Derivatives and Fisher information of bivariate copulas

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1 Statistical Papers manuscript No. will be inserted by the editor Derivatives and Fisher information of bivariate copulas Ulf Schepsmeier Jakob Stöber Received: date / Accepted: date Abstract We provide the rst and second derivatives of log- densities and conditional distribution functions of various bivariate copulas. These derivatives are required in order to calculate e.g. the observed Fisher information in multivariate models based on bivariate copulas. In particular, we obtain all derivatives for the bivariate t-copula, which have not been available until now. All derivatives are implemented in the R package VineCopula, and we demonstrate the accuracy of our implementation by comparing the Fisher information matrices calculated using our functions with known analytical results where available. Keywords Copula Fisher Information derivatives Mathematics Subject Classication 000 6F0 6F 6F99 Introduction A copula is a multivariate distribution function C dened on the unit hypercube 0, ] d, with uniformly distributed marginals. Their importance is due to the famous theorem of Sklar 959, which allows to seperate the description of the joint distribution of multiple random variables into the modelling of Ulf Schepsmeier Lehrstuhl für Mathematische Statistik, Technische Universität München, Parkring 3, Garching-Hochbrück, Germany, Tel.: schepsmeier@ma.tum.de Jakob Stöber Lehrstuhl für Mathematische Statistik, Technische Universität München, Parkring 3, Garching-Hochbrück, Germany, Tel.: stoeber@ma.tum.de

2 Ulf Schepsmeier, Jakob Stöber individual marginal distributions and the dependence structure expressed by a copula. Since copula theory in the d-dimensional case is still a new area of statistics and inference is rather challenging in some cases, copulas are most often considered in the bivariate case. Large classes of bivariate copulas have been developed, and many examples including their stochastic properties are given in Joe 997. The bivariate copulas common in applications can mainly be devided into two categories, namely copulas arising from the dependence structure of elliptical random vectors, having for example a multivariate Gaussian or Student-t distribution and Archimedean copulas. Under a bivariate Archimedean copula, we understand a distribution function given in the form Cu, u = ϕϕ ] u ϕ ] u, where ϕ is the so called generator function and ϕ ] is the pseudo-inverse of ϕ and is dened as follows: ϕ ] : 0, ] 0, ϕ ] x := inf{u : ϕu x}. For necessary and sucient conditions on the generator function ϕ, we refer to McNeil and Neslehová 009. In order to ensure the existence of continuous derivatives, we will assume throughout this note that ϕ : 0, 0, ], with j ϕ j 0, 0 j, where ϕ is continuous and ϕ0 =, lim x ϕx = 0. An important method for the construction of general d-dimensional copulas is to decompose them into bivariate copula building blocks by subsequent conditioning Joe 996, Bedford and Cooke 00, Bedford and Cooke 00, Aas et al. 009, Diÿmann et al. 0. This method is called pair copula construction PCC and the resulting models are refered to as R-vine copulas, since they can completly be organized using the graphical structure of a regular vine R-vine. In these methods, the conditional distributions of U U = u, where U, U C, are required for the pair copulas C to be incorporated in the construction. For notational simplicity, we will refer to the conditional cdfs which are derived as hu u = u Cu, u as h-functions Aas et al. 009, Czado 00. The aim of this note is to calculate the derivatives of bivariate copula families which we will often require in applications. For the bivariate t-copula, we signicantly extend the results of Dakovic and Czado 0 where some equations were incomplete and not all derivatives were calculated. The derivatives are implemented in the framework of the R package VineCopula available on CRAN. Using numerical integration techniques we obtain the expected Fisher information matrix using our implementation. This allows to assess our numerical accurancy for copula families where the analytical form of the Fisher Information is known. It also leads to useful numerical tabulations of the expected information, which we provide.

3 Derivatives and Fisher information of bivariate copulas 3 In particular we will consider the rst and second derivatives of bivariate log- density functions denoted with a small c with respect to the parameters and the rst argument u as well as the mixed derivative u cu θ, u ; θ. The derivatives with respect to u can be omitted because of symmetry for all copula families we consider. Additionally to the derivatives of the density functions c we calculate the rst and second derivatives of the h-functions, i.e. the conditional cdfs. Since the h-function is the rst derivative of C with respect to u, the derivative of h with respect to u is again the density cu, u hu,u u = Cu,u u u = cu, u. The same is true for the corresponding second derivatives, i.e. they are already calculated as derivatives of the density functions. hu, u := hu,u u u = u,u u = cu, u hu, u := hu,u u = u,u u = cu, u. θ hu, u := hu,u θ u = u,u θ = θ cu, u Here, i refers to the derivative with respect to the i-th argument of the function. For the sake of notational shortness, we will often suppress function arguments in the following. Further, we will sometimes introduce short notation for complex expressions of parameters and arguments u, u. Considering a function ft; θ, where t = tu, u ; θ is introduced for notational shortness, the following rule applies: θ ft; θ refers to the derivative of the complete expression with respect to θ, including the dependence on θ through t. In contrast, θ ft; θ refers to the partial derivative f ; θ of f, evaluated at t. When a function f ; θ has one argument a single possibly multiple parameters, we will sometimes also denote the derivative with respect to the argument i.e. h ; θ as h ; θ. The remainder is structured as follows: In Section we illustrate areas of applications of the rst and second derivatives for the copula densities while in Section 3 the dierent elliptical and Archimedean copula families we consider are introduced. Corresponding rotated copulas by 90, 80 and 70 degrees are considered in Section 4. The numerical issues and our implementation are discussed in Section 5. Section 6 concludes. Statistical application areas requiring copula derivatives As we have discussed before, inference for multivariate models and in particular higher dimensional copulas is a far less developed area of statistics than univariate applications. One reason for this is, that the likelihood usually is less tractable. Considering maximum likelihood ML estimation, most of the theory and many numerical methods are based on score functions, i.e. partial derivatives of the log-likelihood with respect to the parameters. Where these are not available, they can be approximated using nite dierences, which fxɛfxɛ amounts to approximating the limit of lim ɛ 0 ɛ by choosing a xed, but small, ɛ > 0. This, however has two big disadvantages:

4 4 Ulf Schepsmeier, Jakob Stöber It can be numerically unstable, especially for higher-order derivatives see McCullough 999 and references therein, in particular Donaldson and Schnabel 987. For p parameters, calculating the score function amounts to at least p evaluations of the likelihood function which can be slow in higher dimensions. Thus, at least the rst derivatives of copula densities under investigation with respect to the parameters are required to develop ecient ML methods. Further, increasingly popular algorithms for inference in the case of regression problems linked by a copula, as maximization by parts Song et al, 005, also rely on derivatives of the likelihood function. While many applications until now have been restricted to the bivariate Gaussian case, the derivatives of the Student-t copula density listed in this paper can be used to extend the range of copulas available. While its rst derivatives were already determined by Dakovic and Czado 0 their paper contained several aws which we have corrected. Considering asymptotic properties of ML estimation, it is well known that under certain regularity conditions c.f. Bickel and Doksum 007, p. 386 or Lehmann and Casella 998, p. 449, the ML estimator ˆθ n R p obtained from n observations is strongly consistent and asymptotically normal: n Iθ / ˆθn θ d N0, Id p, as n, where θ R p is the true parameter and Id p the p p identity matrix. Here, the expected Fisher information matrix Iθ can be obtained as ] ] Iθ = E θ lθ X = E θ lθ X lθ X, θ i θ j i,j=,...,p θ i θ j i,j=,...,p where lθ x is the log-likelihood of θ given an observation of X = x. In a nite sample of n independent observation x,..., x n, it has been argued Efron and Hinkley 978 that the Fisher information should be replaced by the observed information I n ˆθ n at the ML estimate ˆθ n I n ˆθ n = θ i θ j n i= lθ x i i,j=,...,p ] θ= ˆθ n. Thus, we require second derivatives of copula densities in order to study the covariance structure of ML estimators when copulas are involved. While we have considered bivariate models until now, the derivatives we obtain will also be required for higher dimensional copulas. Using the hierarchical nature of the aforementioned PCCs, the derivatives of a d-dimensional R-vine copula can be calculated iteratively using the derivatives of the bivariate building blocks C. As the conditional distribution functions h, = C, are involved in the specication of the likelihood, also their derivatives will occur. The detailed calculations and algorithms for this extension of the bivariate case to dimension d are discussed in Stöber and Schepsmeier 0.

5 Derivatives and Fisher information of bivariate copulas 5 3 Bivariate copula families In this section, we introduce the parametric forms of several well-known bivariate families which we will consider in the sequel. The corresponding derivatives are listed in Appendix A for the elliptical copulas and in Appendix B for Archimedean copulas, respectively. Since the derivatives are hard to obtain especially for the t-copula, we list all derivatives for the elliptical families. In the Archimedean case, all functions of interest are given in closed form such that also computer algebra systems as Maple or Mathematica can be used for the calculation of derivatives. For that reason, we illustrate their calculation only for the Clayton copula and restrict ourselves to rst derivatives for the other considered parametric families. 3. Gaussian copula The Gaussian copula is dened by Cu, u ; ρ = Φ Φ u, Φ u, ρ, where Φ,, ρ is the joint distribution of two standard normal distributed random variables with correlation ρ,, Φ is the N0, cdf and Φ the quantile function is its functional inverse. The density of the bivariate Gaussian copula is given by cu, u ; ρ = { exp ρ x x } ρx x ρ ρ, where x = Φ u and x = Φ u. Following Aas et al. 009 we call the conditional cdf of a copula h-function. For the Gaussian copula the h-function is as follows: hu, u ; ρ = Φ u ρφ u Cu, u ; ρ = Φ u ρ 3. t-copula Beside the Gaussian copula, we investigate the t-copula, also an elliptical copula. Unlike the Gaussian copula the t-copula has a second parameter, the degrees of freedom denoted by > 0. The density of the bivariate t-copula with parameters ρ, is given by cu, u ; ρ, = π x x ρx x ρ dtx, dtx, ρ,

6 6 Ulf Schepsmeier, Jakob Stöber where dtx i, = Γ Γ x i π, i =,, is the density of the univariate t-distribution with degrees of freedom and Γ is the gamma function. Here x i := t u i, u i 0,, i =,, with t being the quantile function of the univariate t-distribution with degrees of freedom. The derivatives see Appendix A. of the pdf can be calculated more easily using the logarithm of the t-copula density. It can be written as lu, u ; ρ, = ln cu, u ; ρ, with = ln ln ln ρ ln Γ ln Γ ] ln x ln x M, ρ, x, x := ρ x x ρx x. The now following decomposition of t x i in the positive and the negative range of x i as well as the calculation of the derivative of x i with respect to see Appendix A. was rst considered by Dakovic and Czado 0. where with I x i x i = t xi, i =,, dtx i, I x t x i =,, xi 0, i,, xi < 0, I x i, x i 0 t t dt = B, i =,, being the regularized β-function Abramowitz and Stegun 99, p Again, the h-function corresponding to the t-copula has been derived in Aas et al. 009: hu, u ; ρ, = t u ρt u u = t x ρx ρ t t x ρ lnm, ρ, x, x,.

7 Derivatives and Fisher information of bivariate copulas Clayton copula The rst Archimedean copula we want to look at is the Clayton/MTCJ copula. The generator function ϕ of this copula is ϕt = t θ such that the copula is given by with Cu, u ; θ = u θ u θ θ = Au, u, θ θ, and corresponding density Au, u, θ := u θ u θ, cu, u ; θ = θu u θ u θ u θ θ = θu u θ Au, u, θ θ, where 0 < θ < controls the degree of dependence. If θ the Clayton/MTCJ copula converges to the monotonicity copula with perfect positive dependence, θ = 0 corresponds to independence. The h-function of the Clayton copula can be calculated as hu, u ; θ = u θ Au, u, θ θ. 3.4 Gumbel copula The Gumbel copula is dened as where Cu, u ; θ = exp{ lnu θ lnu θ } θ ] = expt t θ ], t i = ln u i θ, i =,. Here, θ is the parameter of dependence see Genest and Favre 007. For θ the Gumbel copula converges to the comonotonic copula with perfect dependence, in contrast θ = means independence. The h-function given as the rst derivative of C with respect to u is hu, u ; θ = ett θ t t θ t, u ln u In the bivariate case, this copula is usually referred to as Clayton copula due to its appearance in Clayton 978. It is the copula of the multivariate Pareto distribution Mardia 96 and of the multivariate Burr distribution Takahasi 965. It was rst mentioned as a multivariate copula in Cook and Johnson 98 and as a bivariate copula in Kimeldorf and Sampson 975. However, since many properties were discovered studying the corresponding distribution function and Cook and Johnson 98 mentioned its general form, it is more appropriate to call it MTCJ copula in the multivariate setup.

8 8 Ulf Schepsmeier, Jakob Stöber and the density function is cu, u ; θ = Cu, u ; θu u { lnu θ lnu θ } θ lnu lnu θ { θ lnu θ lnu θ θ } = Cu, u ; θ t t θ lnu lnu θ { θ t t θ }. u u 3.5 Frank copula The Frank copula is dened as Cu, u ; θ = θ ln e θ eθ e θu e θu ], and has the following density function: cu, u ; θ = θ e θ e θuu e θ e θu e θu ], where θ, ]\{0}. Its h-function conditional cdf is e θ e θ u hu, u ; θ = e θ uθ u e θ uθ e θ uθ e θ. Remark: Joe 997, p. 4 denes the Frank copula just for positive parameters 0, ], in which case its generator function arises from the Laplace transform ϕθ = θ ln eθ e t of the log-series distribution. It's parameter range can however be extended to, ]\{0}, see e.g. Trivedi and Zimmer Joe copula The last member of the Archimedean family we consider the Joe copula which has the generator function ϕt = e t θ and using it is dened for θ as t i = u i θ i =,, Cu, u ; θ = u θ u θ u θ u θ θ = t t t t θ,

9 Derivatives and Fisher information of bivariate copulas 9 with density cu, u ; θ = u θ u θ u θ u θ θ u θ u θ θ u θ u θ u θ u θ ] = t t t t θ θ t t t t u θ u θ, and corresponding h-function hu, u ; θ = u θ u θ u u θ θ θ = t t t t θ u θ u θ. u θ u θ 4 Rotations of bivariate copulas Apart from the bivariate families listed in Section 3, we do also consider rotations of these copulas by 90, 80 and 70 degrees. For example, the bivariate Gumbel copula only covers positive dependence and exhibits upper tail dependence i.e. tail dependence in the upper right corner of the unit square. This means, that by considering rotations we can obtain tail dependent copulas for all four possible corners. The respective densities are given by c 90 u, u := c u, u c 80 u, u := c u, u c 70 u, u := cu, u from which the corresponding h-functions can be determined: h 90 u, u = h u, u h 80 u, u = h u, u h 70 u, u = hu, u. From this, the corresponding derivatives with respect to copula parameters are obvious, and the derivatives with respect to u and u can be calculated. For the derivatives of h-functions with respect to the second argument u, we obtain h 90 u, u = h u, u h 80 u, u = h u, u h 70 u, u = hu, u. The form of other derivatives follows similarly.

10 0 Ulf Schepsmeier, Jakob Stöber 5 Numerical issues and implementation in C The derivatives of the bivariate copula models we have discussed are included in the R-package VineCopula by Schepsmeier et al. 0. In order to speed up the calculations, all implementations were done in C, using R for a convenient frontend. Since the derivatives for the bivariate t-copula include derivatives of the regularized incomplete beta function, an ecient calculation of these is a key issue to achieve accuracy as well as low computation times. For this, we employ the algorithm of Boik and Robinson-Cox 998. Where this was possible, the C-code was optimized using the Computer Algebra Software Maple Maplesoft, a division of Waterloo Maple inc. 0. As an example for the usage of our package, let us consider the function BiCopDeriv Function, which calculates rst derivatives of the implemented copula families. Usage BiCopDeriv<-functionu,u,family,par,par=0,deriv="par",log=FALSE Arguments u, u Numeric vectors of equal length with values in 0, ]. family An integer dening the bivariate copula family: 0 = independence copula = Gaussian copula = Student t copula t-copula 3 = Clayton copula 4 = Gumbel copula 5 = Frank copula 6 = Joe copula 3 = rotated Clayton copula 80 degrees; survival Clayton 4 = rotated Gumbel copula 80 degrees; survival Gumbel 6 = rotated Joe copula 80 degrees; survival Joe 3 = rotated Clayton copula 90 degrees 4 = rotated Gumbel copula 90 degrees 6 = rotated Joe copula 90 degrees 33 = rotated Clayton copula 70 degrees 34 = rotated Gumbel copula 70 degrees 36 = rotated Joe copula 70 degrees par Copula parameter. par deriv log Second parameter for the bivariate t-copula default: par=0 Derivative argument "par" = partial derivative with respect to the rst parameter default "par" = partial derivative with respect to the second parameter only available for the t-copula "u" = partial derivative with respect to the rst argument u "u" = partial derivative with respect to the second argument u logical; if TRUE than the derivative of the log-likelihood is returned default: log=false; only available for the derivatives with respect to the copula parameters Value A numeric vector of the bivariate copula derivative with respect to deriv evaluated at u and u with parameters par and par. Function : BiCopDeriv, contained in the R-package VineCopula

