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Copula 汇总表 Copula Gaussian t Gumbel Clayton Frank AMH FGM Joe A12 A14 C(u, v; θ) ?2??(u1),?(u2);??① ?1?1c(u, v; θ) ?2???1(u1),??1(u2);???(u1)?(u2)t2?T??1(u1),T??1(u2);?,??t?(u1)t?(u2)τ ρs 6?arcsin ?26?arcsin ?2θ (-1,1) (-1,1) [1,+?) (0,+?) (-?,+?)\\{0} [-1,1) [-1,1] [1,+?) [1,+?) [1,+?) 2arcsin? ?2arcsin? ?11? ?T2?T(u1),T(u2);?,??② ?1??1?③ exp??w1/?? w?(?lnu1)??(?lnu2)? w?1/? w?u1???u2???1 C(u1,u2;?)?w1/????1? u1u2w2?1/?(lnu1lnu2)1?? 1?12?D1(??)?D2(??)?[1][2] ?(1??)(u1u2)???1w?1/??2 2 ??21?4?D1(??)?1?②[1] ?1?ln?1??e???1??1w1w2? ?wi??e??ui?1? u1u2(1?w)?1 w??(1?u1)(1?u2) u1u2??u1u2(1?u1)(1?u2) ??(e???1)e??(u1?u2)?(e???1)?w1w2?(1?w)32 ?2?u1u2?(1??)w???1?1??(1?2u1)(1?2u2) 2??2(??1)2ln(1??) [3][4]1?23?2?/9 ?/3 1?(w1?w2?w1w2)1/? wi?(1?ui)? 2?w?v2?w2?v?3/2 1?1?2 3? ?1?(w1?w2)1/??wi?(u?1)?1i??1 ???1?(w1?w2)1/?? wi?(ui?1/??1)? 2 1?2?Plackett (??1)?1(w?w2?v)/2 w?1?(??1)(u1?u2) v?4u1u2?(??1) （0,+?)\\{1}

注：①?2??(u1),?(u2);???????1?1??1(u1)???1(u2)??2?s12?2?s1s2?s2?kexp??ds1ds2；②Dk(x)?k?22(1??)x2?1??2??1?x0kxtk，，k = 1,2。 D(?x)?D(x)?dtkkk?1et?1

[1]

Understanding relationships using Copulas pp.10

[2] Stochastic Frontier Models with Correlated Error Components pp.15 [3] Bayesian copula selection pp.814

[4] On the relationship between Spearman’s rho and Kendall’s tau for pairs of continuous random variables

pp.2149

令H为具有边缘分布F、G的联合分布函数，那么存在一个Copula函数C，使得

H(x,y)?C?F(x),G(y)?

如果F，G是连续的，则函数C是唯一的。

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h(x,y)?c?F(x),G(y)?f(x)g(y)

对数正态分布

Copula类型：Gaussian Copula 、Gumbel Copula、Clayton Copula、Frank Copula

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Gaussian Copula

CGa(u1,u2;?)????1?u1??????1?u2???2?x12?2?x1x2?x2?exp???dx1dx2222?1???? 2?1???1??2???1(u1),??1(u2);????2???1(u)?2?2???1(u)??1(u)??2???1(u)?2?1212cGa(u1,u2;?)?exp???22??2(1??)1????

?2???1(u1),??1(u2);???????1(u1)?????1(u2)?1(Continuous Bivariate Distributions pp36)

????1(u)?2?2???1(u)??1(u)????1(u)?21212exp??2?2(1??)?2C(u1,u2;?)?2C(u1,u2;?)?x1?x2?2(x1,x2;?)?c(u1,u2;?)??????u1?u2?x1?x2?u1?u2?(x1)?(x2)2?????1(u1)?????1(u2)?1??2????1(u)?2????1(u)?2???1(u)?2?2???1(u)??1(u)????1(u)?2121212exp???22(1??2)??1??2??????????????1(u)?2?2???1(u)??1(u)????1(u)?2?1212?exp??2??2(1??)??????1(u)?2?1????1(u)?212122?1??exp?????2?exp???2??2??2????????（求

