Friedrich-Alexander-Universität Erlangen-Nürnberg. Wirtschafts-und Sozialwissenschaftliche Fakultät. Diskussionspapier 61 / 2004

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1 Friedrich-Alexander-Universität Erlangen-Nürnberg Wirtschafts-und Sozialwissenschaftliche Fakultät Diskussionspapier 6 / 24 Constructing Symmetric Generalized FGM Copulas by means of certain Univariate Distributions Matthias Fischer and Ingo Klein Lehrstuhl für Statistik und Ökonometrie Lehrstuhl für Statistik und empirische Wirtschaftsforschung Lange Gasse 2 D-943 Nürnberg

2 Working Paper No. 6 manuscript No. will be inserted by the editor Constructing Symmetric Generalized FGM Copulas by means of certain Univariate Distributions Matthias Fischer, Ingo Klein 2 Department of Statistics and Econometrics, University of Erlangen-Nürnberg, Lange Gasse 2, 943 Nürnberg Matthias.Fischer@wiso.uni-erlangen.de 2 Department of Statistics and Econometrics, University of Erlangen-Nürnberg, Lange Gasse 2, 943 Nürnberg Ingo.Klein@wiso.uni-erlangen.de Received: date / Revised version: date

3 Abstract In this paper we focus on symmetric generalized Fairlie-Gumbel- Morgenstern or symmetric Sarmanov copulas which are characterized by means of so-called generator functions. In particular, we introduce a class of generator functions which is based on univariate distributions with certain properties. Some of the generator functions from the literature are recovered. Moreover two new generators are suggested, implying two new copulas. Finally, the opposite way around, it is exemplarily shown how to calculate the univariate distribution which belongs to a given copula generator function. Introduction Analyzing the dependence between the components X,..., X n of a random vector X is subject to various lines of statistical research. For this purpose, copula functions have been introduced by Sklar 959 in order to allow for a separation between the marginal distributions and the dependence structure. Although having no closed form and admitting only symmetric dependence patterns, the Gaussian and the Student-t copula which belong to the class of elliptical copulas have become very popular. Alternatively, several classes of copulas like the Archimedean class have been discussed in the literature. Within this work we focus on the construction of symmetric generalizations of FGM so-called symmetric Sarmonov copulas which can be simply estimated and which are given in closed form. The paper is organized as follows: In Section 2 we briefly review the notion of FGM copulas and some of their generalizations. In particular, the

4 main results of Amblard and Girard 22 are reviewed who discuss generalized FGM copulas depending on specific symmetric generator functions. Section 3 introduces generator functions derived from certain symmetric and asymmetric distributions. Several well-known generator functions will be recovered. Moreover, we introduce two new copula generators based on the Cauchy distribution and on the generalized secant hyperbolic distribution of Vaughan 22. Finally, in Section 4 we exemplarily demonstrate the opposite way around, i.e. how to derive the distribution function from a given copula generator. 2 The FGM copula family and generalizations Whereas copulas can be defined in a multivariate setting, we restrict ourselves to the bivariate case. Loosely speaking, a 2-copula is a two-dimensional distribution function defined on the unit square with uniformly distributed marginals. More formally, a two-dimensional copula is a function C : [, ] [, ] [, ] which satisfies the following properties:. C is 2-increasing, i.e. for u v and u 2 v 2 holds: Cv, v 2 Cv, u 2 Cu, v 2 + Cu, u For all u, v [, ]: Cu, = C, v = and Cu, = C, v = u. Note that every copula is bounded below by C min u, y = max{u + v 2, } and above by C max u, v = min{u, v}, the so-called Fréchet-Hoeffding bounds. Moreover, the copula associated with the joint distribution of two

5 independent uniform variables is given by C u, v = uv. Morgenstern 956, Gumbel 96 and Farlie 96 are connected to the so-called FGM-copula or family B in Joe 2 which is of the form C θ u, v = uv + θ u v, θ and can be seen as a direct generalization of C. More general, u and v can be replaced by suitable functions kernels or copula generators or generator function Au and Bv that are assumed to be differentiable functions on, with Au, Bv for u, v, i.e. C θ u, v uv + θaubv uv + θϕu ψv with ϕu u Au and ψu u Bu. Note that the admissible range of θ depends on the functions A, B see, for example, Bairamov und Kotz, 22. Distributions of that form were discussed by Sarmanov 966 and Lee 996, whereas Shubina and Lee 24 derived bounds for the corresponding dependence measures. Within this work we focus on the symmetric case, i.e. Au = Bv or ψu = ϕv for u = v. That means, equation reduces to C θ u, v = uv + θauav = uv + θϕu ϕv, θ [L, U]. 2 Huang and Kotz 999, for example, discuss the symmetric family generated by Au = u p, p > with max{, p} 2 θ /p, and the family generated by Au = u p, p with θ p + p p. Both models are nested in the Bairamov-Kotz FGM family with Au =

