Modelling Dependence with Copulas - IFOR

5.5 Multivariate Archimedean Copulas
The extension of this idea to n dimensions, with u =(u 1 ,u 2 ,... ,u n ), results in the
n-dimensional product copula Π n in the form
Π n (u) =u 1 u 2 ...u n =exp(−[(− ln u 1 )+(− ln u 2 )+...+(lnu n )]).
This leads naturally to the following generalization of (5.2.2):
C n (u) =ϕ [−1] (ϕ(u 1 )+ϕ(u 2 )+...+ ϕ(u n )). (5.5.1)
The functions C n are the serial iterates of the Archimedean 2-copulas generated by
ϕ, that is, if we set C 2 (u 1 ,u 2 )=C(u 1 ,u 2 )=ϕ [−1] (ϕ(u 1 )+ϕ(u 2 )), then for n ≥ 3,
C n (u 1 ,u 2 ,... ,u n )=C(C n−1 (u 1 ,u 2 ,... ,u n−1 ),u n ). Note that this technique of
composing copulas generally fails. But since Archimedean copulas are symmetric
and associative it seems more likely that C n , given certain additional properties of
ϕ (and ϕ [−1] ) , is a copula for n ≥ 3.
Definition 15. A function g(t) is completely monotonic on the interval J if it is
continuous there and has derivatives of all orders which alternate in sign, i.e., if it
satisfies
(−1) k dk
g(t) ≥ 0 (5.5.2)
dtk for all t in the interior of J and k =0, 1, 2,....
As a consequence, if g(t) is completely monotonic on [0, ∞) andg(c) =0for
some c>0, then g must be identically zero on [0, ∞). So if the pseudo-inverse
ϕ [−1] of an Archimedean generator ϕ is completely monotonic, it must be positive
on [0, ∞), i.e., ϕ is strict and ϕ [−1] = ϕ −1 .
Theorem 5.9. Let ϕ be a continuous strictly decreasing function from I to [0, ∞]
such that ϕ(0) = ∞ and ϕ(1) = 0, andletϕ −1 denote the inverse of ϕ. If C n is
the function from I to I n given by (5.5.1), thenC n is an n-copula for all n ≥ 2 if
and only if ϕ −1 is completely monotonic on [0, ∞).
This theorem can be partially extended to the case when ϕ is non-strict and
ϕ [−1] is m-monotonic on [0, ∞) forsomem ≥ 2, that is, the derivatives of ϕ [−1]
alter sign up to and including the mth order on [0, ∞). Then the function C n given
by (5.5.1) is an n-copula for 2 ≤ n ≤ m.
Corollary 5.3. If the inverse ϕ −1 of a strict generator ϕ of an Archimedean copula
C is completely monotonic, then C ≻ Π.
Proof. We begin by proving that for a strict Archimedean copula C generated by
ϕ in Ω if − ln ϕ −1 is concave on (0, ∞), then C ≻ Π.
Assume that − ln ϕ −1 is concave on (0, ∞). Since − ln ϕ −1 (0) = − ln 1 = 0, − ln ϕ −1
is subadditive according to Lemma 5.3. Hence
− ln ϕ −1 (ϕ(u)+ϕ(v)) ≤−ln ϕ −1 (ϕ(u)) − ln ϕ −1 (ϕ(v)) ⇔
− ln C(u, v) ≤−ln uv ⇔ C(u, v) ≥ uv.
This means that C ≻ Π.
The concavity of − ln ϕ −1 on (0, ∞) is equivalent to requiring that
(ϕ −1′ ) 2 − ϕ −1 ϕ −1′′ ≤ 0, on (0, ∞), and this inequality holds for all completely
monotonic functions.
While it is simple to generate n-copulas of the form given by (5.5.1) they suffer
from a very limited dependence structure since all k-margins are identical. One
would like to have a multivariate extension of the Archimedean 2-copula given by
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