Skip to main content Accessibility help
Hostname: page-component-8bbf57454-hr8xl Total loading time: 0.313 Render date: 2022-01-26T08:41:06.939Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

A Primer on Copulas for Count Data

Published online by Cambridge University Press:  17 April 2015

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


HTML view is 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.

Copyright © ASTIN Bulletin 2007


Avérous, J. and Dortet-Bernadet, J.-L. (2000) LTD and RTI dependence orderings. Canad. J. Statist., 28, 151157.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, 566584.CrossRefGoogle Scholar
Capéraà, P. and Genest, C. (1990) Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles. Canad. J. Statist., 18, 315326.CrossRefGoogle Scholar
Carley, H. (2002) Maximum and minimum extensions of finite subcopulas. Comm. Statist. Theory Methods, 31, 21512166.CrossRefGoogle Scholar
Choulakian, V. and De Tibeiro, J. (2000) Copules archimédiennes et tableaux de contingence à variables qualitatives ordinales. Rev. Statist. Appl., 48, 8396.Google Scholar
Conti, P.L. (1993) On some descriptive aspects of measures of monotone dependence. Metron, 51, 4360 (1994).Google Scholar
Denuit, M. and Lambert, P. (2005) Constraints on concordance measures in bivariate discrete data. J. Multivariate Anal., 93, 4057.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 176223. Cambridge Univ. Press, Cambridge.CrossRefGoogle Scholar
Esary, J.D. and Proschan, F. (1972) Relationships among some concepts of bivariate dependence. Ann. Math. Statist., 43, 651655.CrossRefGoogle Scholar
Fang, Z. and Joe, H. (1992) Further developments on some dependence orderings for continuous bivariate distributions. Ann. Inst. Statist. Math., 44, 501517.Google Scholar
Frees, E.W. and Valdez, E.A. (1998) Understanding relationships using copulas. N. Amer. Act. J., 2, 125.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, 347368.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, 543552.CrossRefGoogle Scholar
Genest, C., Marceau, É. and Mesfioui, M. (2003) Compound Poisson approximations for individual models with dependent risks. Insurance Math. Econom., 32, 7391.CrossRefGoogle Scholar
Genest, C. and Rémillard, B. (2004) Tests of independence and randomness based on the empirical copula process. Test, 13, 335370.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, 732764.Google Scholar
Hoeffding, W. (1940) Maßstabinvariante Korrelationstheorie für diskontinuierliche Verteilungen. Arch. Math. Wirt. Sozialforsch., 7, 470.Google Scholar
Joe, H. (1993) Multivariate dependence measures and data analysis. Comput. Statist. Data Anal., 16, 279297.CrossRefGoogle Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts, volume 73 of Monographs on Statistics and Applied Probability. Chapman & Hall, London.CrossRefGoogle Scholar
Joe, H. (2005) Asymptotic efficiency of the two-stage estimation method for copula-based models. J. Multivariate Anal., 94, 401419.CrossRefGoogle Scholar
Kendall, M.G. (1945) The treatment of ties in ranking problems. Biometrika, 33, 239251.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, 28362850.CrossRefGoogle Scholar
Kimeldorf, G. and Sampson, A.R. (1987) Positive dependence orderings. Ann. Inst. Statist. Math., 39, 113128.CrossRefGoogle Scholar
Kowalczyk, T. and Niewiadomska-Bugaj, M. (2001) An algorithm for maximizing Kendall’s τ. Comput. Statist. Data Anal., 37, 181193.CrossRefGoogle Scholar
Lehmann, E.L. (1966) Some concepts of dependence. Ann. Math. Statist., 37, 11371153.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 213222. 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, 954963.CrossRefGoogle ScholarPubMed
Mesfioui, M. and Tajar, A. (2005) On the properties of some nonparametric concordance measures in the discrete case. J. Nonparametr. Stat., 17, 541554.CrossRefGoogle Scholar
Mikusinski, P., Sherwood, H. and Taylor, M.D. (1992) Shuffles of min. Stochastica, 13, 6174.Google Scholar
Nelsen, R.B. (1987) Discrete bivariate distributions with given marginals and correlation. Comm. Statist. B–Simulation Comput., 16, 199208.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, 544567.CrossRefGoogle Scholar
Oakes, D. (1982) A model for association in bivariate survival data. J. Roy. Statist. Soc. Ser. B, 44, 414422.Google Scholar
Pfeifer, D. and Neslehova, J. (2004) Modeling and generating dependent risk processes for IRM and DFA. Astin Bull., 34, 333360.CrossRefGoogle Scholar
Scarsini, M. (1984) On measures of concordance. Stochastica, 8, 201218.Google Scholar
Schweizer, B. and Sklar, A. (1974) Operations on distribution functions not derivable from operations on random variables. Studia Math., 52, 4352.CrossRefGoogle Scholar
Shih, J.H. and Louis, T.A. (1995) Inferences on the association parameter in copula models for bivariate survival data. Biometrics, 51, 13841399.CrossRefGoogle ScholarPubMed
Sklar, A. (1959) Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8, 229231.Google Scholar
Tchen, A.H. (1980) Inequalities for distributions with given marginals. Ann. Probab., 8, 814827.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, 886893.CrossRefGoogle Scholar
Vandenhende, F. and Lambert, P. (2003) Improved rank-based dependence measures for categorical data. Statist. Probab. Lett., 63, 157163.CrossRefGoogle Scholar
Whitt, W. (1976) Bivariate distributions with given marginals. Ann. Statist., 4, 12801289.CrossRefGoogle Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55, 95115.CrossRefGoogle Scholar
Yanagimoto, T. and Okamoto, M. (1969) Partial orderings of permutations and monotonicity of a rank correlation statistic. Ann. Inst. Statist. Math., 21, 489506.CrossRefGoogle Scholar
You have Access
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A Primer on Copulas for Count Data
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

A Primer on Copulas for Count Data
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

A Primer on Copulas for Count Data
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *