Ulf Schepsmeier and. Jakob Stöber

Transcript

1 Web supplement: Derivatives and Fisher information of bivariate copulas Ulf Schepsmeier and Jakob Stöber Lehrstuhl für Mathematische Statistik, Technische Universität München, Parkring 3, Garching-Hochbrück, Germany, contact phone: Introduction This manuscript contains additional material to the paper "Derivatives and Fisher Information" of Schepsmeier and Stöber 0 and considers partial derivatives corresponding to several parametric copula families In Chapter and Chapter 3 we list the partial derivatives of considered elliptical and Archimedean copulas, respectively Especially the Gauss and Student's t-copula are treated in detail For these two families as well as for the Archimedean Clayton copula the partial derivatives of the density function and the conditional distribution function of rst and second order are derived For the remaining Archimedean copulas, ie Gumbel, Frank and Joe, only the partial derivatives of rst order are considered The results for Student's t-copula are based on Dakovic and Czado 0, where we corrected some aws To improve the standalone readability of this manuscript, denitions and notation from the main paper are repeated where necessary Derivatives corresponding to elliptical copulas The following listed derivatives of the elliptical copula families, Gauss and Student's t, are the prosecution of Chapter 3 in the main article by Schepsmeier and Stöber 0 For better readability we also note the denitions of the copulas, although they are already noted in the manuscript

2 Gaussian copula The cumulative distribution function cdf of the Gaussian copula family is given by Cu, u ; ρ = Φ Φ u, Φ u, ρ, where Φ,, ρ is the cdf of two standard normally distributed random variables with correlation ρ,, Φ is the cdf of N0, standard normal and Φ the quantile function is its functional inverse The density is cu, u ; ρ = { exp ρ x x ρx x ρ ρ where x = Φ u and x = Φ u The cdf of the rst variable U given the second is hu, u ; ρ = Φ u ρφ u Cu, u ; ρ = Φ u ρ }, Derivatives of the density function Taking the rst derivative of the density cu, u ; ρ with respect to the correlation parameter ρ we obtain ρ3 x x ρ ρx ρx ρ x x exp ρ = ρ 5 The derivative with respect to u is u = cu, u ; ρ { ρ x x x u ρx ρ } ρρx ρx x x ρρ u, where x i u i = Further, cu, u = cu, u π exp{φ u i /} i =, Derivatives of the h-function The rst derivative of the conditional cdf hu, u ; ρ with respect to the copula parameter ρ is ρ = φ Φ u ρφ u ρ Φ u ρ Φ u ρφ u ρ ρ ρ

3 The derivative with respect to u is c and the derivative with respect to u is Φ u ρφ u ρ = φ u x ρ ρ u Second derivatives of the density function The second derivative of cu, u ; ρ wrt the correlation parameter ρ is c ρ = ρ 5 ρ 5 3ρ x ρ x ρ x x ρ ρ x x ρ x ρ x ρ x x ρ x ρ x x { ρ x x ρρx ρx } x x x exp ρ ρ ρ { ρ 3 x x ρ ρx ρx ρ x x exp ρx ρx x x ρ ρρx ρx x x ρ 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 We dierentiate twice with respect to u and in the second case we dierentiate with respect to u and the correlation parameter ρ Thus, for the second derivative wrt u we obtain } c = ρ x x x u ρx u ρ u u ρ cu, u the partial derivative wrt ρ and u is c = x x ρu ρ 5 ρ x ρx x u u u ρ x ρx x x ρ ρ ρ u u x u x x u ρx x u exp { ρ Finally, we have the partial derivative of c with respect to u, u : ρ x x x u = cu, u ; ρ ρx u ρ x x x u ρx u u ρ ρ ρρx ρx } x x ρ ρ u, ρ x x u u ρ 3

4 Second derivatives of the h-function The second derivatives of h only need to be determined with respect to the second argument u and the copula parameter ρ, since u = c For the second derivative wrt ρ we obtain h ρ = t φt x t= ρx ρ x ρ x ρx ρ ρ ρ x ρx x ρx ρ ρ x ρ ρ x ρx φ ρ ρ ρ ρ, where and Thus t φt φt = π exp{t /} t φt = π exp{t /}t = tφt x ρx = φ ρ ρ x t= ρx x ρx ρ Secondly, we have the second derivative wrt u h = u t φt ρ x x ρx ρ Φ u ρ where x u = x t= ρx ρ π exp{φ u i /} Φ u i x i u i = ρ x ρ, u xi Φ u i i =,, u i and nally the partial derivative wrt ρ and u h = ρ u t φt x ρ x ρx x t= ρx ρ φ x ρx ρ ρ ρ ρ ρ x ρ u ρ ρ ρ x ρ u 4

