Modelling Dependence with Copulas - IFOR

4 Techniques for Construction of Multivariate Copulas
Consider a two component system where the components are subjects to shocks,
which are fatal to one or both components. Let X 1 and X 2 denote the lifetimes of
the two components. Furthermore assume that the shocks form three independent
Poisson processes with parameters λ 1 ,λ 2 ,λ 12 ≥ 0, where the index indicate whether
the shocks kill only component 1, only component 2 or both. Then the times Z 1 ,Z 2
and Z 12 of occurrence of these shocks are independent exponential random variables
with parameters λ 1 ,λ 2 and λ 12 , respectively. Hence
H(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 )].
The univariate survival functions for X 1 and X 2 are F 1 (x 1 )=exp(−(λ 1 + λ 12 )x 1 )
and F 2 (x 2 )=exp(−(λ 2 +λ 12 )x 2 ). Furthermore max(x 1 ,x 2 )=x 1 +x 2 −min(x 1 ,x 2 )
yielding
H(x 1 ,x 2 ) = exp(−(λ 1 + λ 12 )x 1 − (λ 2 + λ 12 )x 2 + λ 12 min(x 1 ,x 2 ))
= F 1 (x 1 )F 2 (x 2 )min(exp(λ 12 x 1 ), exp(λ 12 x 2 )).
Let α 1 = λ 12 /(λ 1 + λ 12 )andα 2 = λ 12 /(λ 2 + λ 12 ). Then exp(λ 12 x 1 )=F 1 (x 1 ) −α1
and exp(λ 12 x 2 )=F 2 (x 2 ) −α2 , and hence the survival copula is given by
Ĉ(u 1 ,u 2 )=u 1 u 2 min(u −α1
1 ,u −α2
2 ) = min(u 1−α1
1 u 2 ,u 1 u 1−α2
2 ).
The survival copulas for the Marshall-Olkin bivariate exponential distribution yields
a copula family given by
C α1,α 2
(u 1 ,u 2 ) = min(u 1−α1
1 u 2 ,u 1 u 1−α2
2 )=
{ u
1−α 1
1 u 2 , u α1
1 ≥ u α2
2 ,
u 1 u 1−α2
2 , u α1
1 ≤ u α2
This family is known as the Marshall-Olkin family.
The Marshall-Olkin copulas have both an absolutely continuous and a singular
component. Since
∂ 2
∂u 1 ∂u 2
C α1,α 2
(u 1 ,u 2 )=
{
u
−α 1
1 , u α1
1 >u α2
2 ,
u −α2
2 , u α1
1