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Dependence Modeling

Ed. by Puccetti, Giovanni


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Inference for copula modeling of discrete data: a cautionary tale and some facts

Olivier P. Faugeras
  • Toulouse School of Economics - Université Toulouse Capitole, Manufacture des Tabacs, Bureau MF319, 21 Allée de Brienne, 31000 Toulouse, France
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Published Online: 2017-06-27 | DOI: https://doi.org/10.1515/demo-2017-0008

Abstract

In this note, we elucidate some of the mathematical, statistical and epistemological issues involved in using copulas to model discrete data. We contrast the possible use of (nonparametric) copula methods versus the problematic use of parametric copula models. For the latter, we stress, among other issues, the possibility of obtaining impossible models, arising from model misspecification or unidentifiability of the copula parameter.

Keywords : copula; discrete data; parametric model; statistical inference; unidentifiability

MSC 2010: 62A01; 62H20; 62H12

References

  • [1] Bücher, A. and I. Kojadinovic (2015). An overview of nonparametric tests of extreme-value dependence and of related statistical procedures. In Extreme ValueModeling and Risk Analysis Methods and Applications, pp. 377-398. Chapman and Hall/CRC, Boca Raton FL.Google Scholar

  • [2] Bunge, M. (1988). Two faces and three masks of probability. In Probability in the Sciences, pp. 27-50. Kluwer Academic Publishers, Dordrecht.Google Scholar

  • [3] Bunge, M. A. (2006). Chasing Reality: Strife Over Realism. University of Toronto Press, Toronto.Google Scholar

  • [4] Deheuvels, P. (1978). Caractérisation complète des lois extrêmes mutivariées et de la convergence des types extrêmes. Pub. Inst. Stat. Univ. Paris 23(3-4), 1-36.Google Scholar

  • [5] Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés, un test non paramétrique d’indépendance. B. Ac. Roy. Belg. 65(f.6), 274-292.Google Scholar

  • [6] Deheuvels, P. (2009). A multivariate Bahadur-Kiefer representation for the empirical copula process. J. Math. Sci. 163(4), 382-398.Google Scholar

  • [7] Faugeras, O. P. (2013). Sklar’s theorem derived using probabilistic continuation and two consistency results. J.Multivariate Anal. 122, 271-277.Web of ScienceGoogle Scholar

  • [8] Faugeras, O. P. (2015). Maximal coupling of empirical copulas for discrete vectors. J. Multivariate Anal. 137, 179-186.Web of ScienceGoogle Scholar

  • [9] Faugeras, O. P. and L. Rüschendorf (2017). Markov morphisms: a combined copula and mass transportation approach to multivariate quantiles. Math. Appl., to appear. Available at http://dx.doi.org/10.14708/ma.v45i1.2921.CrossrefGoogle Scholar

  • [10] Friedman, M. (1953). The methodology of positive economics. In Essays in Positive Economics, pp. 3-16, 30-43. University of Chicago Press, Chicago IL.Google Scholar

  • [11] Genest, C., J. G. Nešlehová, and B. Rémillard (2014). On the empirical multilinear copula process for count data. Bernoulli 20(3), 1344-1371.CrossrefWeb of ScienceGoogle Scholar

  • [12] Genest, C. and J. Nešlehová (2007). A primer on copulas for count data. Astin Bull. 37(2), 475-515.CrossrefWeb of ScienceGoogle Scholar

  • [13] Hoff, P. D. (2007). Extending the rank likelihood for semiparametric copula estimation. Ann. Appl. Stat. 1(1), 265-283.Web of ScienceGoogle Scholar

  • [14] Keynes, J. M. (1939). Professor Tinbergen’s Method. Econ. J. 49(195), 558-577.Google Scholar

  • [15] Kojadinovic, I. (2017). Some copula inference procedures adapted to the presence of ties. Comput. Statist. Data Anal. 112, 24-41.Google Scholar

  • [16] Lee, L.-f. (2001). On the range of correlation coefficients of bivariate ordered discrete random variables. Economet. Theor. 17(1), 247-256.Google Scholar

  • [17] Li, D. X. (2000). On default correlation: a copula function approach. J. Fix. Income 9(4), 43-54.Google Scholar

