Modelling Dependence with Copulas - IFOR

5.5 Multivariate Archimedean Copulas
ϕ 2 ◦ ϕ −1
3 are in L ∗ ∞ , then (5.5.4) is a proper copula. Note that all trivariate margins
of (5.5.4) have the form (5.5.3) and all bivariate margins have the form (5.2.2).
Clearly the idea of (5.5.3) and (5.5.4) generalizes to higher dimensions.
Example 5.13. Let ϕ θ1 (t) =(− ln t) θ1 and ϕ θ2 (t) =(− ln t) θ2 ,whereθ 1 ,θ 2 ≥ 1.
For θ 1 ≤ θ 2 we get the trivariate extension of the Gumbel family. The copula
expression (5.5.3) becomes
C(u; θ 1 ,θ 2 )=exp{−([(− ln u 1 ) θ2 +(− ln u 2 ) θ2 ] θ1/θ2 +(− ln u 3 ) θ1 ) 1/θ1 }.
We can now present an algorithm for random variate generation from the multivariate
extension of (5.2.2) indicated by (5.5.3) and (5.5.4). Because the recursive
nature of the multivariate extension of (5.2.2) indicated by (5.5.3) and (5.5.4) the
algorithm is basically an extension of the algorithm for random variate generation
from an Archimedean copula of the form (5.2.2) already presented.
Algorithm 4.
• Generate a uniform (0, 1) variate q.
• Set t 1 = K −1
C θ1
(q).
• Generate a uniform (0, 1) variate s 1 independent of q.
• Set a 1 = ϕ −1
θ 1
(s 1 ϕ θ1 (t 1 )) and u n = ϕ −1
θ 1
((1 − s 1 )ϕ θ1 (t 1 ))
.
.
• Set t i = K −1
C θi
(a i−1 ).
• Generate a uniform (0, 1) variate s k independent of q and s 1 ,... ,s i−1 .
• Set a i = ϕ −1
θ i
.
(s i ϕ θi (t k )) and u n−i+1 = ϕ −1
θ i
((1 − s i )ϕ θi (t i ))
• Set t n−1 = K −1
C θn−1
(a n−2 ).
• Generate a uniform (0, 1) variate s n−1 independent of q, s 1 ,... ,s n−2 .
• Set u 1 = ϕ −1
θ n−1
(s n−1 ϕ θn−1 (t n−1 )) and u 2 = ϕ −1
θ n−1
((1 − s n−1 )ϕ θn−1 (t n−1 ))
• (u 1 ,u 2 ,... ,u n ) is the desired random variate from
C n (u 1 ,u 2 ,... ,u n ; θ 1 ,... ,θ n−1 ),
where i ∈{2,... ,n− 2} and
C k (u 1 ,u 2 ,... ,u k ; θ 1 ,... ,θ k−1 ) = C(C k−1 (u 1 ,u 2 ,... ,u k−1 ; θ 2 ,... ,θ k−1 ),u k ; θ 1 )
C(·, ·; θ j ) = C θj (·, ·) =ϕ −1
θ j
(ϕ θj (·)+ϕ θj (·)),
for k =3, 4,... ,n and j =1, 2,... ,k− 1.
Let U 1 ,U 2 ,... ,U n be the uniform (0, 1) random variables given by the Algorithm
4. Then
K Cθ1 (C θ1 (K Cθ2 (C θ2 (...K Cθn−1 (C θn−1 (U 1 ,U 2 )),... ,U n−1 )),U n )) (5.5.5)
is a uniform (0, 1) random variable.
The correctness of the algorithm is stated in the following theorem.
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