COPULA-BASED DENSITY WEIGHTING FUNCTIONS

Transcription

1 COPULA-BASED DENSITY WEIGHTING FUNCTIONS M. S. Khadka, J. Y. Shin, N. J. Park, K.M. George, and N. Park Oklahoma State University Computer Science Stillwater, OK, 74078, USA Phone: , Fax: {mahessk, jiyouns, noh, kmg, a.cs.okstate.edu Abstract In this paper, we propose a method how to construct density weighting functions from Copulas. The notion of Copula was introduced by A. Sklar in A Copula is a dependence function to construct a bivariate distribution function that links joint distributions to their marginals. Other forms of dependence function, based on density weighing functions, have also been developed. The proposed method is demonstrated and validated by showing the derivation of density weighting function from Copula along with numerical simulation. This paper demonstrates how the nature of the density weighting functions of different copulas change with their parameter values and also the similarity and dissimilarity between them. Key Words: Copula, Dependence Function, Density Weighting Function (DWF) 1. INTRODUCTION The study on bivariate distributions has been limited to the normal case for a long time. Recently, various methods of constructing bivariate distributions with specified marginals have been investigated [1, 3, 4, 5, 6, 7, 8, 11, 1, 13]. The notion of Copula was introduced by A. Sklar in At the beginning, Copulas were mainly used in the development of the theory of probabilistic metric spaces. Today, they are widely employed as a method of dependence function to construct a bivariate distribution function that links joint distributions to their marginals. In [], it was stated that in bivariate distributions with respect to the marginal distributions, similarity in their logistics with the normal one can be evaluated through comparison of their properties with those of the normal classical bivariate model. If the logistic distribution is close to the normal one, then they are considered symmetrical. Two logistic bivariate distributions have been studied by Gumbel in []. Kotz and Seeger [3] published a treatise on a class of bivariate densities. Other methods of dependence function, based on density weighing functions, have also been introduced in this study. In 1995, Dou Long and Roman Krzysztofowicz [1] proposed a model, based on regression dependence, for bivariate density which is constructed from specified marginals and a functional dependence structure. Various forms of this functional dependence structure lead to different families of models. In their paper, one of those families is chosen and investigated in depth: its bivariate densities assume a polygonal dependence structure and positive (or negative) mutual regression dependence between variates. In this paper, a general overview of Copula is presented, and their density weighting function (DWF) is derived theoretically.we compared the properties and behaviors of the density weighting functions (DWF) in the family with one parameter Copula, and a numerical simulation has been conducted for validation. Some derivation of density weighting function from Copula is presented. The remainder of this paper is organized as follows. In section, the definition of Copula and Sklar s theorem is reviewed, and the concept of density weighting function (DWF) is presented. A Copulabased derivation of a density weighting function (DWF) is presented in Section 3. Numerical simulations are shown in section 4. In the final section, conclusion and future work are presented.. OVERVIEW OF COPULA AND DENSITY WEIGHTING FUNCTION (DWF).1. Copula A. Sklar has proposed a new class of functions, called Copulas in the univariate margins case, in These new functions are defined in the range of space 0,1 of bivariate distribution functions whose margins

2 are uniform in the interval0,1. In this section, we introduce the definitions of Copula [14] and Sklar s theorem [14]. Even though Copulas and other functions could be defined for n-dimensional space, our discussion is limited to the bivariate case. Definition of Copula [14]: A Copula is a function C : 0,1 0,1 which satisfies: i) for every u, v in 0,1, C u, 0 0 C0, v, and Cu, 1 u and C1, v v ii) for every u 1, u, v 1, v in 0,1 such that u1 u and v1 v, C u, v Cu, v1 Cu1, v Cu1, v1 0. The importance of Copulas in the construction of joint distributions is addressed and demonstrated in the Sklar s Theorem [14]. Sklar s Theorem: Let X and Y be random variables with joint distribution function H and marginal distribution function F and G, respectively. Then there exists a Copula C such that x y CFx Gy H,,, for all x, y in (1) In other words, given a C in which is the set of Copulas and distributions F and G, the function H in (1) is a bivariate distribution with margins F and G. In particular, if X and Y are extended real valued random variables and defined in a common probability space with individual distribution F X and F Y and joint distribution F X, Y, then there exists a C X, Y in such that FX, Y ( u, CX, Y FX ( u), FY (. Further, if F X and FY are continuous, thenc X, is unique. We shall refer to Y C X, Y as a Copula of X and Y. Thus, the Copula of two random variables can be employed in order to identify their dependence structure. For instance, the parametric families of bivariate Copulas are shown in Table 1 [15], as follows: Model Name C( u, where ( u, [0,1] parameter Plackett 1 ( 1)( u 1 ( 1)( u 4 (1 ) C ( u, 1 ( 1) 0 C( u, uv 1 (1 u)(1 1 1 Ali-Mikhail-haq 1 Gumbel a a 1 Cu, v exp [( ln u) ( ln ] Cook-Johnson 1 1 C ( u, u v 1 0 Frank 1 v C( u, ln 1 e 1 e 1 e 1 Cuadra-Auge-1 1 u 1 0 C ( u, min( u, uv 0 1 Frechet-1 C( u, pmax( 0, u v 1) (1 p)min( u, 0 p 1 Morgenstern C( u, uv[1 3 (1 u)(1 ] Product C( u, uv - Table 1. One Parameter Bivariate Copulas.. Density Weighting Function (DWF) Kotz and Seeger [3] published a treatise on a class of bivariate densities of the form 1 3 h( x, y) f ( x) g( y) ( x, y) () 1 3