5 Student's t-copula Derivatives of the density function The log-density of Student's t-copula is given by where lu, u ; ρ, = ln cu, u ; ρ, = ln ln ln ρ ln Γ ln Γ ln x ln x M, ρ, x, x := ρ x x ρx x, x i := t u i, u i 0,, i =, lnm, ρ, x, x, We follow the approach by Dakovic and Czado 0 to write the cdf t x i of the univariate t-distribution in terms of the regularized beta function I: I x t x i =,, xi 0, i,, xi < 0, I x i where I x i, x i 0 t t dt = B, i =,, Abramowitz and Stegun, 99, p 948 Let us denote the density of the univariate t-distribution with degrees of freedom by dtx i ; Then, this implies for the partial derivative u i, which we denote as x i t, that x i = t xi, i =, dtx i ; Expanding the expression for the cdf t x i in Equation we obtain 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 =,, is the partial derivative of the regularized beta function with respect Here, I x i to the rst argument in brackets, ie d dx I t, x evaluated at t i 5

6 The partial derivative of l with respect to ρ is l ρ which implies for the copula density c that ρ = ρ ρ x x M, ρ, x, x, ρ = l ρ cu, u 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 where Ψ denotes the digamma function The partial derivative of the Student's t-copula density with respect to u is = cu, u x ρx u dtx ; ρ x x ρx dtx ;, x u where dtx i u i ; = dt x i ; u i dtx i ; = x i x i Derivatives of the h-function with respect to ρ, and u The partial derivatives with respect to ρ, and u, respectively, are given as follows: ρ = dt x ρx x ρ = t x ρ x x ρ ; x ρx x ρ x x ρ dt x ρx x ρ x ρx x ρ x ρx x ρ ; x 3 ρ x 3 ρ x x,, 6

7 = dt x ρx u x ρ ; dtx ; ρ x ρ x ρx x ρ ρ 3 x These can be evaluated using our previous result for x i, since t x i = I x i, = x i dtx i; Second derivatives of the density function We compute the second partial derivatives of the copula density c Where this leads to simpler expressions, we state the corresponding derivatives of the log density l from which for ρ are related as follows: c ρ = ρ = l ρ ρ ρ c = l ρ c l ρ ρ = l ρ c ρ c ρ, The second derivatives of the log-likelihood function l with respect to ρ and are: l = Ψ Here, l ρ = ρ ρ M, ρ, x, x ρ x x M, ρ, x, x x x x x x x x x M = and x Ψ x x x x x x x x x x x x x x M = x ρ x x x x x x i = dtx i ; t t xi dtx i ; dt x x x x x x x x M M, ρ, M, ρ, x, x x, x M M M, ρ, x, x x ρ x x x x x 4ρ x x ρ x i dt x i ; x i x i ; dt x i ; x i, x x x x x, 7

8 Ψ is the trigamma function, the second derivative of the logarithm of the gamma function Ψ z = d ln Γz d z This expression for x i involves the second partial derivative of the cdf t x i, as well as the partial derivatives of dtx i, with respect to denoted dt and x i denoted dt, respectively: txi = B, x i Ψ Ψ t x i Ψ Ψ dt B, x i x i x i x i 0 x i t t lntdt 8 I x i, Ψ Ψ lnx i x i ln x i x i x i x i t t lnt dt, 4 x i ; = Ψ 0 Γ Γ Γ Γ π x i ln x i x i Γ Ψ Γ Γ Γ π ln x i x i x i dt x i ; = Γ 3 Γ x i x i π x i, Taking the partial derivatives of l with respect to both parameters yields l ρ = ρ ρ x x x x M M ρ ρ M, ρ, x, x M, ρ, x, x where M ρ M, ρ, x, x, M ρ = ρ x x Further, the derivative of the log-density with respect to ρ and u is l ρ x x = x ρ u M, ρ, x, x dtx ; M, ρ, x, x x ρx 8

9 The derivatives of the copula density with respect to and u can be determined similarly: c = u dtx ; cu, u dtx ; dt x ; dt x ; x x ρx dtx ; M, ρ, x, x u cu, u dtx ; x ρx M, ρ, x, x x ρx M x M, ρ, x, x ρ x M, ρ, x, x x x x x x x x x x For the second partial derivative with respect to the argument u we obtain c u = u dtx ; u dtx ; x ρx dtx ; cu, u dtx ; M, ρ, x, x u cu, u dtu ; dtx ; M, ρ, x, x x ρx M, ρ, x, x dtx ;, u where u dtx ; = dtx ; x x x Finally, the second partial derivative with respect to u and u is c cu, u x ρx = dtx ; u u dtx ; dtx ; M, ρ, x, x u x ρx dtx ; M, ρ, x, x u ρ M, ρ, x, x x ρx x ρx M, ρ, x, x Second derivatives of the h-function In this section, we obtain the second partial derivatives of the conditional distribution function h-function Taking derivatives with respect to the association parameter ρ we 9