6 u p q, p >, q. The admissible range of θ is calculated by Bairamov and Kotz 22. Lai and Xie 2 consider ϕu = u b u p, b, p. Restricting the parameter θ in equation 2 to the interval [, ], Amblard and Girard 22 show that C θ,ϕ u, v = uv + θϕuϕv, θ [, ] 3 is a copula if and only if the generator function ϕ satisfies the following conditions: C Boundary condition: ϕ = ϕ =, C2 Lipschitz condition: ϕu ϕv u v u, v [, ] 2. Furthermore, C θ is absolutely continuous. Equivalently, C θ u, v from 3 generates a parametric family of copulas if, and only, if C3 ϕ u for almost every u [, ], C4 ϕu min{u, u} u [, ], C5 ϕ is absolutely continuous. The generator function ϕ can be interpreted as a measure of distance on the main diagonal of the unit square between the copula C ϕ and the independence copula C : ϕu = θ C ϕu, u C u, u = θ C ϕu, u u 2. Let G denote the class of copula generators C θ u, v which satisfy the conditions C3-C5. Amblard and Girard 22 state the following members. Example Copula generators

7 . The generator ϕ u = u u is connected to the standard FGM family. 2. The generator ϕ 2 u = π sinπ u leads to a family with large dependence. 3. The upper bound itself ϕ 3 u = min{u, u} is a generator function. 4. The generator of the Huang-Kotz family is given by ϕ 4 u = u u γ, γ. More general, we next focus on the class of so-called distribution-based generators and show that the generator functions ϕ,..., ϕ 4 can be recovered within this class. 3 Distribution-based generators 3. Generators derived from symmetric distributions Let G G denote the subclass of symmetric, positive generators, i.e. any ϕ G satisfies S ϕ u = ϕu u [,.5], P ϕu u [, ]., Consequently, for any ϕ G C3 ϕ u, for almost every u [,.5], C4 ϕu u, u [,.5]. A subset of G which is derived from symmetric random variables X is the class of ξ f functions which are introduced in the next definition.

8 Definition ξ f function Let X denote a continuous, symmetric random variable with corresponding density function fx, distribution function F x and inverse distribution function F x. Assume further that f is differentiable. Define ξ f u ff u, u [, ] 4 and let Ξ denote the class of functions from that type. It is straightforward to verify that, if f is symmetric, ξ f is monotone increasing on [,.5] and satisfies S and P. In order to ensure that ξ f is actually a copula generator we have to impose some certain restrictions on X which can be expressed using generalized absolute score functions. Definition 2 Generalized absolute score function For a positive, differentiable function g : R R + we define the generalized absolute score function by ψ g x = d dx lngx = g x gx. 5 We now characterize ξ f functions that actually generate a copula by means of generalized absolute score functions. Lemma Let ξ Ξ. Then, C3 ψ f x, x, C4 ψ F x, x.

9 Proof: Firstly, Secondly, ϕ u u [, ] d du ff u u [,.5] ψ f x = f x fx x, ]. ϕu min{u, u} u [, ] ff u u u,.5] fx F x x, ] ψ F x = fx x, ]. F x To sum up, every symmetric and absolute continuous univariate distribution with restricted ψ f function and restricted ψ F function can be used to construct a copula generator ϕ f. The interpretation of the density-based generator is the following: Loosely speaking, the density represents the distribution of the distance between a copula and the independence copula. In the sequel, we investigate for several densities with closed form cdf which meet the requirements of theorem. However, a counterexample is given first. Example 2 Normal generator For the standard normal distribution ψ Norm f = x > for x >. Hence, equation C3 is not satisfied and ξ Norm is not a copula generator. Example 3 Logistic generator The pdf, cdf and quantile functions of the logistic distribution are given by f Log x = e x + e x 2, F Logx = + e x and F Log x = log x x.

10 Hence, ξ Log u = f Log F Log u = u u, i.e. we recover the generator of the FGM family from example. Note that ψ Log exp x F x = and ψlog f x = + exp x exp x + exp x. Example 4 Hyperbolic secant generator For the hyperbolic secant distribution, f HS x = 2/π e x + e x, F HSx = arctanex π/2 πx and F HS tan x = log. 2 Consequently, ξ HS u = π sinπu, i.e. we recover the copula generator from example 2. In this case, ψf HS expx + exp x x = arctane x and ψf HS x = e x e x e x + e x. Example 5 Laplace generator The Laplace or double exponential distribution can be characterized by.5e x, x.5e x, x f Lap x =, F Lap x =,.5e x, x >.5e x, x > and its inverse F Lap x = log2x, x. log, x > 2 2x Consequently, ξ Lap u = min{u, u}, i.e. we recover the upper bound from example 3. In addition, for x ψ Lap F x = and ψlap f x =. Finally, we derive two new generators corresponding to the Cauchy distribution and to the GSH distribution from Vaughan 22. Note that the GSH distribution nests both the logistic and the hyperbolic secant distribution.

11 Example 6 Cauchy generator For the Cauchy distribution, f C x = π + x 2, F Cx = 2 arctan x + π 2π, and F C x = tan πu.5. Hence, ξ C u = π + tan πu.5 2, 6 which we will call a Cauchy generator. It can be verified that ψ C F x = 2 + x 2 2 arctan x + π and ψc f x = 2x + x 2. Example 7 GSH generator The generalized secant hyperbolic GSH distribution of Vaughan 22 is defined by f GSH x; t = sint 2tcoshx+cost for π < t <, e x +e x 2 for t =, sinht 2tcoshx+cosht for t >. It includes the logistic t = and the hyperbolic secant distribution t = π/2 as special cases. Note that t is a kurtosis parameter. Moreover, both the cumulative distribution function and the inverse distribution function are given in closed form by + t arccot expx+cost sint for π < t <, F GSH x = +e for t =, x t arccoth expx+cosht sinht for t > and F GSH u = ln sintu sint u ln u u ln sinhtu sinht u for π < t <, for t =, for t >.

12 It is straightforward but cumbersome to derive the ξ GSH function which is given by ξ GSH u = t sint sintu sint u + sint u +2 sintu cost for π < t <, u u for t =, 7 t sinht sinhtu sinht u + sinht u +2 sinhtu cosht for t >. It is shown in Appendix A that the function ξ GSH from equation 7 generates a copula for t [ π/2,. Figure, below, displays ξ f, ψ f, ψ F for the distributions discussed before. Figure to be inserted here In general, the conditional mean function as one possible dependence function is given by EU V = v = ucu, vdu = 2 + θϕ v uϕ udu. Using partial integration rules und setting K ϕ ϕudu we get EU V = v = 2 θk ϕϕ v, i.e. the conditional mean expectation is determined only by ϕ v i.e. by the score function. Hence copulas based on the logistic, the hyperbolic secant, the Laplace and the GSH distribution imply a stictly monotone increasing relation in mean, whereas the copula generated by the Cauchy distribution allows for non-monotone relations see figure 2, below. Figure 2 to be inserted here

13 3.2 Generators derived from asymmetric distributions We now drop the assumption that X is a random variable with symmetric distribution. Again f X and F X denote the density and the cumulative distribution function, respectively. The copula generator from example 3 can be generalized to the asymmetric case, other examples are thinkable. Example 8 Asymmetric logistic generator The generalized logistic distribution of type I has density function which is given by f AL x; α = α exp x, α >. + exp x α+ Note that α determines the skewness of the distribution. It is negatively skewed for < α <, positively skewed for α > and symmetric for α =. It can be easily verified that F AL x; α = + exp x α and F AL u; α = lnu /α. Consequently, ξ AL u; α = α u /α u +/α = α u /α u. Obviously, setting α = results in the copula generator from example 3. Furthermore, for < α we have α u /α u minu, u, u [, ]. For u = and u = the inequality is trivial. For u.5, α u /α u u /α u u, α <. For u >.5, we have to show that ξ AL u; α = α u /α u uu u.

14 The second inequality is trivial, whereas the first one follows from α u /α u for < α. Note that the last inequality can be proved using ξ AL u; α.5, ξ AL ; α = = u, ξ AL u; α and the continuity from ξ AL ; α. Additionally, ξ ALu; α = α u /α u /α. For < α and u, the absolute value of ξ Al is restricted to, because α u /α is positive and less than and u /α is also positive and less than one for u [, ]. To sum up, ξ AL is a copula generator for < α, which is similar to that of the symmetric modified FGM copula. Different copula generators are plotted in the figure 3, below. Figure 3 to be inserted here 4 Distributions derived from generators Up to now, we derived the copula generator for a given distribution which satisfies the requirements of Corollary. Conversely, given a copula generating function, what is the corresponding distribution? The answer to this question will be exemplarily demonstrated for so-called polynomial copulas see Mari and Kotz, 2. In general, a polynomial copula of order p, q is defined by with < θ kq θ pq C pq u, v = uv + p + q + up v q, 8 min k+q+ q, k+q+ k. In the symmetric case, i.e. setting p = q k in 8, the polynomial copula is a symmetric generalized

15 FGM copula: Comparing 8 and 2, we find φu = k + u u k. According to 4, the corresponding distribution can be derived as the solution of the differential equation F F u = φu F x = φf x for u F x with respect to F. In the case of a symmetric polynomial copula, this differential equation can be transformed to a Bernoulli differential equation see, for example, Bronstein and Semendjajew, 998 which can be solved explicitly. The solution is given by F x = k + Ce k/k+x k, where C is a constant that ensures F to be actually a cumulative distribution function. Proof: Setting y F x and y fx, we have to solve y k + y = k + yk+ y y k+ k + y k = k +. Substituting z y k, we obtain z = ky k+ y and k z k + z = k +. Multiplying both sides with k, a special case of the simple first order differential equation z + P xz = Qx, with P x = Qx = k/k +

16 results. The corresponding solution see Bronstein & Semendjajew, 99 is given by z = e P xdx k = e k/k+x k + Qxe P xdx dx + C e k/k+x dx + C = + Ce k/k+x. Undoing the substitution, y = + Ce k/k+x k. 5 Summary Symmetric generalized Fairlie-Gumbel-Morgenstern FGM copulas can be characterized by means of so-called generator functions or generators. In this paper we introduced generators based on specific univariate distributions. These distributions are characterized, for example, by a limited score function, which rules out the normal distribution as a suitable candidate. Several generators are recovered within this setting. Moreover, we derived two new generator functions from the Cauchy distribution and the GSH distribution of Vaughan 22, implying two new symmetric generalized FGM copulas. Finally, the opposite way around it is exemplarily shown, i.e. how to calculate the univariate distribution which belongs to a given copula generator function.

17 References. Amblard C, Girard S 22 Symmetry and Dependence Properties within a Semiparametric Family of Bivariate Copulas. Nonparametric Statistics 46: Bairamov I, Kotz S 22 Dependence Structure and Symmetry of Huang- Kotz FGM disrtibutions and their Extensions. Metrika 56: Bronstein IN, Semendyayev KA 998 Handbook of Mathematics. Springer, New York 4. Fairlie DJG 96 The Performance of some Correlation Coefficients for a General Bivariate Distribution. Biometrika 47: Gumbel EJ 96 Bivariate Exponential Distributions. Journal of the American Statistical Association 55: Huang JS, Kotz S 999 Modifications of the Fairlie-Gumbel-Morgenstern distributions. Metrika 49: Joe H 2 Multivariate Models and Dependence Concepts. Chapman & Hall, London 8. Lai CD, Xie, M 2 A new family of positive quadrant dependent bivariate distributions. Statistics & Probability Letters, 46: Lee MT 996 Properties and applications of the Sarmanov family of bivariate distributions. Communications in Statistics Theory and Methods 256: Mari DD, Kotz S 2 Correlation and Dependence. Imperial College Press, London.. Morgenstern D 956 Einfache Beispiele zweidimensionaler Verteilungen. Mitteilungsblatt für Mathematische Statistik 8:

18 2. Sarmanov OV 966 Generalized Normal Correlation and Two-dimensional Frechet Classes. Doklady Akademii Nauk SSSR 68: Shubina M, Lee MT 24 On maximum attainable correlation and other measures of dependence for the Sarmanov family of bivariate distributions. Communications in Statistics Theory and Methods 335: Sklar A 959 Fonctions de répartition á n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8: Vaughan DC 22 The Generalized Hyperbolic Secant distribution and its Application. Communications in Statistics Theory and Methods 32: A Proof for GSH generator Theorem GSH copula generator The function ξ GSH from equation 7 generates a copula for t [ π/2,. Proof: For t = π/2 and t = the result was already proved. To verify for what values of t this function generates a copula, we have to check the conditions of theorem. Note that ψ GSH f x; t = sinh x cosh x + cos t for t π, ] First, cosht for all t >. Moreover, sinhx and coshx for x and hence sinh x cosh x + cosh t = sinh x sinh x = tanhx. cosh x + cosh t cosh x A similar inequality is valid because cost for all t [ π/2, ]. For t < π/2, it is no longer valid.