3 They identified and characterized the density weighting function (DWF) for the known classes of bivariate distributions such as Farlie-gumbel-Morgenstern (F.G.M.)[]. In [3], they proposed a method to construct joint distributions from density weighing functions, whereas, in this paper, we propose a method to solve () from a different perspective such that the density weighting functions are employed as a relation among variables. Therefore, we construct the density weighting functions from certain known densities. 3. THEORITICAL DERIVATIONS In the proposed method, the density weighting function is sought by employing the method as shown in (3), where h(x,y) is the partial derivative of the given Copula C(u, function with respect to x and y. Likewise, f(x)= u/ x and g(y)= v/ y. Ф(x,y)=h(x,y)/f(x)g(y) (3) The density weighting functions of the following Copulas are presented in this section. Cook Johnson Copula Morgenstern Copula Gumbel Copula Ali-Mikhail-Haq Copula 3.1. Cook Johnson Copula The Cook Johnson One Parameter Bivariate Copula is of the form C(u,=[u -α +v -α -1] 1/α where u=f (x) and v=g(y) Now, first we partially differentiate C(u, with respect to y and then the result will be again partially differentiated with respect to x to get h(x,y). Then C/ y=(-1/ α) x [ u -α +v -α -1] -(1+ α)/ α x v -(1+ α) x v/ y C/ y= v -(1+ α) v/ y x [ u -α +v -α -(1+ α)/ α -1] C/ x y=v -(1+ α) v/ y x ( 1+α)/ α x [ u -α +v -α -1] -(1+ α)/ α x - α u -(1+ α) u/ x C/ x y=( 1+α)( u -(1+ α) x [ u -α +v -α -1] -(1+ α)/ α x f(x)g(y) This equation can be written in the form of equation (3) where h(x,y)= C/ x y, Ф(x,y)= ( 1+α)( u -(1+ α) x [ u -α +v -α -1] -(1+ α)/ α ; u/ x = f(x) and v/ y = g(y) 3.. Morgenstern Copula The Morgenstern One Parameter Bivariate Copula is of the form C(u,=uv[1+3ρ(1-u)(1-] where u=f (x) and v=g(y) Now, first we partially differentiate C(u, with respect to y and then the result will be again partially differentiated with respect to x to get h(x,y). Then C/ y=u{[1+3ρ(1-u)(1-] v/ y+v x / y([1+3ρ(1-u)(1-])} C/ y=u{[1+3ρ(1-u)(1-]-3ρv(1-u)} v/ y C/ x y=([1+3ρ(1-u)(1-]-3ρ[u(1-+ v(1-u)]) u/ x x v/ y This equation can be written in the form of equation (3) where

