Copula函数与广义Copula函数

20073:2007-03-12:杨益党(1949),男,湖北均县人,副教授,主要从事金融统计和代数方面的研究.CopulaCopula杨益党(昌吉学院数学系新疆昌吉831100):在综述Copula函数性质的基础上,给出了广义Copula函数的概念,并讨论了广义Copula函数的一系列性质:相关系数;Copula;广义Copula:O13:A:1671-6469(2007)03-0001-061,,,:=Cov(x,y)Var(x)Var(y)=E(x-Ex)(y-Ey)Var(x)Var(y)(1)(x1,y1),,(xn,yn),XY(1),,XY,,,,,(),,,,,,[1]:X~N(0,1),Y=X2,E((x-E(x))(y-E(y))=E(xy)-E(x)E(y)=E(x3)-E(x)E(x2)=0,,,,,,CopulaCopula1959[2],CopulaCopula[3]Copula,nCopula,Copula,,Copula,:Copula,;Copula[5],,,,Copula,Copula,,CopulanCopula,5,2Copula,3Copula,4,51200732Copula1)Sklar:X1,,XnH(x1,,xn)F1(x1),,Fn(xn),CopulaC(u1,,un)H(x1,,xn)=C(F1(x1),,Fn(xn))(2)Copulac,NH(x1,,xn)f:f(u1,,un)=c(F1(x1),,Fn(xn))Nn=1fn(xn)(3)c(u1,,un)=nC(u1,,un)u1un,fnFn.(2)(3),,,CopulaC(u1,,un)c(u1,,un),X1,,Xn;Nn=1fn(xn)2)CopulaCopula(Copulat-Copula)ArchimedeanCopula,ArchimedeanCopula,Copula::[0,1]![0,∀]:#,∃u%[0,1]&(u)0,∋[-1](t)=-1(t)0(t((0)0(0)(t(∀C(u,v)=[-1]((u)+(v))CopulaCopula,Copula(t)=(-lnt),%[1,∀]CopulaCumbelCopula,CG(u,v;a)=exp{-(-logu)1/a+(-logv)1/a]a}a%(0,1],a=1,uv;a!0,uv(t)=(t--1)/,)-1∗0,CopulaClaytoncopula,C(u,v)=(u-+v--1)-1/,)-1∗03)CopulaCopula[2]:(!,∀)F(x1,x2),!~G(x),∀~H(y),Copulap(!x,∀y)=F(x,y)=C(G(x),H(y))s(x),t(y)x,y,s-1(x),t-1(y)s(x),t(y),p(s(!)x,t(∀)y)=C(Fs(!)(x),Ft(∀)(y)),(s(!),t(∀))Copula(!,∀)CopulaC(u1,u2)4),Copula[2]:(1)Kendall:#=p((X1-X2)(Y1-Y2)0)-p((X1-X2)(Y1-Y2)0)=2p((X1-X2)(Y1-Y2)0)-1X1,Y1CopulaC(u,v),#=4++0(u,v(1C(u,v)dC(u,v)-1(2)Spearman:=3(p((X1-X2)(Y1-Y3)0)-p((X1-X2)(Y1-Y3)0))X1,Y1CopulaC(u,v),=12+10+10uvdC(u,v)-3(3)Gini:r=1n(,ni=1ri+si-n+1-,ni=1ri-si)220073X1,Y1CopulaC(u,v),r=4(+10C(u,1-u)du-+10(u-C(u,v))du)(4):=2p(x-x)(y-y)0)-1=4F(x,y)-1,x,yx,y,F(x,y)(x,y),X1,Y1CopulaC(u,v),=4C(12,12)-13Copula1)1:Fx1(x1),,Fxn(xn)nX1,,Xn,Fx1(x1)Fxn(xn)n(X1,,Xn),m(1(m(n)pn,X,,,X,Fx1(x),,Fxm(x)X,2:Fx1(x),,Fxm(x)Xm,a1,,am)0,,mi=1ai=1,:F(x)∃a1F1(x)++amFm(x)(4)X,m(1(m(n)p,Sklar,C(F1(x1),,F1(xn))n;C(F1(x1),,F1(xn)F1(x1),,F1(xn)CopulaC(u1,,un)=u1un,C(F1(x)1),,Fn(xn))=F1(x1)Fn(xn),Copula,,,,,Copula,1:F1(x1)X1,F2(x2),,Fn(xn)X2,,Xn,F1(x1)Fn(xn)F1(x1)F2(x2),,Fn(xn)[0,1],F1(x1)F∃1(x1;x2,,xn)∃F1(x1)x2xn,F∃1(x1;x2,,xn)F∃1(x1),3:F1(x1)F∃1(x1)nX1,,Xn4:nX1,,Xnm(m(n)X1,,Xn2:F(x1,xn):(1)0(F(x1,,xn)(1;(2)F(x1,,xn);(3)F(x1,,xn);F(x1,,xn)n,nn,nn5:F1(x1,,xn),F2(x1,,xn)nX1,,Xn,a1F1(x1,,xn)+a2F2(x1,,xn)nX1,,Xna1,a2)0,a1+a2=1:6:F1(x1),F2(x2)H(x1,x2),h(x1,x2)H(x1,x2),CCopula,C(F1(x1),F2(x2))=H(x1,x2),:320073C∃(u1,v2;x3,,xn)∃C∃(F1(x1),F2(x2))∃C(F1(x1),F2(x2))x3xn,H∃(x1,x2)∃+x1-∀+x2-∀h(x1,x2)+x3-∀+xn-∀dx3dxn,X3,,Xn,[0,1],C∃(u1,v2)=H∃(x1,x2