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Derivatives and Fisher information of bivariate copulas


Data sets with complex relationships between random variables are increasingly studied in statistical applications. A popular approach to model their dependence is the use of copula functions. Our contribution is to derive expressions for the observed and expected information for several bivariate copula families, in particular for the Student’s \(t\)-copula. Further likelihood derivatives which are required for numerical implementations are computed and a numerically stable implementation is provided in the R-package VineCopula. Using a real world data set of stock returns, we demonstrate the applicability of our approach for the routinely calculation of standard errors. In particular, we illustrate how this prevents overestimating the time-variation of dependence parameters in a rolling window analysis.

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  1. 1.

    Maple 13.0. Maplesoft, a division of Waterloo Maple Inc., Waterloo, ON, Canada.

  2. 2.

    Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, IL, USA (2012).

  3. 3.

    R Development Core Team (2012). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL


  1. Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula construction of multiple dependence. Insur Math Econ 44:182–198

    Google Scholar 

  2. Abramowitz M, Stegun I (1992) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publishing, New York

    Google Scholar 

  3. Acar EF, Genest C, Nešlehová J (2012) Beyond simplified pair-copula constructions. J Multivar Anal 110:74–90

    Article  MATH  Google Scholar 

  4. Bedford T, Cooke R (2001) Probability density decomposition for conditionally dependent random variables modeled by vines. Ann Math Artif Intell 32:245–268

    Article  MathSciNet  Google Scholar 

  5. Bedford T, Cooke R (2002) Vines—a new graphical model for dependent random variables. Ann Stat 30:1031–1068

    Article  MATH  MathSciNet  Google Scholar 

  6. Bedford T, Daneshkhah A (2010) Approximating multivariate distributions with vines. Available online

  7. Berger JO, Sun D (2008) Objective priors for the bivariate normal model. Ann Stat 36(2):963–982

    Article  MATH  MathSciNet  Google Scholar 

  8. Berntsen J, Espelid TO, Genz A (1991) An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans Math Softw 17(4):437–451

    Article  MATH  MathSciNet  Google Scholar 

  9. Bickel PJ, Doksum KA (2007) Mathematical statistics: basic ideas and selected topics, vol 1, 2nd edn. Pearson Prentice Hall, Upper Saddle River

    Google Scholar 

  10. Boik RJ, Robinson-Cox JF (1998) Derivatives of the incomplete beta function. J Stat Softw 3(1):1–20

    Google Scholar 

  11. Chan NH, Chen J, Chen X, Fan Y, Peng L (2009) Statistical inference for multivariate residual copula of GARCH models. Statistica Sinica 19:53–70

    MATH  MathSciNet  Google Scholar 

  12. Czado C, Kastenmeier R, Brechmann EC, Min A (2012) A mixed copula model for insurance claims and claim sizes. Scand Actuarial J 2012(4):278–305

  13. Demarta S, McNeil AJ (2005) The t copula and related copulas. Int Stat Rev 73:111–129

    Article  MATH  Google Scholar 

  14. Donaldson JR, Schnabel RB (1987) Computational experience with confidence regions and confidence intervals for nonlinear least squares. Technometrics 29(1):67–82

    Article  MATH  MathSciNet  Google Scholar 

  15. Efron B, Hinkley DV (1978) Assessing the accuracy of the maximum likelihood estimator: observed versus expected fisher information. Biometrika 65(3):457–482

    Article  MATH  MathSciNet  Google Scholar 

  16. Embrechts P, McNeil A, Straumann D (2002) Correlation and dependence in risk management: properties and pitfalls. In: Dempster M (ed) Risk management: value at risk and beyond. Cambridge University Press, Cambridge, pp 176–223

    Chapter  Google Scholar 

  17. Genz AC, Malik AA (1980) An adaptive algorithm for numeric integration over an n-dimensional rectangular region. J Comput Appl Math 6(4):295–302

