内容发布更新时间 : 2024/5/3 15:35:02星期一 下面是文章的全部内容请认真阅读。

（UNDERSTANDING RELATIONSHIPS USING COPULAS）

fFr(y1,y2;?)?cFr?F1(y1),F2(y2)?f1(y1)f2(y2)

??????y1??1??y2??2?L(y1,y2)???????????????fFr(y1,y2;?)dy1dy2 2??2???L(x1,x2)?????????x1x2fFr(x1,x2;?)dx1dx2

???????????x1x2cFr??(x1),?(x2);???(x1)?(x2)dx1dx2?(u11S1,u2)??L(u1,u2)?12?0?0u1u2dCFr(u1,u2;?)?3

?12?1?100CFr(u1,u2;?)du1du2?3

Clayton Copula

Cu???Cl(1,u2;?)?(u??1?u2?1)?1/

Fy?y?????Cl(y1,2;?)??F1(1)???F2(y2)??1??1/

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

1?2（Copula 及其在金融分析上的应用，pp23）

（UNDERSTANDING RELATIONSHIPS USING COPULAS）

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

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

???????????x1x2cCl??(x1),?(x2);???(x1)?(x2)dx1dx2??1?1S(u1,u2)?L(u1,u2)?12?00u1u2dCCl(u1,u2;?)?3

?12?1?100CCl(u1,u2;?)du1du2?3(6)

Gumbel-Barnett Copula1??(1??(1?u)(1?v))?2uv?

?1??(1?u)(1?v)?3CGB(u1,u2;?)?u1?u2?1?(1?u1)(1?u2)exp???ln(1?u1)ln(1?u2)? ??[0,1]

FGB(y1,y2;?)?CGB?F1(y1),F2(y2);??

cGB(u1,u2;?)???1??ln(1?u1)??1??ln(1?u2)????exp???ln(1?u1)ln(1?u2)?

（Continuous Bivariate Distributions，pp.95）

fGB(y1,y2;?)?cGB?F1(y1),F2(y2)?f1(y1)f2(y2)

?L(y1,y2)???L(x1,x2)???????y1??1??y2??2?????fGB(y1,y2;?)dy1dy2 ??????2??2?????????????????????x1x2fGB(x1,x2;?)dx1dx2x1x2cGB??(x1),?(x2);???(x1)?(x2)dx1dx211

???S(u1,u2)??L(u1,u2)?12??12?110000?uu12dCGB(u1,u2;?)?3

?CGB(u1,u2;?)du1du2?3AMH Copula

CAMH(u1,u2;?)?u1u2 ??[?1,1]

1??(1?u1)(1?u2)FAMH(y1,y2;?)?CAMH?F1(y1),F2(y2);??

cAMH(u1,u2;?)??1??(u1u2?u1?u2?2)??2(1?u1)(1?u2)?1??(1?u1)(1?u2)?331????(1??)(1?u1)(1?u2)?2?u1u2

?1??(1?u1)(1?u2)?（Bivariate Distribution Modeling of Tree Diameters and Heights:

Dependency Modeling Using Copulas pp6）

??1?2?(??1)ln(1??)? 错误 ??1?2?6???3?2??Using copulas to measure association between ordinal measures of health and income pp14

??6(??1)2ln(1??)错误 ??1?29?The health-economic applications of copulas:methods in applied econometric research pp48

2(??1)2ln(1??)?2?正确 ??1?3?2Bayesian copula selection，pp814

On the relationship between Spearman’s rho and Kendall’s tau for pairs of continuous random variables pp.2149

fAMH(y1,y2;?)?cAMH?F1(y1),F2(y2)?f1(y1)f2(y2)

?L(y1,y2)???L(x1,x2)???????y1??1??y2??2?fAMH(y1,y2;?)dy1dy2 ?????????2???2?????????????????????x1x2fAMH(x1,x2;?)dx1dx2x1x2cAMH??(x1),?(x2);???(x1)?(x2)dx1dx211

???S(u1,u2)??L(u1,u2)?12??12??S(u1,u2)?z1110000?uu12dCAMH(u1,u2;?)?3

?CAMH(u1,u2;?)du1du2?312(1??)24(1??)3(??12)Dilog(1??)?log(1??)? ?2?2?其中：Dilog(z)??log(t)(1?t)?1dt，?S(u1,u2)?[?0.2711,0.4784]

Stochastic Frontier Models with Correlated Error Components pp.6

FGM Copula

CFGM(u1,u2;?)?u1u2?1??(1?u1)(1?u2)? ??[?1,1]

FFGM(y1,y2;?)?CFGM?F1(y1),F2(y2);?? cFGM(u1,u2;?)?1??(1?2u1)(1?2u2)

（Bivariate Distribution Modeling of Tree Diameters and Heights: Dependency Modeling Using Copulas pp6）

??2? 9fFGM(y1,y2;?)?cFGM?F1(y1),F2(y2)?f1(y1)f2(y2)

?L(y1,y2)???y1??1??y2??2?fFGM(y1,y2;?)dy1dy2 ???????????2???2??????L(x1,x2)????????????????????x1x2fFGM(x1,x2;?)dx1dx2x1x2cFGM??(x1),?(x2);???(x1)?(x2)dx1dx211

???S(u1,u2)??L(u1,u2)?12??12?110000?uu12dCFGM(u1,u2;?)?3

?CFGM(u1,u2;?)du1du2?3