),,C∃(u1,v2;x3,,xn)C∃(u1,v2):H∃(x1,x2)=+x1-∀+x2-∀h(x1,x2)+x3-∀+xn-∀dx3dxn=H(x1,x2)x3xn3:Cij(ui,vj)=Cij(Fi(xi),Fj(xj))=Fij(xi,xj)nX1,,XnCopula,C∃ij(ui,vj)=C∃ij(Fi(xi),Fj(xj))=F∃ij(xi,xj)Copula[4]CopulaMCopula,FrankCopula,GumbelCopulaClaytonCopulaCopula,tGARCHMCopulaMCopulatGARCH,[6],,,,Copula,,Copula,,,Copula4:F1(x1),,Fn(xn)n,C,C(F1(x1),,F1(xn))n,CF1(x1),,F1(xn)nCopula,CopulaCopula,CopulaCopulaCopula:1:C∃ij(ui,vj)Copula.:C∃ij(ui,vj)=C∃ij(Fi(xi),Fj(xj)),34,C∃ij(Fi(xi),Fj(xj))n,CopulaC∃ij(ui,vj)nCopula,m(2(m(n)CopulaCopulaCopula:C(F1(x1))=F1(x1)C∃(F1(x1))C(F1(x1)),RnF1(x1),,F1(xn)nCopula,R=C|C(F1(x1),,F1(xn))=F(x1,,xn),F(x1,,xn)n.R:p,q,p,q)0,p+q=1!C1,C2%R,145pC1+qC2%R,:2:R3:!C1,C2%R,:C1∀C2∃C1C2;:!C%R,!k%Z+,Z+=0,1,,n,,k#C∃Ck,1%R(1,1=(F1(x1),,F1(xn))0,:1)!C1,C2%R,C1∀C2%R;!C1%R,!k%Z+,k∀C%R;2)!C%R,1#C=C,(kL)#C=k#(L#C);3)C1#C2=C2∀C1,C1∀(C2∀C3)=(C1∀C2)∀C3;4)k#(C1∀C2)=(k#C1)∀(k#C2),(k+L)#C=(k#C)∀(L#C).:1)4),2)3)1),C1∀C2=C1(F1(x1),,F1(xn))C2(F1(x1),,F1(xn)),1,C1∀C2%R;k#C=Ck(F1(x1),,Fn(xn)),,k#C%R.4)k#(C1∀C2)=(C1(F1(x1),,F1(xn))C2(F1(x1),,F1(xn))k=C1(F1(x1),,F1420073(xn))kC2(F1(x1),,F1(xn))k=(k#C1)∀(k#C2);(k+L)#C=C(F1(x1),,F1(xn))k+l=C(F1(x1),,F1(xn))kC(F1(x1),,F1(xn))2=(k#C)∀(L#C).,C1C2C%R,kL%Z+,RR,Z+F,,4:F1(x1),,Fn(xn)n,,nk=1akF(xk)=a1u1++anun=C(F1(x1),,Fn(xn))(5)Copulaa1,,an)0,,nk=1ak=1.uk=Fk(xk),k=1,,n:F1(x1),,Fn(xn)n,,,,nk=1aKF(xk)n,C(F1(x1),,Fn(xn))Copula,,nk=1akF(xk)Copula5:Fij(xi,xj)m2,,m21(i,j(maijFij(xi,xj)n,aij)0,,m21(i,j(maij=1:3Fk(xk)Xk,uk=Fk(xk),k=1,,n;CijFi(xi)Fj(xj)Copula,Cij(Fi(xi),Fj(xj))=Fij(xi,xj),Copula:C1∃C11(u1,un)C1n(u1,un)Cn1(un,u1)Cnn(un,un)=F11(x1,x1)F1n(x1,xn)Fn1(xn,x1)Fnn(xn,xn)∃F1,(5)[a1,an]C1a1an=[a1,,an]F1a1an(6)a1,,an)0,,ni=1ai=1,(,ni=1ai)2=1,C*=[a1,an]C1a1an,F*=[a1,,an]F1a1an,2,F*n,6:C*(u1,,un)=F*(x1,,xn),C*(u1,,un)nCopula,uk=Fk(xk),k=1,,n2CopulanCopulaCopula,2Copula,nCopula,nCopula,Copula,,,4Sklar,Copula,,,4,n[7],,Copula,520073,,k,:[3][7],,,,,,,;F(1)x1(xk)F(2)xk(xk),F(1)xk(xk)kaKbK,k:Fxk(xk)=akF(1)xk(xk)+bkF(2)xk(xk),(0(aK,bk(1,ak+bK=1,k=1,2,,n)ak=1,bk=0Fxk(xk)=F(1)xk(xk),kak=0,bk=1,k;ak)bk,Fxk(xk)k;ak(bk,Fxk(xk)k1=a(1)ka(2)ka(m)k=0,0=b(1)kb(2)kb(m)k=1,kF(1)xk(xk),,F(m)xk(xk)2,Fxk(xk)=akF(1)xk(xk)+bkF(2)xk(xk),(0(ak,bk(1,ak+bk=1,k=1,2,,n)kak=1-kh,bk=kh,k=0,1,,m,h=1m;akbk(4),Xkm+1,(6),5(m+1)nn,%2,5,,,,,,,,,,,,,,Copula1959,Copula,,,,,Copula,,,,Copulan,CopulaCopula,,,:[1]张尧庭.我们应该选用什么样的相关性指标?[J].统计研究,2002,(9):41-44.[2]张尧庭.连接函数(copula)技术与金融风险分析[J].统计研究,2002,(9):48-51.[3]韦艳华.张世英,孟利锋.Copula理论在金融上的应用[J].西北农林科技大学学报(社会科学版)2003,98-101.[4]韦艳华,张世英,