11 Derivatives and Fisher information of bivariate copulas With similar usage, input arguments and output values, the functions BiCopDeriv, BiCopHfuncDeriv and BiCopHfuncDeriv are implemented for the second derivatives of densities and the rst and second derivatives of h- functions, respectively. For more detailed information, we refer to the manual of the package VineCopula. To obtain a benchmark for evaluating the numerical accuracy of our implementation, we consider the Fisher information with respect to the parameter which is analytically available for several parametric copula families. Using that the information with respect to the dependence parameters is independent of the marginal distribution see Smith 007, the Fisher information for the Gumbel family can be determined as 5 K 0 = 6 π 8 E θ = θ u eu du Exponential integral; Abramowitz and Stegun 99, p. 8 Iθ = θ 4 θ 3 π θ K 0 9 θ θ 3 θ K 0 θ K 0 K ] 0 E θ e θ θ Oakes and Manatunga 99. For the bivariate Gaussian copula, it is well known see e.g. Berger and Sun 008 that Iρ = ρ ρ, while the Fisher information for the Clayton family is with Iθ = θ θθ θ 4θ 3θ θ ρθ, θ ρθ = 3θ θ ] θ θ Ψ Ψ 3θ θ θ θ θ 0 ] θ θ Ψ Ψ, 3θ θ θ θ θ 0 where Ψ is the trigamma function see Oakes 98. To calculate the Fisher information numerically using our analytical derivatives of copula densities, we employ the adaptive integration routines supplied by Steven G. Johnson and Balasubramanian Narasimhan in the cubature package, available on CRAN based on Genz and Malik 980 and Berntsen et al. 99, with a maximum relative tolerance of e 5.

12 Ulf Schepsmeier, Jakob Stöber Comparing the results from the numerical integration to the known values of the Fisher information, we obtain a maximum relative error in the order of e 6, demonstrating the accuracy of our implementation. The Fisher information with respect to the standard parametrization of dierent copula families is illustrated in Figure with corresponding values in Table and Figure Table. For better comparison, the parameter values on the x-axis are transformed to the respective values of Kendall's τ. Note, that while the Fisher information is increasing with the absolute value of Kendall's τ for the Gaussian and Student-t copula, the same is not true for the Archimedean families. This, however, is a mere consequence of the standard parametrization for these families. If we consider the Fisher information with respect to a parametrization in the form of Kendall's τ, the shapes look similar as for the Gaussian copula. Note that for this reparametrization we obtain ] I τ τ = E τ lτ X lτ X τ i τ j i,j=,...,p = E θτ lθτ X lθτ X τ i τ j i,j=,...,p = E θτ lθτ X θ i lθτ X θ j θ i τ i θ j τ j i,j=,...,p = I θ,i,j θτ θ i θ j, τ i τ j i,j=,...,p ] ] where I θ,i,j θτ is the i, j element of the information I θ θτ with respect to θ. τ Gauss Clayton Gumbel Frank Joe Table Fisher information with respect to Kendall's τ for selected values of τ.

13 Derivatives and Fisher information of bivariate copulas 3 τ ρ ρ ρ ρ ρ ρ Table t-copula: Fisher information with respect to Kendall's τ for selected values of τ and degrees of freedom. 6 Summary and conclusion The asymptotic properties of estimation procedures for copula based models are well-known, though not very much used in practical applications. In particular, most authors do not state standard errors for their parameter estimates in multivariate copula models. One of the main reasons for this is, that the analytical derivatives which are needed to calculate the observed information are usually not available, and that other numerical methods can be highly unreliable. With the derivatives for bivariate copulas implemented in the package VineCopula, we provide - to the best of our knowledge - the rst implementation of the various derivatives which are needed in practice in a statistical software package. Comparing the results derived from integrating our derivatives with analytically known results about the Fisher information of bivariate families, we demonstrate that our implementation is numerically accurate, making it suitable in particular for the use with hierarchical copula models. Acknowledgment 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 95/99- FUGG. The rst 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.

14 4 Ulf Schepsmeier, Jakob Stöber Fisher information Fisher information Fisher information τ Fisher information Fisher information τ τ τ τ a Fisher information for the Gauss copula over a Kendall's τ range of 0.8, 0.8 b Fisher information for the Archimedean copulas over a Kendall's τ range of 0.05, 0.8. Top left: Clayton, top right: Gumbel, bottom left: Frank, bottom right: Joe Fisher information Fisher information Fisher information τ Fisher information Fisher information τ τ τ τ c see above d see above Figure Fisher information with respect to the standard parametrization upper panel, ab and with respect to Kendall's τ lower panel, cd.

