Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T06:33:05.425Z Has data issue: false hasContentIssue false
  • English
  • Français

A Primer on Copulas for Count Data

Published online by Cambridge University Press:  17 April 2015

Christian Genest  and
Johanna Nešlehová
Christian Genest
Affiliation:
Département de mathématiques et de statistique, Université Laval, 1045, avenue de la Médecine Québec, Canada, G1V 0A6
Johanna Nešlehová
Affiliation:
Department of Mathematics, ETH Zurich, CH-8092 Zurich, Switzerland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.

Keywords

Bernoulli distributionCopulasCount dataDiscrete distributionGeometric distributionKendall’s tauPoisson distributionSpearman’s ρSubcopula
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 37 , Issue 2 , November 2007 , pp. 475 - 515
Copyright
Copyright © ASTIN Bulletin 2007

References

Avérous, J. and Dortet-Bernadet, J.-L. (2000) LTD and RTI dependence orderings. Canad. J. Statist., 28, 151–157.CrossRefGoogle Scholar
Cameron, A.C., Li, T., Trivedi, P.K. and Zimmer, D.M. (2004) Modeling the differences in counted outcomes using bivariate copula models: With application to mismeasured counts. Econom. J., 7, 566–584.CrossRefGoogle Scholar
Capéraà, P. and Genest, C. (1990) Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles. Canad. J. Statist., 18, 315–326.CrossRefGoogle Scholar
Carley, H. (2002) Maximum and minimum extensions of finite subcopulas. Comm. Statist. Theory Methods, 31, 2151–2166.CrossRefGoogle Scholar
Choulakian, V. and De Tibeiro, J. (2000) Copules archimédiennes et tableaux de contingence à variables qualitatives ordinales. Rev. Statist. Appl., 48, 83–96.Google Scholar
Conti, P.L. (1993) On some descriptive aspects of measures of monotone dependence. Metron, 51, 43–60 (1994).Google Scholar
Denuit, M. and Lambert, P. (2005) Constraints on concordance measures in bivariate discrete data. J. Multivariate Anal., 93, 40–57.CrossRefGoogle Scholar
Embrechts, P., McNeil, A.J. and Straumann, D. (2002) Correlation and dependence in risk management: Properties and pitfalls. In Risk Management: Value at Risk and Beyond (Cambridge, 1998), pages 176–223. Cambridge Univ. Press, Cambridge.CrossRefGoogle Scholar
Esary, J.D. and Proschan, F. (1972) Relationships among some concepts of bivariate dependence. Ann. Math. Statist., 43, 651–655.CrossRefGoogle Scholar
Fang, Z. and Joe, H. (1992) Further developments on some dependence orderings for continuous bivariate distributions. Ann. Inst. Statist. Math., 44, 501–517.CrossRefGoogle Scholar
Frees, E.W. and Valdez, E.A. (1998) Understanding relationships using copulas. N. Amer. Act. J., 2, 1–25.CrossRefGoogle Scholar
Genest, C. and Favre, A.-C. (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J. Hydrologic Eng., 12, 347–368.CrossRefGoogle Scholar
Genest, C., Ghoudi, K. and Rivest, L.-P. (1995) A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82, 543–552.CrossRefGoogle Scholar
Genest, C., Marceau, É. and Mesfioui, M. (2003) Compound Poisson approximations for individual models with dependent risks. Insurance Math. Econom., 32, 73–91.CrossRefGoogle Scholar
Genest, C. and Rémillard, B. (2004) Tests of independence and randomness based on the empirical copula process. Test, 13, 335–370.CrossRefGoogle Scholar
Genest, C., Rémillard, B. and Beaudoin, D. (2007) Omnibus goodness-of-fit tests for copulas: A review and a power study. Insurance Math. Econom., 42, in press.Google Scholar
Goodman, L.A. and Kruskal, W.H. (1954) Measures of association for cross classifications. J. Amer. Statist. Assoc., 49, 732–764.Google Scholar
Hoeffding, W. (1940) Maßstabinvariante Korrelationstheorie für diskontinuierliche Verteilungen. Arch. Math. Wirt. Sozialforsch., 7, 4–70.Google Scholar
Joe, H. (1993) Multivariate dependence measures and data analysis. Comput. Statist. Data Anal., 16, 279–297.CrossRefGoogle Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts, volume 73 of Monographs on Statistics and Applied Probability . Chapman & Hall, London.Google Scholar
