Modelling Dependence with Copulas - IFOR

5 Archimedean Copulas
Theorem 5.10. Let the random variables U 1 ,... ,U n
Algorithm 4. Then
be generated according to
(U 1 ,U 2 ) have copula C θn−1 ,
(U 1 ,U 3 ), (U 2 ,U 3 ) have copula C θn−2 ,
.
(U 1 ,U n ), (U 2 ,U n ),... ,(U n−1 ,U n ) have copula C θ1 .
Hence (U 1 ,U 2 ,... ,U n ) have copula C n (u 1 ,u 2 ,... ,u n ; θ 1 ,... ,θ n−1 ).
is strictly in-
Proof. From the algorithm it follows that:
⋆ (U 1 ,U 2 ) have the copula C θn−1 .
⋆ (K Cθn−1 (C θn−1 (U 1 ,U 2 )),U 3 ) have copula C θn−2 , and since K Cθn−1
creasing on I, (C θn−1 (U 1 ,U 2 ),U 3 ) have copula C θn−2 .
U 1 = ϕ −1
θ n−1
(S n−1 ϕ θn−1 (C θn−1 (U 1 ,U 2 ))
= f n−1 (C θn−1 (U 1 ,U 2 )),
U 2 = ϕ −1
θ n−1
((1 − S n−1 )ϕ θn−1 (C θn−1 (U 1 ,U 2 ))
= g n−1 (C θn−1 (U 1 ,U 2 )),
where f n−1 and g n−1 are strictly increasing on I. Hence (U 1 ,U 3 )and(U 2 ,U 3 )have
copula C θn−2 .
⋆ (K Cθn−2 (C θn−2 (A n−2 ,U 3 )),U 4 ) have copula C θn−3 ,
where A n−2 = K Cθn−1 (C θn−1 (U 1 ,U 2 )). Since K Cθn−2 is strictly increasing on I,
(C θn−2 (K Cθn−1 (C θn−1 (U 1 ,U 2 )),U 3 ),U 4 ) have copula C θn−3 .
A n−2 = ϕ −1
θ n−2
(S n−2 ϕ θn−2 (C θn−2 (A n−2 ,U 3 ))
= f n−2 (C θn−2 (A n−2 ,U 3 )),
U 3 = ϕ −1
θ n−2
((1 − S n−2 )ϕ θn−2 (C θn−2 (A n−2 ,U 3 ))
= g n−2 (C θn−2 (A n−2 ,U 3 )),
where f n−2 and g n−2 are strictly increasing on I. Hence (A n−2 ,U 4 )and(U 3 ,U 4 )
have copula C θn−3 . Furthermore
U 1 = f n−1 (C θn−1 (U 1 ,U 2 ))
= f n−1 (K −1
C θn−1
(K Cθn−1 (C θn−1 (U 1 ,U 2 ))))
= f n−1 (K −1
C θn−1
(A n−2 ),
U 2 = g n−1 (C θn−1 (U 1 ,U 2 ))
= g n−1 (K −1
C θn−1
(K Cθn−1 (C θn−1 (U 1 ,U 2 ))))
= g n−1 (K −1
C θn−1
(A n−2 ),
and hence (U 1 ,U 4 )and(U 2 ,U 4 ) have copula C θn−3 .
By continuing this way, it follows that
(U 1 ,U 2 ) have copula C θn−1 ,
(U 1 ,U 3 ), (U 2 ,U 3 ) have copula C θn−2 ,
.
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