15 Derivatives and Fisher information of bivariate copulas Fisher information 3e Fisher information e04 Fisher information e nu tau nu tau nu tau Figure Fisher Information of the Student t-copula over a Kendall's τ range of 0.5, 0.5 and degrees of freedom 7, 0 with respect to the correlation parameter ρ left the degrees of freedom middle and both right.

16 6 Ulf Schepsmeier, Jakob Stöber References Aas et al. 009] Aas, K., C. Czado, A. Frigessi, and H. Bakken 009. Paircopula construction of multiple dependence. Insurance: Mathematics and Economics 44, 898. Abramowitz and Stegun 99] Abramowitz, M. and I. Stegun 99. Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Dover Publisher. Bedford and Cooke 00] Bedford, T. and R. Cooke 00. Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 3, Bedford and Cooke 00] Bedford, T. and R. Cooke 00. Vines - a new graphical model for dependent random variables. Annals of Statistics 30, Berger and Sun 008] Berger, J. O. and D. Sun 008. Objective priors for the bivariate normal model. The Annals of Statistics 36, Berntsen et al. 99] Berntsen, J., T. O. Espelid, and A. Genz 99. An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Soft. 7 4, Bickel and Doksum 007] Bickel, P. J. and K. A. Doksum 007. Mathematical Statistics: Basic Ideas and selected Topics second ed., Volume. Pearson Prentice Hall, Upper Saddle River. Boik and Robinson-Cox 998] Boik, R. J. and J. F. Robinson-Cox 998. Derivatives of the incomplete beta function. Journal of Statistical Software 3. Clayton 978] Clayton, D. G A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrica 65, 45. Cook and Johnson 98] Cook, R. D. and M. E. Johnson 98. A family of distributions for modeling non-elliptically symmetric multivariate data. J. Roy. Statist. Soc. B 43, 08. Czado 00] Czado, C. 00. Pair-Copula Constructions of Multivariate Copulas. In P. e. a. Jaworski Ed., Copula Theory and Its Applications, Lecture Notes in Statistics, Volume 98, Berlin Heidelberg, pp Springer-Verlag. Dakovic and Czado 0] Dakovic, R. and C. Czado 0. Comparing point and interval estimates in the bivariate t-copula model with application to nancial data. Statistical Papers 5, Diÿmann et al. 0] Diÿmann, J., E. C. Brechmann, C. Czado, and D. Kurowicka 0. Selecting and estimating regular vine copulae and application to nancial returns. submitted. Donaldson and Schnabel 987] Donaldson, J. R. and R. B. Schnabel 987. Computational Experience with Condence Regions and Condence Intervals for Nonlinear Least Squares. Technometrics 9, 678.

17 Derivatives and Fisher information of bivariate copulas 7 Efron and Hinkley 978] Efron, B. and D. V. Hinkley 978. Assessing the accuracy of the maximum likelihood estimator: Observed versus expected sher information. Biometrika 65 3, pp Genest and Favre 007] Genest, C. and A. Favre 007. Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, Genz and Malik 980] Genz, A. C. and A. A. Malik 980. An adaptive algorithm for numeric integration over an n-dimensional rectangular region. J. Comput. Appl. Math. 6 4, Joe 996] Joe, H Families of m-variate distributions with given margins and mm-/ bivariate dependence parameters. In L. Rüschendorf and B. Schweizer and M. D. Taylor Ed., Distributions with Fixed Marginals and Related Topics, Volume 8, Hayward, CA, pp. 04. Inst. Math. Statist. Joe 997] Joe, H Multivariate Models and Dependence Concepts. Chapman und Hall, London. Kimeldorf and Sampson 975] Kimeldorf, G. and A. R. Sampson 975. Uniform representations of bivariate distributions. Comm. Statist. 4, Lehmann and Casella 998] Lehmann, E. L. and G. Casella 998. Theory of Point Estimation second ed.. Springer, New York. Mardia 96] Mardia, K. V. 96. Multivariate Pareto distributions. Ann. Math. Statist. 33, McCullough 999] McCullough, B. D Assessing the reliability of statistical software: Part ii. American Statistician 53, 49. McNeil and Neslehová 009] McNeil, A. J. and J. Neslehová 009. Multivariate archimedean copulas, d-monotone functions and l-norm symmetric distributions. The Annals of Statistics 37, Oakes 98] Oakes, D. 98. A model for association in bivariate survival data. Journal of the Royal Statistical Society. Series B Methodological 44 3, pp Oakes and Manatunga 99] Oakes, D. and A. K. Manatunga 99. Fisher information for a bivariate extreme value distribution. Biometrika 79 4, pp Schepsmeier et al. 0] Schepsmeier, U., J. Stöber, and E. C. Brechmann 0. VineCopula: Statistical inference of vine copulas. R package version.0. Sklar 959] Sklar, M Fonctions de répartition á n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 93. Smith 007] Smith, M. D Invariance theorems for Fisher information. Communications in Statistics - Theory and Methods 36, 3. Stöber and Schepsmeier 0] Stöber, J. and U. Schepsmeier 0. Ecient maximum likelihood inference for regular vine copulas. in preperation.

18 8 Ulf Schepsmeier, Jakob Stöber Takahasi 965] Takahasi, K Note on the multivariate Burr's distribution. Ann. Inst. Statist. Math. 7, Trivedi and Zimmer 007] Trivedi, P. and D. Zimmer 007. Copula modeling: an introduction for practitioners. Now Publishers.

19 Derivatives and Fisher information of bivariate copulas 9 A Web supplement: Derivatives corresponding to elliptical copulas A. Gaussian copula Derivatives of the density function with respect to ρ and u Taking the rst derivative of the density with respect to the correlation parameter ρ we obtain { ρ 3 x x ρ ρx ρx ρ x x exp ρ = ρ 5 } ρρx ρx x x ρρ. Furthermore, one can calculate the derivatives of c with respect to u and u. In the nonrotated case these derivatives are the same because of the symmetry. For the Gaussian copula the derivative with respect to u is = cu, u ; ρ ρ x x x ρx u u u ρ, with x i π = u i exp{φ u i /} i =,. Derivatives of the h-function with respect to ρ and u The rst derivative of the h-function conditional cdf with respect to the copula parameter is h ρ = φ Φ u ρφ Φ u u ρ Φ u ρφ u ] ρ ρ ρ ρ, where φ is the pdf of the standard normal distribution and Φ as above the standard normal cdf. The derivative with respect to u is the pdf and the derivative with respect to u is h = φ u Φ u ρφ u ρ ρ ρ x u.

20 0 Ulf Schepsmeier, Jakob Stöber Second derivatives of the density function with respect to ρ, u and mixed In the following, we list the second derivatives of the pdf. The rst one is the derivative with respect to the correlation parameter ρ. c ρ = ρ 5 ρ 5 3ρ x ρ x ρ x x ρ ρ m x x ρ x ρ x ρ x x ρ x ρ x x ρ x x x ρ ] exp { ρρx ρx x } x ρ ρ { ρ 3 x x ρ ρx ρx ρρx ρ x x exp ρx x } x ρ ρ ρx ρx x x ρ x x ρx ρ x x ρx ρ x ρ x x ] ρ x ρ ρ ρρx ρx x x ρ ρ ] /ρ ρ { ρ 3 x x ρ ρx ρx ρρx ρ x x exp ρx x } x ρ 3 5ρ ρ ρ The next two derivatives involve the rst copula argument u. In the rst case we dierentiate twice with respect to u and in the second case we dierentiate with respect to u and the correlation parameter ρ. c u = u ρ x x u ρx x u ρ ρ cu, u x u x x ρx x u u ρ x c = x ρu ρ 5 ρ x ρx x u u u ρ x ρx x x ρ ρ ρ u u exp Finally, we have the derivative of c with respect to both arguments u, u : { ρρx ρx x } x ρ ρ = cu, u ; ρ ρ x x x ρx u u ρ x x x ρx u u u u ρ ρ ρ x x u u ρ. Second derivatives of the h-function with respect to ρ, u and mixed We continue with the second derivatives of the h-function. As noted above, these only need to be determined with respect to the second argument u and the copula parameter ρ. h ρ = t φt t= x ρx ρ x ρ ρ x ρx ρ ρ x x ρx ρx ρ ρ x φ ρ ρ x ρx ρ ρ ρ ρ,

21 Derivatives and Fisher information of bivariate copulas with φt = π exp{t /} and Thus Secondly, we have t φt t= h = u t φt t= t φt = exp{t /}t = tφt. π x ρx ρ x ρx ρ x ρx = φ ρ ρ ρ x Φ u x ρx ρ x ρx ρ. ρ x ρ, u with x π = u exp{φ u i /} Φ u i x i xi = Φ u i i =,, u i u i and nally h = ρ u t φt x ρ x ρx ρ ρ x t= ρx ρ φ x ρx ρ ρ ρ ρ ρ ρ x u. ρ x ρ u

22 Ulf Schepsmeier, Jakob Stöber A. t-copula Derivatives of the density function with respect to ρ, and u Writing l for the log density and c for the bivariate copula density, the rst derivative of c with respect to ρ is ρ = l ρ cu, u, where the rst derivative of l with respect to ρ is l ρ = ρ ρ ρ x x M, ρ, x, x. Similarly, the derivative with respect to the degrees of freedom is l = Ψ Ψ ln ρ ln x x x x x ] x ln x x ρ x x x x M, ρ, x, x ρ x x lnm, ρ, x, x, x x with Ψ being the digamma function. To obtain the partial derivative of l or c with respect to we need the derivative of x i = t u i, i =,. x i = dtx i, B, x i x i ] x i t t lntdt 0 4dtx i, I x i, ] Ψ Ψ = dtx, I x i, ] x i x i, u i, i =,. Here, I x, is the rst derivative of the regularized beta function with respect to i the rst argument in brackets. Beside the derivatives with respect to the parameters, we will also require the rst derivative with respect to u. Because of symmetry, the derivative with respect to u has the same functional form with exchanged arguments. where = cu, u u dtx, x ρx ρ x x ρx dtx,, x u dtx i, = dt x i, u i dtx i, = x i. x i Derivatives of the h-function with respect to ρ, and u For the following calculations of the derivatives of the h-function we need the derivative of t with respect to and evaluated at x i. t x i = I, = x i dtx i,. x i

