• Marius Hofert
  • Ivan Kojadinovic
  • Martin Mächler
  • Jun Yan
Part of the Use R! book series (USE R)


Assume that one is given the two bivariate data sets displayed in Fig. 1.1 and asked to compare them in terms of the “dependence” between the two underlying variables. The first (respectively, second) data set, denoted by (xi1, xi2), i ∈{1, …, n} (respectively, (yi1, yi2), i ∈{1, …, n}), is assumed to consist of n = 1000 independent observations (that is, a realization of independent copies) of a bivariate random vector (X1, X2) (respectively, (Y1, Y2)). Roughly speaking, comparing the two data sets in terms of dependence means comparing the way X1 and X2 are related with the way Y1 and Y2 are related.


  1. Cherubini, U., Luciano, E., & Vecchiato, W. (2004). Copula methods in finance. Chichester: Wiley.CrossRefGoogle Scholar
  2. Cherubini, U., Mulinacci, S., Gobbi, F., & Romagnoli, S. (2011). Dynamic copula methods in finance. Chichester: Wiley.CrossRefGoogle Scholar
  3. Devroye, L. (1986). Non-uniform random variate generation. New York: Springer.CrossRefGoogle Scholar
  4. Durante, F., & Sempi, C. (2010). Copula theory: An introduction. In P. Jaworski, F. Durante, W. K. Härdle & W. Rychlik (Eds.), Copula theory and its applications (Warsaw, 2009). Lecture notes in statistics (pp. 3–32). Berlin: Springer.CrossRefGoogle Scholar
  5. Durante, F., & Sempi, C. (2015). Principles of copula theory. Boca Raton, FL: CRC Press.CrossRefGoogle Scholar
  6. Embrechts, P. (2009). Copulas: A personal view. Journal of Risk and Insurance, 76, 639–650.CrossRefGoogle Scholar
  7. Embrechts, P., & Hofert, M. (2013). A note on generalized inverses. Mathematical Methods of Operations Research, 77(3), 423–432.MathSciNetCrossRefGoogle Scholar
  8. Embrechts, P., McNeil, A. J., & Straumann, D. (2002). Correlation and dependency in risk management: Properties and pitfalls. In M. Dempster (Ed.), Risk Management: Value at Risk and Beyond (pp. 176–223). London: Cambridge University Press.CrossRefGoogle Scholar
  9. Genest, C., & Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrological Engineering, 12, 347–368.CrossRefGoogle Scholar
  10. Genest, C., Gendron, M., & Bourdeau-Brien, M. (2009). The advent of copulas in finance, European Journal of Finance, 15, 609–618.CrossRefGoogle Scholar
  11. Genest, C., Rémillard, B., & Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics, 44, 199–213.MathSciNetzbMATHGoogle Scholar
  12. Hoeffding, W. (1940). Massstabinvariante Korrelationstheorie, Schriften des mathematischen Seminars und des Instituts für Angewandte Mathematik der Universität Berlin, 5, 181–233.Google Scholar
  13. Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.CrossRefGoogle Scholar
  14. Joe, H. (2014). Dependence modeling with copulas. Boca Raton, FL: Chapman & Hall/CRC.CrossRefGoogle Scholar
  15. Mai, J.-F., & Scherer, M. (2014). Financial engineering with copulas explained. London: Palgrave Macmillan.CrossRefGoogle Scholar
  16. McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts, techniques and tools (2nd ed.). Princeton, NJ: Princeton University Press.zbMATHGoogle Scholar
  17. Nelsen, R. B. (2006). An introduction to copulas. New York: Springer.zbMATHGoogle Scholar
  18. Patton, A. J. (2013). Copula methods for forecasting multivariate time series. In G. Elliott & A. Timmermann (Eds.), Handbook of economic forecasting (pp. 899–960). New York: Springer.Google Scholar
  19. Rémillard, B. (2013). Statistical methods for financial engineering. Boca Baton, FL: Chapman & Hall/CRC.CrossRefGoogle Scholar
  20. Salvadori, G., De Michele, C., Kottegoda, N. T., & Rosso, R. (2007). Extremes in nature: An approach using copulas. Water science and technology library (Vol. 56). Berlin: Springer.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Marius Hofert
    • 1
  • Ivan Kojadinovic
    • 2
  • Martin Mächler
    • 3
  • Jun Yan
    • 4
  1. 1.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Laboratory of Mathematics and its ApplicationsUniversity of Pau and Pays de l’AdourPauFrance
  3. 3.Seminar for StatisticsETH ZurichZurichSwitzerland
  4. 4.Department of StatisticsUniversity of ConnecticutStorrsUSA

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