    Article  MATH  Google Scholar 

  18. Hobæk Haff I, Segers J (2012) Non-parametric estimation of pair-copula constructions with the empirical pair-copula. ArXiv e-prints 1201.5133

  19. Hofert M, Mächler M, McNeil AJ (2012) Likelihood inference for Archimedean copulas in high dimensions under unknown margins. J Multivar Anal 110:133–150

    Article  MATH  Google Scholar 

  20. Huard D, E’vin G, Favre AC (2006) Bayesian copula selection. Comput Stat Data Anal 51:809–822

    Article  MATH  MathSciNet  Google Scholar 

  21. Joe H (1996) Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependence parameters. In: L Rüschendorf, Schweizer B, Taylor MD (eds) Distributions with fixed marginals and related topics. Institute of Mathematical Statistics, Hayward, vol 28, pp 120–141

  22. Joe H (1997) Multivariate models and dependence concepts. Chapman und Hall, London

    Book  MATH  Google Scholar 

  23. Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. Springer, New York

    MATH  Google Scholar 

  24. Mai J, Scherer M (2012) Simulating copulas: stochastic models, sampling algorithms and applications. Series in quantitative finance. World Scientific Publishing Company Incorporated, Singapore

    Book  Google Scholar 

  25. McCullough BD (1999) Assessing the reliability of statistical software: part ii. Am Stat 53(2):149

    Google Scholar 

  26. McNeil AJ, Nešlehová J, (2009) Multivariate archimedean copulas, d-monotone functions and l1-norm symmetric distributions. Ann Stat 37:3059–3097

    Google Scholar 

  27. Nadarajah S (2006) Fisher information for the elliptically symmetric pearson distributions. Appl Math Comput 178(2):195–206

    Article  MATH  MathSciNet  Google Scholar 

  28. Nelsen R (2006) An introduction to copulas. Springer, New York

    MATH  Google Scholar 

  29. Oakes D (1982) A model for association in bivariate survival data. J R Stat Soc Ser B 44(3):414–422

    MATH  MathSciNet  Google Scholar 

  30. Oakes D, Manatunga AK (1992) Fisher information for a bivariate extreme value distribution. Biometrika 79(4):827–832

    Article  MATH  MathSciNet  Google Scholar 

  31. Patton AJ (2006) Estimation of multivariate models for time series of possibly different lengths. J Appl Econ 21(2):147–173

    Article  MathSciNet  Google Scholar 

  32. Schepsmeier U, Stöber J (2012) Web supplement: derivatives and Fisher information of bivariate copulas. Tech. rep. TU München. Available online

  33. Schepsmeier U, Stöber J, Brechmann EC (2012) VineCopula: statistical inference of vine copulas. Available online

  34. Sklar M (1959) Fonctions de répartition á n dimensions et leurs marges. Publ Inst Stat Univ Paris 8:229–231

    MathSciNet  Google Scholar 

  35. Smith MD (2007) Invariance theorems for Fisher information. Commun Stat Theory Methods 36(12):2213–2222

    Article  MATH  Google Scholar 

  36. Song PXK, Fan Y, Kalbfleisch JD (2005) Maximization by parts in likelihood inference. J Am Stat Assoc 100(472):1145–1158

    Article  MATH  MathSciNet  Google Scholar 

  37. Stöber J, Czado C (2011) Detecting regime switches in the dependence structure of high dimensional financial data. ArXiv e-prints 1202.2009

  38. Stöber J, Schepsmeier U (2012) Is there significant time-variation in multivariate copulas? ArXiv e-prints 1205.4841

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We acknowledge substantial contributions by our working group at Technische Universität München. Numerical calculations were performed on a Linux cluster supported by DFG grant INST 95/919-1 FUGG. The first author gratefully acknowledges the support of the TUM Graduate School’s International School of Applied Mathematics, the second author is supported by TUM’s TopMath program and a research stipend provided by Allianz Deutschland AG.

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Correspondence to Ulf Schepsmeier.