23 Derivatives and Fisher information of bivariate copulas 3 Using this, we obtain the derivatives with respect to ρ, and u, respectively. h ρ = dt x ρx, x x ρx x ρ h = t x ρ x x ρ h = dt x ρx u x ρ x ρx x ρ x ρ dt x ρx x ρ, dtx, x ρx x ρ 3 ρ ρ x ρ x ρ, x x 3 ρ x ] x x ρx x ρ ρ 3 x Second derivatives of the density function with respect to ρ,, u and mixed For the derivatives of the density function c, we have c ρ = ρ = l ρ ρ ρ c = l ρ c l ρ ρ = l ρ c ρ c ρ, i.e. they can be calculated from the derivatives of the log-likelihood function, which we will now determine. Exempli gratia, for the second derivative of the pdf c with respect to ρ we need the second derivative of l with respect to ρ. l ρ l = Ψ = ρ ρ M, ρ, x, x ρ x x M, ρ, x, x Ψ x x x x x x x x x x x x x x x x x x x x x x x x with x x x x x x M M, ρ, x, x M M, ρ, x, x M M, ρ, x, x, M = x ρ x x x ρ x x x x

24 4 Ulf Schepsmeier, Jakob Stöber and M = and x x x x i = dtx i, t x i dtx i, x x x 4ρ x x ρ x x x t x i dt x i, x i dt x i, dt x i, x ] i. x Here, Ψ is the trigamma function, which is the second derivative of the logarithm of the gamma function Ψ z = d d ln Γ z. z In the calculation of x i we need the second derivative of tx i with respect to and the derivatives of dtx i, with respect to and x i, respectively. tx i = B, x i x i ]] x i t t lntdt 0 Ψ Ψ ] t x i Ψ Ψ ] 8 I x i, ] Ψ Ψ and dt B, x i x i x i lnx i x i ln x i x i x i ] x i t t lnt dt 4 0 x i, = Ψ Γ Γ Γ π ln ] x i x i Γ Γ Ψ Γ Γ ] Γ π x i ln x i x i x i x i and dt x i, = Γ 3 Γ x i x i. π l ρ = ρ ρ x x x x ρ M, ρ, x, x M M ρ M, ρ, x, x M ρ M, ρ, x, x,

25 Derivatives and Fisher information of bivariate copulas 5 with M ρ = ρ x x. l ρ x x = x ρ u M, ρ, x, x dtx, M, ρ, x, x x ρx. The derivatives of the copula density involving u can be determined similarly: c = u dtx, dtx, cu, u x ρx M, ρ, x, x cu, u dtx, x dt u dtx, x ρx M, ρ, x, x x ρx M, ρ, x, x x x x x x x, dt x, x ] x x M x ρ x M, ρ, x, x x and c u = u dtx, cu, u dtu, dtx u, dtx, cu, u dtx, x ρx M, ρ, x, x dtx, u M, ρ, x, x x ρx ] M, ρ, x, x dtx, u, with dtx, = u dtx, x x x. The last derivative is with respect to u and u. c cu, u = u u dtx, dtx, x ρx M, ρ, x, x x ρx M, ρ, x, x dtx, u dtx, u ρ M, ρ, x, x x ρx x ρx M, ρ, x, x ]

26 6 Ulf Schepsmeier, Jakob Stöber Second derivatives of the h-function with respect to ρ,, u and mixed In this section, we obtain the required second derivatives of the h-function. h ρ = dt x ρx, x x ρx x ρ dt x ρx x ρ, ρ x x ρx h = t x ρ x x ρ dt x ρx x ρ x ρx x ρ x ρ x ρ x x ρ dt x ρx x ρ x ρ x x ρ 3 x x ρx x ρ, x ρx x ρ, dt 3 ρ 3 ρ x ρ 3 3 x ρx x ρx x ρ x ρ x x x x 3 ρ x, 5 ρ x x ] ] x x ρ x x ρ 3 3 x ρx x x 4 x ρ 5 ρ x x x ρ ] x x ρ x x ρx x ρ x ρ 3 ρ x x x x x x 3 ]

27 Derivatives and Fisher information of bivariate copulas 7 dt h = u x ρx x ρ dtx, dt x ρx x ρ dt x ρx ρ x ρ, x ρ dtx, dtx,, dt x, dtx, 3, dtx, dtx, ρ x ρ ρ x ρ x ρx x ρ x ρx x ρ ρ 3 x ρ 3 x ρ x ρ 3 3 x ρx ρ x ρ 5 x x ρx ρx x ρ 3 h ρ u = dt x ρx x ρ ρ dtx, x ρ, x ρx x ρ dt x ρx, x ρ dtx, x ρx x ρ ρ x ] ρx 3 3 x x ρ ρ 3 x x ρ x ρx x ρ x ρx x ρ x x ρ 5 ρ x 3 ρ x 3 ρ ρ x ρ 3

28 8 Ulf Schepsmeier, Jakob Stöber h ρ = dt x ρx x ρ x x ρ x ρ, dt x ρx x ρx x ρ 3 ρ x ρ x x, ] x x x ρx x ρ x ρ 3 ρ x dt x ρx, x x ρ x ρ 3 3 x ρx x ρ x x ρ ] x x x ρ x ρ x x ρ 3 ρ x x x ρx x x 3 ρ x ρ 5 ρ x

29 Derivatives and Fisher information of bivariate copulas 9 dt h = u x ρ x x ρ dt x ρx x ρ dtx, x ρx x ρ dtx,, x ρx x ρ, dt x ρx x ρ 3 ρ dtx, x x dt x, dt x, x, ] x ρ x ρx ρ x ρ x ρ 3 x dt x ρx, x ρ ρ x x dtx, x ρ 3 ρ x x ρ x ρ x ρ 3 x x ρx x ρ ρ 3 x 3 x ρx x ρ x ρx x ρ ρ x 3 5 ρ x x x x

