Modelling Dependence with Copulas - IFOR

2 Copulas
.
• Simulate a value u n from C n (u n |u 1 ,... ,u n−1 ).
The correctness of the algorithm follows from the fact that for independent
random U(0, 1) variables Q 1 ,... ,Q n ,
(Q 1 ,C2 −1 (Q 2|Q 1 ),... ,Cn
−1 (Q n|Q 1 ,C2 −1 (Q 2|Q 1 ),...))
has joint distribution function C.
To simulate a value from C k (u k |u 1 ,... ,u k−1 ) in general means simulating q from
U(0, 1) from which u k = C −1
k
(q|u 1,... ,u k−1 ) can be obtained from the equation
q = C k (u k |u 1 ,... ,u k−1 ) by numeric rootfinding. However we will present situations
where C −1
k
(·|u 1,... ,u k−1 ) exists in closed form and hence there is no need for
numeric rootfinding. In these situation this algorithm can be recommended. We
will refer to this method as the conditional method for random variate generation.
Example 2.5. Consider the bivariate copula family given by
for θ>0.
C(u, v) =(u −θ + v −θ − 1) −1/θ (2.4.1)
C 2 (v|u) = ∂C
∂u (u, v) =− 1 θ (u−θ + v −θ − 1) −1/θ−1 (−θu −θ−1 )
= (u θ ) −1−θ
θ (u −θ + v −θ − 1) −1/θ−1 =(1+u θ (v −θ − 1)) −1−θ
θ .
Solving the equation q = C 2 (v|u) forv yields
C −1
2 (q|u) =v = (
(q −θ
1+θ − 1)u −θ +1) −1/θ
.
Thus the following algorithm generates a random variate from the copula given by
(2.4.1).
• Generate a value u from U(0, 1),
• Generate a value q from U(0, 1),
Set v =((q −θ
1+θ − 1)u −θ +1) −1/θ .
(u, v) is the desired random variate.
10