Modelling Dependence with Copulas - IFOR

• Generate independent uniform (0, 1) variates q and u.
Assume that this gives q =0.6 andu =0.2.
• Choose the index j such that
j =min{i ≥ 1|
i∑
λ k ≥ q}.
k=1
Since λ 1 qwe get j =2.
• The desired random variate from C is:
(a 2 1(u), (a 2 2(u), (a 2 3(u)), where
(a 2 1(u),a 2 2(u), (a 2 3(u)) = (u, u, u) − (0, 0, 1)(2u − 1)
= (u, u, 1 − u) =(0.2, 0.2, 0.8).
Note the similarity with the binary representation of j − 1=1(u ∼ 0,
1 − u ∼ 1).
When many random variates are produced according to the algorithm, the rank
correlation matrix of the data is very close to ρ.
Example 8.3. Consider the following problem: We are given margins F 1 ,... ,F 4
and rank correlation matrix ρ given by
⎛
⎞
1 0.1 −0.1 −1
⎜ 0.1 1 0.7 −0.1
⎟
⎝ −0.1 0.7 1 0.1 ⎠ .
−1 −0.1 0.1 1
This is a positive-semi-definite matrix (however not positive-definite since det(ρ) =
0) and hence a proper rank correlation matrix. We want to find a distribution
with this rank correlation matrix, which belongs to the family of mixtures of extremal
distributions obtained from convex combinations of extremal distributions
with margins F 1 ,... ,F 4 . In other words we want to find a convex decomposition
of ρ in the eight extremal rank correlation matrices. Simply solving the resulting
equation system
⎛
⎜
⎝
1 1 1 1 −1 −1 −1 −1 0.1
1 1 −1 −1 1 1 −1 −1 −0.1
1 −1 1 −1 1 −1 1 −1 −1
1 1 −1 −1 −1 −1 1 1 0.7
1 −1 1 −1 −1 1 −1 1 −0.1
1 −1 −1 1 1 −1 −1 1 0.1
1 1 1 1 1 1 1 1 1
given by (8.0.8) and only allowing nonnegative solutions yields
⎞
,
⎟
⎠
λ 1 =0,λ 2 =0.425,λ 3 =0,λ 4 =0.125,λ 5 =0,λ 6 =0.25,λ 7 =0,λ 8 = 0425,
as the only solution (note that in general the decomposition is not unique). From
Theorem 8.3 it is clear that the resulting copula, C, isgivenby
C(u 1 ,u 2 ,u 3 ,u 4 ) = 0.425 max(min(u 1 ,u 2 ,u 3 )+u 4 − 1, 0) +
0.125 max(min(u 1 ,u 2 ) + min(u 3 ,u 4 ) − 1, 0) +
0.25 max(min(u 1 ,u 3 )+min(u 2 ,u 4 ) − 1, 0) +
0.425 max(u 1 + min(u 2 ,u 3 ,u 4 ) − 1, 0).
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