Operational Risk Modeling Analytics phần 7

Gumbel copula copula Gumbel đã được thảo luận trước đây là một ví dụ về một copula Archimedean. Đây cũng là một copula giá trị cực với chức năng phụ thuộcTừ đó, bằng cách thiết lập w = l / 2, copula Gumbel được xem có chỉ số phụ thuộc vào đuôi phía trên của 2 - 2'1 '. Galambos copula copula Galambos [42] có chức năng phụ thuộc | EXTREME VALUE COPULAS 259 from setting A w to its upper bound A w 1. At the other extreme if A w max w 1 w then there is perfect correlation and hence perfect dependency with c tí Ù u. It is convenient to write the index of upper tail dependence in terms of the dependence function A w . The result is that 1 2u C u u Xu lim--------------- U- 1 1 u l-2 2 1 2 lim------ --------- u 1 1 u lim2-2A l 2 u2 1 2 -1 u 1 2-24 1 2 . If a copula is specified through 4 w then the index of upper tail dependency is easily calculated. There are several well-known copulas in this class. Gumbel copula The Gumbel copula was discussed previously as an example of an Archimedean copula. It is also an extreme value copula with dependence function 4 w wớ 1 w e 0 0. From this by setting w 1 2 the Gumbel copula is seen to have index of upper tail dependence of 2 21 0. Galambos copula The Galambos copula 42 has the dependence function 4 w 1 w ớ 1 w ổ 1 Zé 0 0. Unlike the Gumbel copula it is not Archimedean. It has index of upper tail dependence of 2-1 ớ. The bivariate copula is of the form c 1 2 W 2 exp I ị ln i -ỡ ln 2 -ỡj 5 j- An asymmetric version of the Galambos copula with three parameters has dependence function 4 w 1 I aw w I 0 a 3 l. It has index of upper tail dependence of aTe The one-parameter version is obtained by setting a 3 bivariate asymmetric Galambos copula has the form fr e -ơV1 ỡì C 1 U2 wiu2exp alnuj 31nu2 260 MULTIVARIATE MODELS Fig. Galambos copula density 9 0 0 4 1 Fig. Galambos copula pdf Ớ EXTREME VALUE COPULAS 261 Figures and demonstrate the clear upper tail dependence. Hiisler and Reiss copula The Hiisler and Reiss copula 57 has dependence function 1 9 w A w w4 1 w ỉ 1 ỡ w ớ 0 where z is the cdf of the standard normal distribution. When w 1 2 1 1 2 l ớ resultingin an index of upper tail dependence of 2 2Ỉ 1 Ớ . Tawn copula The Gumbel copula can be extended to a three-parameter asymmetric version by introducing two .