In this section we give proofs for Theorem 1 and Corollary 1 in Sect. 3.1.


(Theorem 1) This proof is based on an earlier attempt by Nadarajah (2006). (1) Fisher information with respect to \(\rho \): For the transformed variables \(U\) and \(V\), corresponding to \(\rho = 0\),

$$\begin{aligned} \begin{pmatrix}X \\ Y\\ \end{pmatrix} = \frac{1}{2}\begin{pmatrix} \sqrt{1+\rho }+\sqrt{1-\rho }&\sqrt{1+\rho }-\sqrt{1-\rho } \\ \sqrt{1+\rho }-\sqrt{1-\rho }&\sqrt{1+\rho }+\sqrt{1-\rho } \\ \end{pmatrix}\begin{pmatrix}U \\ V \\ \end{pmatrix}, \end{aligned}$$

the moments are given as

$$\begin{aligned} E_N[U^pV^q] \!=\! \frac{m^{(p+q)/2}(N\!-\!1)}{\pi } B\left(\frac{p+q}{2}\!+\!1,N\!-\!\frac{p+q}{2}\!-\!1\right)B\left(\frac{p\!+\!1}{2},\frac{q\!+\!1}{2}\right),\nonumber \\ \end{aligned}$$

if both \(p\) and \(q\) are even integers (Nadarajah 2006). If either \(p\) or \(q\) is odd, Expression (1) is equal to zero.

For the calculation of the Fisher Information of the Pearson VII distribution with respect to \(\rho \) we need the second partial derivative of the log-density with respect to \(\rho \).

$$\begin{aligned} \frac{\partial \log L}{\partial \rho }&= \frac{\rho }{1-\rho ^2} - \frac{N}{\left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)} \cdot \frac{-2xy(1-\rho ^2) + 2\rho (x^2+y^2-2\rho xy)}{m(1-\rho ^2)^2},\\ \frac{\partial ^2 \log L}{\partial ^2 \rho }&= \frac{1+\rho ^2}{(1-\rho ^2)^2} - \left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)^{-1} \\&\cdot \left[ \frac{2N(x^2+y^2-2\rho xy)}{m(1-\rho ^2)^2} + \frac{8N(\rho x^2 + \rho y^2 -(1-\rho ^2)xy) \rho }{m(1-\rho ^2)^3} \right] \\&+ \left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)^{-2} \frac{4N (\rho x^2 + \rho y^2 -(1-\rho ^2)xy)^2}{m^2(1-\rho ^2)^4} \end{aligned}$$

Using the equations in the Appendix of Nadarajah (2006) for \(E[X^pY^q]\) in terms of \(E[U^pV^q]\) and Expression (1), we can determine the Fisher Information of Pearson VII distribution with respect to \(\rho \):

$$\begin{aligned} \mathcal{I }_{\rho }&= E_N\left[ -\frac{\partial ^2 \log L}{\partial ^2 \rho } \right] = \frac{1+\rho ^2}{(1-\rho ^2)^2} - E_N\left[ \left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)^{-1} \right. \\&\cdot \frac{2N ( (x^2+y^2-2\rho xy)(1-\rho ^2) + 4\rho (\rho x^2+\rho y^2-(1+\rho ^2)xy)}{m(1-\rho ^2)^3} \\&+ \left. \left(1+\frac{x^2+y^2-2\rho xy}{m(1-\rho ^2)}\right)^{-2} \frac{4N\left(\rho x^2 + \rho y^2 - (1-\rho ^2)xy\right)}{m^2(1-\rho ^2)^4} \right] \\&= \frac{1+\rho ^2}{(1-\rho ^2)^2} + E_{N+2}\left[ \frac{4N(N-1)\left(\rho x^2 + \rho y^2 - (1-\rho ^2)xy\right)}{(N+1)m^2(1-\rho ^2)^4} \right] \\&- E_{N+1}\left[ \frac{2(N\!-\!1) ( (x^2\!+\!y^2\!-\!2\rho xy)(1\!-\!\rho ^2) \!+\! 4\rho (\rho x^2\!+\!\rho y^2\!-\!(1\!+\rho ^2)xy)}{m(1-\rho ^2)^3} \right] \\ \!&= \!\frac{1\!+\!\rho ^2}{(1-\rho ^2)^2} \!+\! \frac{N(N\!-\!1)\rho ^2}{(1\!-\!\rho ^2)^2} B(3,N-1) \!+\! \frac{N(N-1)(2-3\rho ^2+\rho ^6)}{4(1\!-\!\rho ^2)^4} B(3,N\!-\!1) \\&- \frac{2N(N-1)(1+\rho ^2)}{(1-\rho ^2)^2} B(2,N-1), \end{aligned}$$