  • [18] Lindner, A. M. and A. Szimayer (2005). A limit theorem for copulas. Discussion Paper 433, Sonderforschungsbereich 386, Available at https://epub.ub.uni-muenchen.de/1802/1/paper_433.pdf.Google Scholar

  • [19] Marshall, A.W. (1996). Copulas,marginals, and joint distributions. In Distributions with FixedMarginals and Related Topics, pp. 213-222. Institute of Mathematical Statistics, Hayward CA.Google Scholar

  • [20] Mikosch, T. (2006a). Copulas: Tales and facts. Extremes 9(1), 3-20.Google Scholar

  • [21] Mikosch, T. (2006b). Copulas: Tales and facts-rejoinder. Extremes 9(1), 55-62.Google Scholar

  • [22] Moore, D. S. and M. C. Spruill (1975). Unified large-sample theory of general chi-squared statistics for tests of fit. Ann. Stat. 3(3), 599-616.CrossrefGoogle Scholar

  • [23] Nelsen, R. B. (2006). An Introduction to Copulas. Second edition. Springer, New York.Google Scholar

  • [24] Nešlehová, J. (2007). On rank correlation measures for non-continuous random variables. J. Multivariate Anal. 98(3), 544-567.Web of ScienceCrossrefGoogle Scholar

  • [25] Nikoloulopoulos, A. K. and D. Karlis (2009). Finite normal mixture copulas for multivariate discrete data modeling. J. Stat. Plan. Infer. 139(11), 3878-3890.Web of ScienceGoogle Scholar

  • [26] Nikoloulopoulos, A. K. and D. Karlis (2010). Regression in a copula model for bivariate count data. J. Appl. Stat. 37(9), 1555-1568.CrossrefWeb of ScienceGoogle Scholar

  • [27] Pappadà, R., F. Durante, and G. Salvadori (2016). Quantification of the environmental structural risk with spoiling ties: is randomization worthwhile? Stoch. Environ. Res. Risk Assess, to appear. Available at http://dx.doi.org/10.1007/s00477-016-1357-9.CrossrefGoogle Scholar

  • [28] Poincaré, H. (1965[1901]). Letter to Léon Walras. In Correspondence of Léon Walras and Related Papers. Vol. I, 1857-1883. Vol. II, 1884-1897. Vol. III, 1898-1909, and Indexes by W. Jaffé, pp. 164-165. North Holland Publishing Co., Amsterdam.Google Scholar

  • [29] Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Ann. Stat. 4(5), 912-923.CrossrefGoogle Scholar

  • [30] Rüschendorf, L. (1981). Stochastically ordered distributions and monotonicity of the OC-function of sequential probability ratio tests. Math. Operationforsch. Stat. Ser. Statist. 12(3), 327-338.Google Scholar

  • [31] Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process. J. Stat. Plan. Infer. 139(11), 3921-3927.Google Scholar

  • [32] Rüschendorf, L. (2013). Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer, Berlin.Google Scholar

  • [33] Salmon, F. (2009). Recipe for disaster: the formula that killed Wall Street. Wired Magazine 17(3).Google Scholar

  • [34] Sempi, C. (2004). Convergence of copulas: critical remarks. Rad. Mat. 12(2), 241-249.Google Scholar

  • [35] Tao, T. (2014). Analysis. Volume I. Third edition. Hindustan Book Agency, New Delhi.Google Scholar

  • [36] van Ophem, H. (1999). A general method to estimate correlated discrete random variables. Economet. Theor. 15(2), pp. 228-237.Google Scholar

  • [37] Vapnik, V. N. (2000). The Nature of Statistical Learning Theory. Second edition. Springer, New York.Google Scholar

  • [38] Yan, L., L. Yang, Q. Yichen, and Y. Jun (2016). Copula modeling for data with ties. Available at: https://arxiv.org/abs/1612.06968.Google Scholar

About the article

Received: 2016-12-20

Accepted: 2017-06-05

Published Online: 2017-06-27

Published in Print: 2017-01-26


Citation Information: Dependence Modeling, Volume 5, Issue 1, Pages 121–132, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2017-0008.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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