4 h(x,y)= C/ x y, Ф(x,y)=([1+3ρ(1-u)(1-]-3ρ[u(1-+ v(1-u)]); u/ x = f(x), and v/ y = g(y) 3.3. Gumbel Copula The Gumbel One Parameter Bivariate Copula is of the form C(u,=exp{-[(-ln u) α +(-ln α ] 1/ α } where u=f (x) and v=g(y) Now, first we partially differentiate C(u, with respect to y and then the result will be again partially differentiated with respect to x to get h(x,y). Then C/ y= exp{-[(-ln u) α +(-ln α ] 1/ α } x - / y[(-ln u) α +(-ln α ] 1/ α C/ y=(-ln ( α-1) /v x exp{-[(-ln u) α +(-ln α ] 1/ α } x [(-ln u) α +(-ln α ] ( 1- α)/ α x v/ y let B=exp{-[(-ln u) α +(-ln α ] 1/ α } and A=[(-ln u) α +(-ln α ] (1- α)/ α then C/ x y=(-ln ( α-1) /v[a B/ x+b A/ x] (a) B/ x=(-b/ α) [(-ln u) α +(-ln α ] ( α -1)/ α x / x[(-ln u) α +(-ln α ] B/ x=(-ln u) ( α-1) /u x exp{-[(-ln u) α +(-ln α ] 1/ α } x [(-ln u) α +(-ln α ] ( 1- α)/ α x u/ x Again, A/ x=((1- α)/α) [(-ln u) α +(-ln α ] (1- α)/ α x α(-ln u) α-1 x (-1/u) x u/ x A/ x =(-(1- α) (-ln u) α-1 )/u) [(-ln u) α +(-ln α ] (1- α)/ α x u/ x Now putting the values of B/ x and C/ x in eq(a); we get, C/ x y=(-ln ( α-1) /v[[(-ln u) α +(-ln α ] ( 1- α)/ α x (-ln u) ( α-1) /u x exp{-[(-ln u) α +(-ln α ] 1/ α } x [(-ln u) α +(-ln α ] (1--α1)/ α x u/ x + exp{-[(-ln u) α +(-ln α ] 1/ α } x (-(1- α) (-ln u) α-1 )/u) [(-ln u) α +(-ln α ] (1- α)/ α x u/ x] C/ x y =([(-ln u)(-ln ] ( α-1) / u x exp{-[(-ln u) α +(-ln α ] 1/ α } x {[(-ln u) α +(-ln α ] (1- α)/ α (1- α) [(-ln u) α +(-ln α ] (1- α)/ α } u/ x x v/ y This equation can be written in the form of equation (3) where h(x,y)= C/ x y, Ф(x,y)=([(-ln u)(-ln ] ( α-1) / α) x exp{-[(-ln u) α +(-ln α ] 1/ α } x {[(-ln u) α +(-ln α ] ( α -1)/ α (1- α) [(-ln u) α +(-ln α ] (1- α1)/ α ; u/ x = f(x), and v/ y = g(y) 3.4. Ali-Mikhail-Haq Copula The Ali-Mikhail-Haq One Parameter Bivariate Copula is of the form C(u,=uv[1- α(1-u)(1-] -1 where u=f (x) and v=g(y) Now, first we partially differentiate C(u, with respect to y and then the result will be again partially differentiated with respect to x to get h(x,y). Then C/ y=u[1- α(1-u)(1-] -1 {1- αv(1-u) [1- α(1-u)(1-] -1 } v/ y C/ x y={ αuv[1- α(1-u)(1-] - [1-α](1-u)(1- [1- α(1-u)(1-] -1 ]+ [1- α(1-u)(1-] -1 [1-α(1- [1- α(1-u)(1-] - 1 ][1- αv(1-u)[1- α(1-u)(1-] -1 ]} u/ x x v/ y This equation can be written in the form of equation (3) where h(x,y)= C/ x y, Ф(x,y)= C/ x y={ αuv[1- α(1-u)(1-] - [1-α](1-u)(1- [1- α(1-u)(1-] -1 ]+ [1- α(1-u)(1-] - 1 [1-α(1- [1- α(1-u)(1-] -1 ][1- αv(1-u)[1- α(1-u)(1-] -1 ]}; u/ x = f(x), and v/ y = g(y)

5 4. NUMERICAL SIMULATIONS AND COMPARATIVE STUDY In this section, the different graphs for density weighting function of different Copulas as derived in the previous section are demonstrated against different parameters in three-dimensional space using Matlab over the domain [0, ]. These graphs show that how the nature of the copula changes with the change in their parameter values. The test data for these simulations are obtained from [9] and [10]. 4.1 Simulation of Cook-Johnson Copula Fig 1.1.DWF of Cook-Johnson Copula with α = 1 Fig 1.. DWF of Cook-Johnson Copula with α = 4 Fig 1.3.DWF of Cook-Johnson Copula with α = 8 Fig 1.4. DWF of Cook-Johnson Copula with α = Simulation of Ali-Mikhali-Haq Copula Fig.1. DWF of Ali-Makhail-Haq Copula Fig.. DWF of Ali-Makhail-Haq Copula with with α = -1 with α = 0

6 Fig.5. DWF of Ali-Makhail-Haq Copula Fig.6. DWF of Ali-Makhail-Haq Copula with with α = 0.3 with α = Simulation of Morgenstern Copula Fig 3.1. DWF of Morgenstern Copula with ρ =-1/3 Fig 3.. DWF of Morgenstern Copula with ρ =-1/6 Fig 3.3. DWF of Morgenstern Copula with ρ = 0 Fig 3.4. DWF of Morgenstern Copula with ρ = 1/3