19 Case : π/2 < t <. Next it remains to show that sint ψf GSH 2tcoshx+cost x; t = + t arccot expx+cost sint Substituting u e x x lnu, we have to show that ψ GSH F u; t = sint 2t u 2 + 2u +cost + t arccot u+cost sint To prove this, we show in the lemma below that ψ GSH F for t π/2,, x, ]. for any t π/2,, u, ]. lim u + ψgsh F u; t = and is decreasing in u for all t π/2,. Noting that ψf GSH u; t is continuous on [, ], the proof is completed. Lemma 2 Define fu; t Then lim u + ψgsh F sint 2t u 2 + 2u +cost and gu; t + t arccot u+cost sint fu; t u; t = lim =. 9 u + gu; t d u ψgsh F u; t, < u. Proof: Assume that t π/2, is arbitrary but fix. Again, sint < and cost > for t π/2,. As can apply the rule of l Hopital lim On the one hand, u + ψgsh F lim f u; t = lim u + u + 2 = lim fu; t = and lim gu; t =, we u + u + fu; t u; t = lim u + gu; t = lim f u; t u + g u; t. sin t t sin t t lim u + 2 u2 + 2u 2 u 2 + 2u + cos t 2 u 2 u2 + 2 u u cos t 2 = sin t. t.

20 On the other hand, lim u + g u; t = lim u + t sin t sint =. + u+cost2 t sint 2 Thus, equation 9 is verified. Secondly, we have to show that dψ F u du = f u; tgu; t fu; tg u; t gu; t 2, or, equivalently, that sint u 2 tu u cost 2 + arccot u+cost sint t sint u 2 u + cost t 2 t + π/2 + arctan sint u sin t 2 t 2 u u cos t 2 u sin t2 t 2 u + cost u 2 t + π/2 + arctan u sin t sint for u, ], t π/2,. Define Au, t t + π/2 + arctan u+cost sint and Uu, t = u sint. From A, t = t+π/2+arctan tanπ/2+t = t+π/2 arctantanπ/2+t = and A u, t = sint + u 2 + 2u cost we conclude that Au, t for all t π/2,. Consequently, for u, ], u 2 and u 2 Au, t Au, t. Finally, U, t = and U u, t = sint sint +u 2 +2u cost = A u, t. Together with the continuity of A and U, Au, t Uu, t and the proof of the lemma is completed.

21 Case 2: t >. Now it has to be proved that ψ GSH F x; t = sinht 2tcoshx+cosht t arccoth expx+cosht sinht or, substituting u = expx again, ψ GSH F u; t = Defining f u, t sinht 2t u 2 + 2u +cosht t arccoth u+cosht sinht sinht 2t u 2 + 2u ψf GSH f u, t ; t = lim u + g u, t = lim u + Remains to verify that or, dψ F u du for t >, x, ]. for t >, u, ]. and gu, t +cosht t arccoth u+cosht sinht f u, t = lim u, t g u + sinht u 2 tu 2 +2 u cosht+ 2 sinht tu 2 +2 u cosht+ = f u; tg u; t f u; tg u; t g u; t 2, f u; tg u; t f u; tg u; t, =. This is equivalently to Define Au, t u 2 u + cosht t arccoth u sinht sinht t arccoth u+cosht sinht and Uu, t = u sinhu. Then u 2 Au, t Au, t Uu, t. The second inequality follows from A, t = U, t and U u, t = sinht sinh t u u cosh t + = A u, t.

22 ξu ψfx.3.2. ψfx u x 5 b ψflog x a ξlog u 5 x c ψflog x ξu ψfx.3.2. ψfx u x 5 e ψfhs x d ξhs u 5 x f ψfhs x ξu ψfx.3.2. ψfx u x 5 h ψflap x g ξlap u 5 x i ψflap x ξu ψfx.3.2. ψfx u x 5 k ψfc x j ξc u 5 x l ψfc x t =.5 t= t=5.4 ξu ψfx.3.2 ψfx t =.5 t= t= u m ξgsh u Fig. Copula generator t=.5 t= t=5 5 x n ψfgsh x 5 5 x o ψfgsh x 5

23 EU V=v EU V=v v v.7.65 a Logistic.7.65 t=.5 t= t=5 b Laplace EU V=v.5 EU V=v v v c Cauchy Fig. 2 Conditional Mean Functions d GSH Boundary alpha=.25 alpha=.5 alpha= Fig. 3 Asymmetric logistic generator