelle, Crédit Lyonnais. [3] Bretthorst, G.L., 1996. An introduction to model selection using probability theory as logic. In: Heidbreger, G. (Ed.), Maximum Entropy and Bayesian Methods, pp. 1-42. · Zbl 0895.62004 [4] Chen, X.; Fan, Y., Pseudo-likelihood ratio tests for semiparametric multivariate copula model selection, La revue canadienne de statistique, 33, 3, 389-414, (2005) · Zbl 1077.62032 [5] Chen, X., Fan, Y., Patton, A., 2003. Simple tests for models of dependence between multiple financial time series, with applications to U.S. equity returns and exchange rates. Discussion Paper 483, Financial Markets Group, International Asset Management. [6] Cherubini, U.; Luciano, E.; Vecchiato, W., Copula methods in finance. wiley finance series, (2004), Wiley Chichester · Zbl 1163.62081 [7] 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, 5ème Série 65, pp. 274-292. · Zbl 0422.62037 [8] De Michele, C.; Salvadori, G., A generalized Pareto intensity-duration model of storm rainfall exploiting 2-copulas, J. geophys. res., 108, (2003) [9] Dobrić, J., Schmidt, F., 2004. Testing goodness of fit for parametric families of copulas—application to financial data. Seminar of economic and social statistics, University of Cologne. [10] Durrleman, V., Nikeghbali, A., Roncalli, T., 2000. Which copula is the right one? Working document, Groupe de Recherche Opérationnelle, Crédit Lyonnais. [11] Embrechts, P., McNeil, A., Straumann, D., 2002. Correlation and dependence in risk management: properties and pitfalls. Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge, pp. 176-223. [12] Embrechts, P., Lindskog, F., McNeil, A., 2003. Modelling dependence with copulas and applications to risk management. Handbook of Heavy Tailed Distributions in Finance. Elsevier, Amsterdam, pp. 329-384. [13] Evin, G., 2004. Choix de la meilleure famille de copule en hydrologie. Internship Report, École Nationale de la Statistique et de l’Analyse de l’Information. [14] Favre, A.-C.; El Adlouni, S.; Perreault, L.; Thiémonge, N.; Bobée, B., Multivariate hydrological frequency analysis using copulas, Water resources res., 40, W01101, (2004) [15] Fermanian, J.-D., Goodness-of-fit tests for copulas, J. multivariate anal., 95, 119-152, (2005) · Zbl 1095.62052 [16] Frahm, G.; Junker, M.; Szimayer, I., Elliptical copulas: applicability and limitations, Statist. probab. lett., 63, 275-286, (2003) · Zbl 1116.62352 [17] Frees, E.W.; Valdez, E.A., Understanding relationships using copulas, North American actuarial J., 2, 1, 1-25, (1998) · Zbl 1081.62564 [18] Genest, C.; MacKay, R.J., The joy of copulas: bivariate distributions with uniform marginals, Amer. statist., 40, 280-283, (1986) [19] Genest, C.; Rivest, L.-P., Statistical inference procedures for bivariate Archimedean copulas, J. amer. statist. assoc., 88, 1034-1043, (1993) · Zbl 0785.62032 [20] Genest, C.; Verret, F., Locally most powerful rank tests of independence for copulas models, J. nonparametric statist., 17, 521-535, (2005) · Zbl 1065.62081 [21] Genest, C., Quessy, J.-F., Rémillard, B., 2005a. Local efficiency of a Cramér – von Mises test of independence. J. Multivariate Anal., in press. [22] Genest, C., Quessy, J.-F., Rémillard, B, 2005b. Goodness-of-fit procedures for copula models based on the probability integral transformation. Scand. J. Statist., 32, in press. [23] Jaynes, E.T.; Bretthorst, G.L., Probability theory: the logic of science, (2003), Cambridge University Press Cambridge, UK, New York [24] Joe, H., Multivariate models and dependence concepts. monographs on statistics and applied probability, (1997), Chapman & Hall London [25] Juri, A.; Wüthrich, M., Tail dependence from a distributional point of view, Extremes, 6, 3, 213-246, (2003) · Zbl 1049.62055 [26] Justel, A.; Pena, D.; Zamar, R., A multivariate kolmogorov – smirnov test of goodness of fit, Statist. probab. lett., 35, 251-259, (1997) · Zbl 0883.62054 [27] Kass, R.E.; Wasserman, L., The selection of prior distributions by formal rules, J. amer. statist. assoc., 91, 1343-1370, (1996) · Zbl 0884.62007 [28] Kendall, M., Stuart, A., 1983. The Advanced Theory of Statistics, fourth ed., vol. 2. Oxford University Press, New York. · Zbl 0416.62001 [29] Kruskal, W., Ordinal measures of association, J. amer. statist. assoc., 53, 814-861, (1958) · Zbl 0087.15403 [30] Nelsen, R.B., An introduction to copulas. lecture notes in statistics, (1999), Springer New York [31] Pollard, D., General chi-square goodness-of-fit tests with data-dependent cells, Z. wahrscheinlichkeitstheorie und verwandte gebiete, 50, 317-331, (1979) · Zbl 0404.62023 [32] Rosenblatt, M., Remarks on a multivariate transformation, Ann. math. statist., 23, 470-472, (1952) · Zbl 0047.13104 [33] Saïd, M., 2004. Méthodes statistiques pour tester la dépendance entre les variables latentes pour des risques concurrents. Ph.D. Thesis, Université Laval. [34] Schmidt, R., Tail dependence for elliptically contoured distributions, Math. meth. oper. res., 55, 2, 301-327, (2002) · Zbl 1015.62052 [35] Sklar, A., 1959. Fonctions de répartition à $$n$$ dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris 8, pp. 229-231. · Zbl 0100.14202 [36] Whelan, N., Sampling from Archimedean copulas, Quantitative finance, 4, 339-352, (2004)
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