Modelling Dependence with Copulas - IFOR

5 Archimedean Copulas
Example 5.2. Let ϕ(t) =(t −θ − 1)/θ, whereθ ∈ [−1, ∞)\{0}. This gives the
Clayton family
C θ (u, v) =max([u −θ + v −θ − 1] −1/θ , 0). (5.2.6)
For θ>0 the copulas are strict and the copula expression simplifies to
C θ (u, v) =(u −θ + v −θ − 1) −1/θ . (5.2.7)
The Clayton family has lower tail dependence for θ ≥ 0 and is comprehensive, i.e.,
C −1 = W , C ∞ = M and lim θ→0 C θ = Π. Since most of the following results are
results for strict Archimedean copulas we will refer to (5.2.7) as the Clayton family
and refer to (5.2.6) as the extension to negative dependence of the Clayton family.
Example 5.3. Let ϕ(t) =− ln e−θt −1,whereθ ∈ R\{0}. This gives the Frank
e θ −1
family
C θ (u, v) =− 1 (
θ ln 1+ (e−θu − 1)(e −θv )
− 1)
e −θ .
− 1
The Frank copulas are strict copulas and are comprehensive, i.e., C −∞ = W , C ∞ =
M and lim θ→0 C θ = Π. Members of the Frank family are the only Archimedean
copulas which satisfy the equation C(u, v) =Ĉ(u, v) for so called radial symmetry.
Example 5.4. Let ϕ(t) =1−t for t in [0, 1]. Then ϕ [−1] (t) =1−t for t in [0, 1] and 0
for t>1; i.e., ϕ [−1] (t) =max(1−t, 0). Hence C(u, v) =max(u+v−1, 0) = W (u, v).
Hence W is Archimedean.
The results in the following theorem will enable multivariate extensions of Archimedean
copulas.
Theorem 5.2. Let C be an Archimedean copula with generator ϕ. Then:
1. C is symmetric; i.e., C(u, v) =C(v, u) for all u, v in I;
2. C is associative; i.e., C(C(u, v),w)=C(u, C(v, w)) for all u, v, w in I.
Proof. The first part follows directly from (5.2.2).
C(C(u, v),w) = ϕ [−1] (ϕ(C(u, v)) + ϕ(w))
= ϕ [−1] (ϕ(ϕ [−1] (ϕ(u)+ϕ(v))) + ϕ(w))
= ϕ [−1] (ϕ(u)+ϕ(v)+ϕ(w))
= ϕ [−1] (ϕ(u)+ϕ(ϕ [−1] (ϕ(v)+ϕ(w))))
= ϕ [−1] (ϕ(u)+ϕ(C(v, w))) = C(u, C(v, w)),
and hence C is associative.
The associativity property of Archimedean copulas is not shared by copulas in
general as indicated by the following example.
Example 5.5. Let C θ be a member of the bivariate Farlie-Gumbel-Morgenstern
family of copulas, i.e., C θ (u, v) =uv + θuv(1 − u)(1 − v), for θ ∈ [−1, 1]. Then
( ( 1 1
C θ
4 ,C θ
2 , 1 ( ( 1
≠ C θ C θ
3))
4 , 1 )
, 1 )
2 3
for all θ ∈ [−1, 1] except 0. Hence the only member of the bivariate Farlie-Gumbel-
Morgenstern family of copulas that is Archimedean is Π.
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