7 4.4 Simulation of Gumbel Copula Fig 4.1. DWF of Gumbel Copula with α = 1 Fig 4.. DWF of Gumbel Copula with α = 4 Fig 4.3. DWF of Gumbel Copula with α = 8 Fig 4.3. DWF of Gumbel Copula with α=16 5. CONCLUSION Through the numerical simulation results obtained for the DWF of different Copulas against various parameter values, the following observations and conclusion are drawn. In Cook-Johnson Copula whenever the parameter value increases, the DWF value also increases and vice versa. Similarly, in case of Ali-Mikhali-Haq Copula, when the value of parameter is zero, it gives the product Copula while the negative parameter value yields a concave DWF and positive parameter value yields convex DWF. We obtain the same result in Morgenstern Copula as in Ali-Mikhali-Haq Copula. In case of Gumbel Copula, as the parameter value increases, the DWF value decreases and vice versa. So from this we can conclude that though the four Copulas that are discussed in this paper are one parameter Copulas, they behave differently. From simulation results, it can be easily observed that Cook-Johnson Copula and Gumbel Copula behave reverse because Cook-Johnson Copula DWF increases with increase in parameter value while Gumbel Copula DWF decreases with increase in parameter value. However, Morgenstern Copula and Ali-Mikhali-Haq Copula behave somewhat similar according to the value of the parameter. The algorithms used in the numerical simulation part of this paper that provide the comparison of the density weighting functions for different one parameter copulas are developed in MatLab. In the Appendix, we have included one of the algorithms (Cook Johnson Copula) that generates the density weighting function in MatLab.

8 References [1] D. Long, R. Krzysztofowicz, A family of bivariate densities constructed from marginals, , Jounal of the American Statistical Association, Vol. 90, No. 430, [] E. J. Gumbel, Bivariate Logistic Distributions, Journal of the American Statistical Association, vol. 56, no. 94, pp , [3] S. Kotz, J. P. Seeger, A new approach to dependence in multivariate distributions. In: Advances in probability distributions with given marginals, , Mathematics and its Applications 67, Kluwer Acad. Publ., Dordrecht, [4] M. E. Johnson, A. Tenenbein, A Bivariate Distribution Family with Specified Marginals, Journal of the American Statistical Association, vol. 76, no. 373, pp , [5] R.L. Plackett, A Class of Bivariate Distributions, Journal of the American Statistical Association, vol. 60, no. 310, pp , [6] M. Shaked, A Family of Concepts of Dependence for Bivariate Distributions, Journal of the American Statistical Association, vol. 7, no. 359, pp , [7] W. Whitt, Bivariate Distributions with Given Marginals, The Annals of Statistics, vol. 4, no. 6, pp , [8] A. W. Marshall, I. Olkin, Families of Multivariate Distributions, Journal of the American Statistical Association, vol. 83, no. 403, pp , [9] creation date: Not Available, date accessed: Febrary 1, 006. [10] creation date: Not Available, date accessed: Febrary, 006. [11] M.J. Frank, Convolutions for dependent random variables, Advances in Probability Distributions with Given Marginals, Kluwer, 1991, [1] G. Wu, R. Gonzalez, Curvature of the Probability Weighting Function, Management Science, vol. 4, no. 1, pp , [13] R. T. Clemen, T. Reilly, Correlations and Copulas for Decision and Risk Analysis, Management Science, vol. 45, no., pp. 08-4, [14] R. B. Nelsen, An Introduction to Copulas, Springer, New York, [15] Carri`ere, J. F. (004). Copulas. Encyclopedia of Actuarial Science, Vol. 1-3, Wiley. New York, NY. Appendix function final = diffc_cj(inx, iny) % differential c Cook-Johnson - data in inx = sort(inx); iny = sort(iny); mux = mean(inx); muy = mean(iny); stdx = std(inx); stdy = std(iny); u = evcdf(inx, mux, stdx); v = evcdf(iny, muy, stdy); [u, v] = meshgrid(u, ; final = phi(u, ; surf(inx, iny, final); zlabel('phi') xlabel('r.v. for KOSPI') ylabel('r.v. for SP500') title('cook-johnson Copula'); function result = phi(u, alpha = 16; result = (1+alpha).*((u.*.^(-(alpha+1))).*((u.^(-alpha)+v.^(-alpha)-1).^(-(1+.*alpha)./alpha));