30 30 Ulf Schepsmeier, Jakob Stöber B Web supplement: Derivatives corresponding to Archimedean copulas B. Clayton copula Derivatives of the density function with respect to θ and u For the rst derivative of the density with respect to the copula parameter θ we get θ = u u θ u θ u θ θ θu u θ lnu u u θ u θ θ θu u θ u θ u θ θ lnu θ u θ θ θ uθ lnu u θ = cu, u lnu u u θ u θ lnau, u, θ θ and the derivative with respect to u is lnu θ uθ lnu u θ Au, u, θ lnu θ = θ u u θ θ u θ u θ u θ u u θ u θ u θ u θ θ u θ θu u θ u θ = cu, u θ cu, u θ θ u u θ Au, u, θ. Derivatives of the h-function with respect to θ and u Additionally, we calculate the derivatives of the h-function with respect to the copula parameter and u, respectively: and h θ = uθ u θ u θ θ θ θ u θ u θ u θ θ θ u θ u θ v θ θ θ θ u θ u θ u θ θ θ = h u θ Au, u, θ θ θ θ u h u = θ u θ Au, u, θ θ θ Au, u, θ θ A u, with A u = θu θ. Second derivatives of the density function with respect to θ, u and mixed Using A θ = uθ lnu u θ lnu

31 Derivatives and Fisher information of bivariate copulas 3 and A θ = uθ lnu u θ lnu we obtain the second derivative with respect to θ c θ = θ lnu lnau, u, θ A θ θ θ Au, u, θ A θ Au cu, u,u,θ θ lnau, u, θθ θ 4 With and we have and A θ θ θ A Au θ, u, θ θ A θ Au, u, θ. A = θu θ u A = θθ u θ u c θ u u θ cu, u = u u θ A cu u u, u A cu u, u θ A u Au, u, θ ] c u θ = θ θ cu, u u θ Au, u, θ θ θ cu, u u u θ Au, u, θ ] cu, u θ u θ lnu Au, u, θ u θ A θ u θ. Au, u, θ c = u u u θ u θ u u θ Au, u, θ cu, u θ u θ Au, u, θ A u Second derivatives of the h-function with respect to θ, u and mixed We will now determine the second derivatives of the h-function. h θ = h θ lnu lnau, u, θ A θ θ θ Au, u, θ A θ Au hu, u,u,θ θ lnau, u, θθ θ 4 A θ θ θ A Au θ, u, θ θ A θ Au, u, θ

32 3 Ulf Schepsmeier, Jakob Stöber h u = θ θ u θ3 Au, u, θ θ h = θ θ u θ u θ θ Au, u, θ θ A u Au, u, θ 3 θ θ θ Au, u, θ A θ θ u h u u hu, u u u θ Au, u, θ 3 θ A θ u A u θ u θ3 Au, u, θ θ

33 Derivatives and Fisher information of bivariate copulas 33 B. Gumbel copula Derivatives of the density function with respect to θ and u The derivative of the pdf with respect to the copula parameter can be calculated as θ = cu, u t t θ ln t t θ t ln ln u t ln ln u θ t t ln t t θ ] t ln ln u t ln ln u ln ln u ln u θ t t Cu, u t t θ ln u ln u θ t t θ θ t t θ u u ln t t θ t ln ln u t ln ln u θ t t Similarly, the derivative of c with respect to u is = cu, u t t θ t u u lnu ] t θ θ θ u u u lnu Cu, u t t θ ln u ln u θ θ t t θ t u u u lnu.. Derivatives of the h-function with respect to θ and u The derivatives of the h-function with respect to the parameter θ and u are: h θ = hu, u t t θ ln t t ln t t θ θ t ln ln u t ln ln u θ t t θ ] t ln ln u t ln ln u ln ln u t t h t t θ θ θ = hu, u t ] θ u u lnu u lnu t t. u

34 34 Ulf Schepsmeier, Jakob Stöber B.3 Frank copula Derivatives of the density function with respect to θ and u For the rst derivative of the pdf with respect to the copula parameter we get with θ = cu, u θ eθ e θ u u e θ ] t t e θ t θ t t t, θ t i := e θu i, i =, and t i θ = u ie θu i, i =,. The derivative with respect to u is t ] = cu, u θ t e θ t t, u u where t i u i = θe θu i, i =,. Derivatives of the h-function with respect to θ and u The calculation of the derivatives of the h-function with respect to the copula parameter and u is straightforward. h θ = hu, u ; θ u e θu e θu u u e θ u θ u u e θ uθ u e θ uθ e θ e θ u θ u e θ uθ e θ uθ e θ, h θ e θ u θ u θ e θ u θ = hu, u ; θ u e θ u θ u e θ uθ e θ uθ e θ.

35 Derivatives and Fisher information of bivariate copulas 35 B.4 Joe copula Derivatives of the density function with respect to θ and u For the derivative of the density with respect to θ we get θ = cu, u ln t t t t θ θ t θ t θ t θ t t t θ ln u ln u t t t t t t t t θ u θ u θ t θ t θ t θ t t t, θ with t i θ = t i ln u i i =,. While the derivative with respect to the rst copula argument u is θ = cu, u t θ t θ t θ u t t t t u u u t t t t θ u θ u θ t θ u t θ t u. Derivatives of the h-function with respect to θ and u Again as before we determine the derivatives of the h-function with respect to the copula parameter and u, respectively. h θ = hu, u ln t t t t θ t t t t θ u θ t θ, θ t θ t θ t θ t t t θ ln u t t t t h θ = hu,u t θ t t θ θ. u t t t t u u u