Modelling Dependence with Copulas - IFOR

5.4 Order
Theorem 5.7. Let ϕ be a strict generator. The coefficient of lower tail dependence
for the copula C(u, v) =ϕ −1 (ϕ(u)+ϕ(v)) is equal to
λ L = 2 lim
s→∞ [ϕ−1′ (2s)/ϕ −1′ (s)].
Proof. The proof is similar to that of Theorem 5.6.
Example 5.10. Consider the Clayton family given by
C θ (u, v) =(u −θ + v −θ − 1) −1/θ
for θ ≥ 0. This strict copula family has generator ϕ(t) =(t −θ − 1)/θ. It follows
that ϕ −1 (s) =(1+θs) −1/θ . Using Theorem 5.7 shows that the Clayton family has
lower tail dependence.
λ L = 2 lim [ϕ −1′ (2s)/ϕ −1′ (s)]
s→∞
=
−(1 + 2θs)−1/θ−1
2 lim [
s→∞ −(1 + θs) ]
−1/θ−1
= 2 2 −1/θ−1 =2 −1/θ .
Example 5.11. Consider the Frank family given by
C θ (u, v) =− 1 (
θ ln 1+ (e−θu − 1)(e −θv )
− 1)
e −θ − 1
for θ ∈ R\{0}. This strict copula family has generator ϕ(t) =− ln e−θt −1. It follows
e θ −1
that ϕ −1 (s) =− 1 θ ln(1 − (1 − e−θ )e −s )andϕ −1′ (s) =− 1 θ (1 − e−θ )e −s )/(1 − (1 −
e −θ )e −s ). Since
ϕ −1′ (0) = − 1 θ
1 − e −θ
1 − (1 − e −θ ) = − 1 1 − e −θ
θ e −θ
is finite the Frank family does not have upper tail dependence according to Theorem
5.6. Furthermore
and hence
ϕ −1′ (2s)
ϕ −1′ (s) = e−s 1 − (1 − e−θ )e −s
1 − (1 − e −θ )e −2s
ϕ −1′ (2s)
lim
s→∞ ϕ −1′ (s) =0.
Thus the Frank family does not have lower tail dependence.
5.4 Order
Recall the concordance ordering of copulas suggested by the Fréchet-Hoeffding inequality.
In this section we will give conditions in terms of the generators, that
enables us to determine among other things which parametric families are positively
ordered.
Definition 14. A function f defined on [0, ∞) is subadditive if for all x, y in [0, ∞),
f(x + y) ≤ f(x)+f(y). (5.4.1)
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