where \(B(a,b)\) is the Beta-function.

(2) and (3) of Theorem 1: (Nadarajah (2006), pp .198–200) \(\square \)


(Corollary 1) The \(t\)-distribution with parameters \(\rho \in (-1,1)\) and \(\nu >0\) results from the Pearson VII distribution for \(N=\frac{m+2}{2}\) and \(m=\nu \). The log density is

$$\begin{aligned} l(x,y;\rho ,\nu ) = -\log (2\pi ) - \frac{1}{2}\log (1-\rho ^2) - \frac{\nu +2}{2}\log \left(1+\frac{x^2+y^2-2\rho xy}{\nu (1-\rho ^2)}\right). \end{aligned}$$

Thus, (1) follows directly from Theorem 1 (1).

Since the Fisher Information for the Pearson VII distribution with respect to \(N, m\) and \(\rho \) is known from Theorem 1 ,we can compute the elements for (2) and (3) of the Fisher Information matrix easily. Denoting the Fisher Information of Pearson VII by \(\mathcal{I }^{P7}\) and the Fisher Information of the \(t\)-distribution by \(\mathcal{I }^{t}\) we obtain

$$\begin{aligned} \mathcal{I }^{t}_m&= \mathcal{I }^{P7}_m + 2\mathcal{I }^{P7}_{mN} \underbrace{\left(\frac{\partial N}{\partial m}\right)}_{1/2} + \mathcal{I }^{P7}_N \underbrace{\left(\frac{\partial N}{\partial m}\right)^2}_{1/4} \\&= \frac{1}{m} B\left(2,\frac{m}{2}\right) - \frac{m+2}{4m} B\left(3,\frac{m}{2}\right) \end{aligned}$$


$$\begin{aligned} \mathcal{I }^{t}_{\rho m}&= \mathcal{I }^{P7}_{\rho m} + \mathcal{I }^{P7}_{\rho N} \underbrace{\left(\frac{\partial N}{\partial m}\right)}_{1/2} \\&= -\frac{\rho }{2(1-\rho ^2)} \left( B\left(2,\frac{m}{2}\right) - \frac{m+2}{2} B\left(3,\frac{m}{2}\right) \right). \end{aligned}$$

Here, \(\mathcal{I }_{m} = E[-\frac{\partial ^2 \log l}{\partial ^2 m}], \mathcal{I }_{mN} = E[-\frac{\partial ^2 \log l}{\partial N\partial m}], \mathcal{I }_{N} = E[-\frac{\partial ^2 \log l}{\partial ^2 N}], \mathcal{I }_{\rho m} = E[-\frac{\partial ^2 \log l}{\partial \rho \partial m}]\) and \(\mathcal{I }_{\rho N} = E[-\frac{\partial ^2 \log l}{\partial \rho \partial N}]\). \(\square \)

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Schepsmeier, U., Stöber, J. Derivatives and Fisher information of bivariate copulas. Stat Papers 55, 525–542 (2014).

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  • Copula
  • Expected information
  • Observed information
  • Derivatives

Mathematics Subject Classification (2000)

  • 62F10
  • 62F12
  • 62F99