Modelling Dependence with Copulas - IFOR

7 Extreme Value Copulas
It follows from Theorem 5.6 that
λ U =
( )
ϕ −1′ (2s)
2− 2 lim
s→0 ϕ −1′ (s)
=
( (1 − e −2s ) 1/θ−1 e −2s )
2− 2 lim
s→0 (1 − e −2s ) 1/θ−1 e −2s
= {e x =1+x + O(x 2 )}
( (−2s + O(s 2 )) 1/θ−1 )
= 2− 2 lim
s→0 (−s + O(s 2 )) 1/θ−1
= 2− 22 1/θ−1 =2− 2 1/θ .
The upper extreme value limit of C θ is the Gumbel family with coefficient of upper
tail dependence 2 − 2 1/θ as shown in Example 3.4.
Three one-parameter bivariate families of extreme value copulas are
C θ (u, v) = exp(−[(− ln u) θ +(− ln v) θ ] 1/θ ), (7.2.3)
C θ (u, v) = uv exp((− ln u) −θ +(− ln v) −θ ) −1/θ ), (7.2.4)
C θ (u, v) = exp(ln uΦ( 1 θ + θ 2 ln(ln u
ln v )) + ln vΦ(1 θ + θ ln v
ln( ))). (7.2.5)
2 ln u
for θ ≥ 1, θ ≥ 0andθ ≥ 0 respectively. (7.2.3) is the Gumbel family, (7.2.4)
is the Galambo family and (7.2.5) is the Hüsler and Reiss family. All three have
upper tail dependence and C θ = Π for θ = 1, 0 and 0 for (7.2.3), (7.2.4) and
(7.2.5) respectively. Furthermore C ∞ = M for all three. Multivariate extensions of
these bivariate extreme value copulas are discussed in Joe (1997) [6]. One of these
extensions is the multivariate extension of the Gumbel family presented in chapter
5.5.
Consider a multivariate distribution function F . Finding the multivariate limiting
distribution is often much more difficult than in the univariate case since
it means finding sequences {a jn }, {b jn } for j = 1,... ,m such that ((M 1n −
a 1n )/b 1n ,... ,(M mn − a mn )/b mn ) converges in distribution. Results concerning domains
of attraction of multivariate extreme value distributions can be found in
Marshal and Olkin (1983) [11]. One example is the following theorem.
Theorem 7.2. Let G be a k-dimensional (max) extreme value distribution such
−1
that G i is of type Λ for i = 1,... ,k. Let φ i (t) = F i (F1 (t)), i = 2,... ,k,
r i (t) =F i ′(t)/F
i(t) and x 0 1 =sup{x : F (x) < 1}. Then F ∈ MDA(G) if
1 − F ( x1
r
lim
+ t, x 2
1(t)
t→x 0 1
for all x such that G(x) > 0.
r + φ 2(φ 2(t)) 2(t),... ,
1 − F 1 (t)
x k
r k (φ k (t)) + φ k(t))
An application of this result is shown in the following example.
Example 7.4. Consider the bivariate dependence model
X 1 = min(Z 1 ,Z 12 ),X 2 =min(Z 2 ,Z 12 ),
= − ln G(x) (7.2.6)
where Z 1 ,Z 2 and Z 12 are independent exponentially distributed random variables
with parameters λ 1 ,λ 2 and λ 12 respectively. Let X 1 and X 2 have joint distribution
function F ,whereF has margins F 1 and F 2 .
F (x 1 ,x 2 ) = P[X 1 >x 1 ,X 2 >x 2 ]
= P[Z 1 >x 1 ]P[Z 2 >x 2 ]P[Z 12 > max(x 1 ,x 2 )]
56