Joe, H. (2005) Asymptotic efficiency of the two-stage estimation method for copula-based models. J. Multivariate Anal., 94, 401–419.CrossRefGoogle Scholar
Kendall, M.G. (1945) The treatment of ties in ranking problems. Biometrika, 33, 239–251.CrossRefGoogle ScholarPubMed
Kim, G., Silvapulle, M.J. and Silvapulle, P. (2007) Comparison of semiparametric and parametric methods for estimating copulas. Comput. Statist. Data Anal., 51, 2836–2850.CrossRefGoogle Scholar
Kimeldorf, G. and Sampson, A.R. (1987) Positive dependence orderings. Ann. Inst. Statist. Math., 39, 113–128.CrossRefGoogle Scholar
Kowalczyk, T. and Niewiadomska-Bugaj, M. (2001) An algorithm for maximizing Kendall’s τ. Comput. Statist. Data Anal., 37, 181–193.CrossRefGoogle Scholar
Lehmann, E.L. (1966) Some concepts of dependence. Ann. Math. Statist., 37, 1137–1153.CrossRefGoogle Scholar
Marshall, A.W. (1996) Copulas, marginals, and joint distributions. In Distributions with Fixed Marginals and Related Topics (Seattle, WA, 1993), volume 28 of IMS Lecture Notes Monogr. Ser. , pages 213–222. Inst. Math. Statist., Hayward, CA.CrossRefGoogle Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton, NJ.Google Scholar
Meester, S.G. and Mackay, R.J. (1994) A parametric model for cluster correlated categorical data. Biometrics, 50, 954–963.CrossRefGoogle ScholarPubMed
Mesfioui, M. and Tajar, A. (2005) On the properties of some nonparametric concordance measures in the discrete case. J. Nonparametr. Stat., 17, 541–554.CrossRefGoogle Scholar
Mikusinski, P., Sherwood, H. and Taylor, M.D. (1992) Shuffles of min. Stochastica, 13, 61–74.Google Scholar
Nelsen, R.B. (1987) Discrete bivariate distributions with given marginals and correlation. Comm. Statist. B–Simulation Comput., 16, 199–208.CrossRefGoogle Scholar
Nelsen, R.B. (1999) An Introduction to Copulas, volume 139 of Lecture Notes in Statistics . Springer, New York.CrossRefGoogle Scholar
Neslehova, J. (2004) Dependence of Non-Continuous Random Variables. Doctoral dissertation, Universität Oldenburg, Oldenburg, Germany.Google Scholar
Neslehova, J. (2007) On rank correlation measures for non-continuous random variables. J. Multivariate Anal., 98, 544–567.CrossRefGoogle Scholar
Oakes, D. (1982) A model for association in bivariate survival data. J. Roy. Statist. Soc. Ser. B, 44, 414–422.Google Scholar
Pfeifer, D. and Neslehova, J. (2004) Modeling and generating dependent risk processes for IRM and DFA. Astin Bull., 34, 333–360.CrossRefGoogle Scholar
Scarsini, M. (1984) On measures of concordance. Stochastica, 8, 201–218.Google Scholar
Schweizer, B. and Sklar, A. (1974) Operations on distribution functions not derivable from operations on random variables. Studia Math., 52, 43–52.CrossRefGoogle Scholar
Shih, J.H. and Louis, T.A. (1995) Inferences on the association parameter in copula models for bivariate survival data. Biometrics, 51, 1384–1399.CrossRefGoogle ScholarPubMed
Sklar, A. (1959) Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8, 229–231.Google Scholar
Tchen, A.H. (1980) Inequalities for distributions with given marginals. Ann. Probab., 8, 814–827.CrossRefGoogle Scholar
Trégouët, D.-A., Ducimetière, P., Bocquet, V., Visvikis, S., Soubrier, F. and Tiret, L. (2004) A parametric copula model for analysis of familial binary data. Am. J. Hum. Genet., 64, 886–893.CrossRefGoogle Scholar
Vandenhende, F. and Lambert, P. (2003) Improved rank-based dependence measures for categorical data. Statist. Probab. Lett., 63, 157–163.CrossRefGoogle Scholar
Whitt, W. (1976) Bivariate distributions with given marginals. Ann. Statist., 4, 1280–1289.CrossRefGoogle Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55, 95–115.CrossRefGoogle Scholar
Yanagimoto, T. and Okamoto, M. (1969) Partial orderings of permutations and monotonicity of a rank correlation statistic. Ann. Inst. Statist. Math., 21, 489–506.CrossRefGoogle Scholar