×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.