Modelling Dependence with Copulas - IFOR

7.2 Multivariate Extreme Value Theory
and hence F 1 (x) =1− exp(−(λ 1 + λ 12 )x) andF 2 (x) =1− exp(−(λ 2 + λ 12 )x). It
follows that F 1 ,F 2 ∈ MDA(Λ). With the notation used in Theorem 7.2 we have
r 1 (t) =F 1(t)/F ′ 1 (t) =(λ 1 + λ 12 )exp(−(λ 1 + λ 12 )t)/ exp(−(λ 1 + λ 12 )t)
= λ 1 + λ 12 ,
r 2 (t) =...= λ 2 + λ 12 ,
−1
φ 2 (t) =F 2 (F1 (t)) = − 1
λ 2+λ 12
ln e −(λ1+λ12)t = λ1+λ12
λ 2+λ 12
t,
x 0 1 =sup{x : F 1 (x) < 1} = ∞.
Furthermore
F (x 1 ,x 2 )=F (x 1 ,x 2 )+F 1 (x 1 )+F 2 (x 2 ) − 1,
1 − F (x 1 ,x 2 )=F 1 (x 1 )+F 2 (x 2 ) − F (x 1 ,x 2 )
= {F (x 1 ,x 2 )=F 1 (x 1 )F 1 (x 1 )exp(λ 12 min(x 1 ,x 2 ))}
= F 1 (x 1 )+F 2 (x 2 ) − F 1 (x 1 )F 2 (x 2 )exp(λ 12 min(x 1 ,x 2 )).
Hence
where
1 − F
(
x1+(λ 1+λ 12)t
)
, x2+(λ1+λ12)t
λ 2+λ 12
1 − F 1 (x 1 )
λ 1+λ 12
= e −x1 + e −x2 − e −x1−x2 h(t),
h(t) =exp(min(−λ 1 t + λ 12x 1
λ 1 + λ 12
, −λ 1 t − λ 12 t + λ 12x 2
λ 2 + λ 12
+ λ 12
λ 1 + λ 12 t
λ 2 + λ 12
t)).
Hence the right side of equation (7.2.6) is
e −x1 + e −x2 − e −x1−x2 lim h(t).
t→∞
Consider the case when max(λ 1 ,λ 2 ) > 0. Then lim t→∞ h(t) = 0. From Theorem
7.2 it follows that
− ln G(x 1 ,x 2 ) = e −x1 + e −x2 =⇒
G(x 1 ,x 2 ) = exp(−e −x1 − e −x2 )=Π(e −e−x 1
,e −e−x 2
).
Consider the case when λ 1 = λ 2 =0,λ 12 > 0. Then lim t→∞ h(t) =exp(min(x 1 ,x 2 )).
From Theorem 7.2 it follows that
− ln G(x 1 ,x 2 ) = e −x1 + e −x2 − e −x1−x2+min(x1,x2)
= e −x1 + e −x2 − e − max(x1,x2)
= e −x1 + e −x2 − e min(−x1,−x2)
= e −x1 + e −x2 − min(e −x1 ,e −x2 )
= max(e −x1 ,e −x2 ) =⇒
G(x 1 ,x 2 ) = exp(− max(e −x1 ,e −x2 ))
= exp(min(−e −x1 , −e −x2 ))
= min(e −e−x 1
,e −e−x 2
)
= M(e −e−x 1
,e −e−x 2
).
Thus F ∈ MDA(G) whereG is given by
{
G(x 1 ,x 2 )=
Π(e −e−x 1
,e −e−x 2
) max(λ 1 ,λ 2 ) > 0
,e −e−x 2
) λ 1 = λ 2 =0,λ 12 > 0
M(e −e−x 1
This shows that bivariate vectors with a Marshall-Olkin copula gives perfect positive
dependence or independence in the limit for componentwise maxima, depending on
the parameterization of the copula.
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