10 get h ρ = dt dt x ρx x ρ x ρx x ρ ; ; ρ x x ρx x ρ 3 x x ρ x x ρ x 3 x ρx x ρ 3 3 x ρx x ρ ρ x 5 ρ x Using similar notation and techniques as for the derivatives of the log-likelihood function we obtain h = t x ρx dt x ρx ; x ρ x x ρ dt x ρx x ρ x ρ x ρ x x ρ x ρx x ρ ; x ρx x ρ x ρ x 3 ρ x x 3 ρ x x x 0

11 dt ρ x ρx x ρ ρ x x x ρ x x x x x ; x x ρx x ρ x x 5 ρ x ρ x x ρ x x x 3 x ρx x ρ Also the second partial derivatives involving u follow similarly: dt xρx ; x h ρ = ρ x ρx u dtx ; dt dt ρ x ρx x ρ x ρx x ρ ; dt x ; dtx ; 3 ; x ρ dtx ; dtx ; x ρ ρ x ρ ρ x ρ 3 3 x ρx ρx 3 x ρ x ρ x 3 x ρx x ρ x ρx x ρ dtx ;, 5 3 ρ ρ x 3 ρ x ρ x

12 h ρ u = dt xρx x ρ ρ dtx ; x ρ dt xρx x ρ x ρx dtx ; x ρ 3 ρ x ; x ρx x ρ ; ρx 3 x x ρ 3 ρ x x ρ x ρx x ρ 5 x ρx x ρ x x ρ ρ x 3 3 ρ x ρ ρ x ρ Finally, we obtain the partial derivatives with respect to, and ρ and u, respectively h ρ = dt x ρx ; dt x ρx ; x ρ x x ρ x x ρ x ρ x ρx x ρ x ρx x ρ 3 x ρ x 3 ρ x ρ x 3 x

13 dt ρ x ρx x ρ x x x x ρ x ρx x ρ 3 ; x x ρ x x ρ x x ρ 3 ρ x ρ x x 3 x 3 x ρx x ρ 5 ρ x h u = dt x dt ρ x x ρ xρx x ρ ρ xρx x ρ dtx ; dtx ; x ρ ; x ρx x ρ ; dt xρx x ρ dtx ; ; x 3 ρ x dt x ; dt x ; x x ρx x ρ 3 ρ x x 3

14 dt xρx x ρ dtx ; ; ρ x ρ x 3 ρ x x x ρ x x ρ x ρx x ρ 3 3 ρ x ρ x 3 x ρx x ρ 3 x ρx x ρ ρ x 5 ρ x x x x 3 Derivatives corresponding to Archimedean copulas In the class of Archimedean copulas we consider the Clayton, Gumbel, Frank and Joe copula, mentioned in Chapter 3 For the Clayton copula we will exemplarily list all rst and second partial derivatives of the density function as well as of the conditional distribution function h-function, while in the other case we restrict ourself to the rst partial derivatives The remaining partial derivatives not listed here can easily be computed by a computer algebra system such as Mathematica or Maple 3 Clayton copula The Clayton copula, or sometimes referred to as MTCJ copula, is dened as Cu, u ; θ = u θ u θ θ = Au, u, θ θ, with Au, u, θ := u θ u θ The density of the Clayton copula is 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 h-function is given by Using the same notation, the hu, u ; θ = u θ Au, u, θ θ Wolfram Research, Inc, Mathematica, Version 80, Champaign, IL 0 Maple 30 Maplesoft, a division of Waterloo Maple Inc, Waterloo, Ontario 4

15 Derivatives of the density function The partial derivative of the density c with respect to the parameter θ is 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 = cu, u lnu u lnu θ uθ lnu u θ u θ u θ lnau, u, θ θ and the derivative with respect to u is θ uθ lnu u θ Au, u, θ lnu, = θ u u θ θ u θ u θ θ u θ u u θ u θ u u θ 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 We calculate the derivatives of the h-function with respect to the copula parameter θ and u, respectively: where θ = uθ u θ u θ u θ = θ θ u θ v θ θ θ θ u θ h u θ Au, u, θ θ θ u θ u θ u θ u θ θ θ u θ θ, u θ θ θ = θ u θ Au, u, θ θ u θ Au, u, θ A θ, u A u = θu θ Second derivatives of the density function For the following, we use the partial derivative of Au, u, θ to shorten the notation A θ = uθ lnu u θ lnu, A = θu θ, u A θ = uθ lnu u θ lnu, A = θθ u θ u 5

16 The second partial derivative of the Clayton copula density with respect to θ is c θ = θ lnu lnau, u, θ A θ θ θ Au, u, θ A θ cu, u Au,u,θ θ lnau, u, θθ A θ θ θ A θ θ 4 Au, u, θ θ A θ Au, u, θ Further, the partial derivative with respect to u is c = u u θ u θ cu, u u θ A u u cu, u A u cu, u θ A u Au, u, θ, and nally, we obtain the derivatives with respect to u, and θ and u, respectively 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 u θ u θ Au, u, θ cu, u θ u θ Au, u, θ A u Second derivatives of the h-function The second partial derivatives of the conditional distribution function with respect to θ and u are given as follows, h θ = θ lnu lnau, u, θ A θ θ θ Au, u, θ A θ hu, u Au,u,θ θ lnau, u, θθ A θ θ θ A θ θ 4 Au, u, θ θ A θ Au, u, θ, 6

17 h = θ θ u θ3 Au, u, θ θ u θ u θ θ θ θ h θ u = θ θ Au, u, θ θ u u hu, u u Au, u, θ A θ u Au, u, θ 3 θ A u, u θ Au, u, θ 3 A θ θ u A u θ u θ3 Au, u, θ θ 3 Gumbel copula The Gumbel copula is given by Cu, u ; θ = exp{ lnu θ lnu θ } θ = expt t θ, where t i := ln u i θ, i =, and θ The h-function and the density are as follows: hu, u ; θ = et t θ t t θ t, u ln u cu, u ; θ = Cu, u ; θu u { lnu θ lnu θ } θ lnu lnu θ θ = cu, u { θ lnu θ lnu θ θ } = Cu, u ; θ t t θ lnu lnu θ { θ t t θ } u u The derivative of the density c with respect to the copula parameter θ is t t θ ln t t ln t t θ θ Cu, u t t ln u ln u θ θ u u t t θ θ t t θ θ t ln ln u t ln ln u θ t t t ln ln u t ln ln u t t ln ln u ln u ln t t θ t ln ln u t ln ln u θ t t 7

18 Similarly, the derivative of c with respect to u is obtained as = cu, u t t θ t u u lnu θ t θ θ u u u lnu Cu, u t t θ ln u ln u θ u u θ t t θ t u lnu Finally, the partial derivatives of the h-function wrt θ and u are: θ = hu, 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 θ ln ln u, t t t t θ θ = hu, u u u lnu θ t θ u lnu t t u 33 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 θu u e θ e θu e θu, where θ, \{0} The corresponding h-function is e θ e θ u hu, u ; θ = e θ u θ u e θ u θ e θ uθ e θ The partial derivative of the density function c with respect to the copula parameter θ is given by θ = cu, u θ eθ e θ u u e θ t t e θ t θ t t t θ Here, we use t i := e θu i, i =, which implies t i θ = u ie θu i, i =, 8

19 The derivative with respect to u is u = cu, u θ t t u e θ t t, where t i = θe θu i, i =, u i Similarly, we obtain the partial derivatives of the h-function 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 u θ e θ uθ e θ, θ e θ u θ u θ e θ u θ = hu, u ; θ u e θ u θ u e θ u θ e θ uθ e θ 34 Joe copula The Joe copula is dened for θ as Cu, u ; θ = u θ u θ u θ u θ θ where t i := u i θ = t t t t θ, i =, Its density and h-function are 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, hu, u ; θ = u θ u θ u u θ θ θ u θ u θ = t t t t θ u θ u θ Using the partial derivative of t i u i, θ, t i θ = t i ln u i i =,, the derivative of the density with respect to θ can be determined as 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 9

20 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 t θ t u u Finally, we determine the derivatives of the h-function with respect to the copula parameter and u, respectively θ = hu, u ln t t t t θ θ t θ t θ t θ t t t θ ln u t t t t t t t t θ u θ t θ, θ t θ t t θ t t t t u u u = hu, u θ u References Abramowitz M, Stegun I 99 Handbook of mathematical functions with formulas, graphs, and mathematical tables Dover Publishing, New York Dakovic R, Czado C 0 Comparing point and interval estimates in the bivariate t-copula model with application to nancial data Statistical Papers 5:70973 Schepsmeier U, Stöber J 0 Derivatives and Fisher information of bivariate copulas 0