??2???1(u)?2?2???1(u1)??1(u2)??2???1(u)?212exp???2(1??2)1??2?1导）

FGa(y1,y2;?)????1?F1(y1)??????1?F2(y2)???2?x12?2?x1x2?x2??exp??dx1dx2222??1?? 2?1????1??2???1?F1(y1)?,??1?F2(y2)?;???2FGa(y1,y2;?)fGa(y1,y2;?)??cGa?F1(x1),F2(x2)?f1(x1)f2(x2)?y1?y2???1?F1(y1)????1?F2(y2)???2???F1(y1)?,??F2(y2)?;???y1?y1?1?1

??2???1?F1(y1)?,??1?F2(y2)?;??x1???1?F1(y1)?x2???1f1(y1)f2(y2)????1?F1(y1)??????1?F2(y2)??其中

?F2(y2)?，则

fGa(y1,y2;?)??2?x1,x2;??????f1(y1)f2(y2)

?(x1)?(x2)?L(y1,y2)??f1(y1)f(y2)?y1??1??y2??2??dy1dy2x,x;???????212????????(x)?(x)?1??2?12 (3)

?????????????y1??1????1??y2??2????2???2?x1,x2;??dx1dx2????L(u1,u2)?12????????u1?0.5??u2?0.5??2(x1,x2;?)dx1dx2?12??1?100u1u2dCGs(u1,u2;?)?0.5?E(u1)?E(u2)??0.25?

?12?1?100u1u2dCGs(u1,u2;?)?3??S(u1,u2)?(u)?4?111,u20?0CGa(u1,u2;?)dCGa(u1,u2;?)?1

?(x??????L1,x2)?2sin??6?S(u1,u2)???sin??2?(u1,u2)??

Gumbel Copula

CGum(u1,u2;?)?exp???(?lnu1/?1)?(?lnu2)1/????

FGum(y1,y2;?)?exp?????ln?F1(y1/?1/?1)?????ln?F2(y2)?????

cuCGum(u1,u2;?)(lnu1?lnu2)1/??1?1/?1/??1?Gum(1,u2;?)?u1/?1/?2???(?lnu)???1? 1u2?(?lnu1)?(?lnu2)???1(?lnu2)???（Copula 及其在金融分析上的应用，pp22）

（UNDERSTANDING RELATIONSHIPS USING COPULAS）

f?)??2FGum(y1,y2;?)Gum(y1,y2;?y?cGum?F1(y1),F2(y2)?f1(y1)f2(y2)

1?y2??????y1??1??L(y1,y2)????????????y2??2???fGum(y1,y2;?)dy1dy2 2??2????L(x1,x2)?????????x1x2fGum(x1,x2;?)dx1dx2

???????????x1x2cGum??(x1),?(x2);???(x1)?(x2)dx1dx2(4)

11?S(u1,u2)??L(u1,u2)?12?0?0u1u2dCGum(u1,u2;?)?3

?12?1?100CGum(u1,u2;?)du1du2?3Clayton Copula

C,u????Cl(u12;?)?(u1?u2?1)?1/?

FCl(y1,y2;?)???F1(y1)?????F?2(y2)????1??1/

c?2CCl(u1,u2;?)1??Cl(u1,u2;?)??u?(1??)(u1u2)???(u??1?u2?1)?2?1/??u

12（Copula 及其在金融分析上的应用，pp23）

（UNDERSTANDING RELATIONSHIPS USING COPULAS） （Bivariate Distribution Modeling of Tree Diameters and Heights:

Dependency Modeling Using Copulas pp6）

fCl(y1,y2;?)?cCl?F1(y1),F2(y2)?f1(y1)f2(y2)

???????y1??1??y2??2?L(y1,y2)?????????2?????2??fCl(y1,y2;?)dy1dy2 ???L(x1,x2)?????????x1x2fCl(x1,x2;?)dx1dx2

???????????x1x2cCl??(x1),?(x2);???(x1)?(x2)dx1dx2?11S(u1,u2)??L(u1,u2)?12?0?0u1u2dCCl(u1,u2;?)?3

?12?1?100CCl(u1,u2;?)du1du2?3Frank Copula

C1??exp(??u1)?1??exp(??u2)?1??Fr(u1,u2;?)???ln?1?1? ??(??,??)\\{0}?exp(??)??FFr(y1,y2;?)?CFr?F1(y1),F2(y2);??

;?)??2CFr(u1,u2;?)??(e???1)e??(u1?u2)cFr(u1,u2?u? 1?u2?(e???1)?(e??u1?1)(e??u2?1)?2（Copula 及其在金融分析上的应用，pp.24）

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