Optimization Letters

, Volume 6, Issue 1, pp 99–125 | Cite as

Copulas with maximum entropy

  • Julia PiantadosiEmail author
  • Phil Howlett
  • Jonathan Borwein
Original Paper


We shall find a multi-dimensional checkerboard copula of maximum entropy that matches an observed set of grade correlation coefficients. This problem is formulated as the maximization of a concave function on a convex polytope. Under mild constraint qualifications we show that a unique solution exists in the core of the feasible region. The theory of Fenchel duality is used to reformulate the problem as an unconstrained minimization which is well solved numerically using a Newton iteration. Finally, we discuss the numerical calculations for some hypothetical examples and describe how this work can be applied to the modelling and simulation of monthly rainfall.


Copula Maximum entropy Grade correlation Fenchel duality 


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  1. 1.
    Azzalini A.: Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. R. Stat. Soc. B 65(2), 367–389 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bazaraa M.S., Shetty C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (1979)zbMATHGoogle Scholar
  3. 3.
    Ben-Israel A., Greville T.N.E.: Generalized Inverses, Theory and Applications. Pure & Applied Mathematics. Wiley, New York (1974)Google Scholar
  4. 4.
    Birkhoff G.: Tres observaciones sobre el algebra lineal. Univ. Nac. Tucumán Rev. Ser. A 5, 147–151 (1946)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Borwein J.M., Lewis A.S.: Convex Analysis and Nonlinear Optimization, Theory and Examples. CMS Books in Mathematics. 2nd edn. Springer, New York (2006)Google Scholar
  6. 6.
    Borwein J.M., Lewis A.S., Nussbaum R.: Entropy minimization, DAD problems and stochastic kernels. J. Funct. Anal. 123, 264–307 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chen X., Fan Y., Tsyrennikov V.: Efficient estimation of semiparametric multivariate copula models. J. Am. Stat. Assoc. 101(475), 1228–1240 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Genest C., Ghoudi K., Rivest L.P.: A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3), 543–552 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Genest C., Remillard B., Beaudoin D.: Goodness-of-fit tests for copulas: a review and a power study. Insur. Math. Econ. 44, 199–213 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Holm H., Alouini M.S.: Sum and difference of two squared correlated Nakagami variates in connection with the Mckay distribution. IEEE Trans. Commun. 52(8), 1367–1376 (2004)CrossRefGoogle Scholar
  11. 11.
    Jaworski P.: On copulas and their diagonals. Inform. Sci. 179(17), 2863–2871 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Katz R.W., Parlange M.B.: Overdispersion phenomenon in stochastic modelling of precipitation. J. Clim. 11, 591–601 (1998)CrossRefGoogle Scholar
  13. 13.
    Mathworks: Matlab Mathworks version 6.5. (2003)Google Scholar
  14. 14.
    Nelsen, R.B.: An introduction to copulas. Lecture Notes in Statistics, Springer, New York (1999)Google Scholar
  15. 15.
    Piantadosi J., Boland J.W., Howlett P.G.: Simulation of rainfall totals on various time scales—daily, monthly and yearly. Environ. Modeling Assess. 14(4), 431–438 (2009)CrossRefGoogle Scholar
  16. 16.
    Piantadosi J., Howlett P.G., Boland J.W.: Matching the grade correlation coefficient using a copula with maximum disorder. J Indust Management Opt. 3(2), 305–312 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Rosenberg K., Boland J.W., Howlett P.G.: Simulation of monthly rainfall totals. ANZIAM J. 46(E), E85–E104 (2004)MathSciNetGoogle Scholar
  18. 18.
    Schmid F., Schmidt R.: Multivariate extensions of Spearman’s rho and related statistics. Stat. Probability Lett. 77(4), 407–416 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Srikanthan R., McMahon T.A.: Stochastic generation of annual, monthly and daily climate data: A review. Hydr. Earth Sys. Sci. 5(4), 633–670 (2001)Google Scholar
  20. 20.
    Stern R.D., Coe R.: A model fitting analysis of daily rainfall. J. Roy. Stat. Soc. A 147(Part 1), 1–34 (1984)CrossRefGoogle Scholar
  21. 21.
    Sun W., Rachev S., Stoyanov S., Fabozzi F.: Multivariate Skewed Student’s t Copula in Analysis of Nonlinear and Asymmetric Dependence in German Equity Market. Stud. Nonlinear Dynam. Econ. 12(2/3), 1–35 (2008)MathSciNetGoogle Scholar
  22. 22.
    Venter G., Barnett J., Kreps R., Major J.: Multivariate Copulas for Financial Modeling. Variance Casualty Actuarial Society-Arlington, Virginia 1(1), 103–119 (2007)Google Scholar
  23. 23.
    Wilks D.S., Wilby R.L.: The weather generation game: a review of stochastic weather models. Prog. Phys. Geog. 23(3), 329–357 (1999)Google Scholar
  24. 24.
    Zakaria R., Metcalfe A.V., Howlett P.G., Piantadosi J., Boland J.: Using the skew t-copula to model bivariate rainfall distribution. ANZIAM J. 51, C231–C246 (2010)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Julia Piantadosi
    • 1
    Email author
  • Phil Howlett
    • 1
  • Jonathan Borwein
    • 2
  1. 1.School of Mathematics and Statistics, CIAMUniversity of South AustraliaMawson LakesAustralia
  2. 2.School of Mathematical and Physical Sciences, CARMAThe University of NewcastleCallaghanAustralia

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