Modelling Dependence with Copulas - IFOR

5 Archimedean Copulas
Proof. Let U and V be uniform (0, 1) random variables with joint distribution
function C, and let K C denote the distribution function of C(U, V ). Then from
(3.3.6) we have
τ C = 4E(C(U, V )) − 1,
= 4
∫ 0
1
t dK C (t) − 1,
= 4 ( [tK C (t)] 1 0 −
= 3−
∫ 1
0
∫ 1
0
K C (t)dt.
K C (t)dt ) − 1,,
From Theorem 5.3 and Corollary 5.1 it follows that the distribution function K C
of C(U, V )is
and hence
τ C =3− 4
∫ 1
0
K C (t) =t −
(
t −
ϕ(t)
ϕ ′ (t + )
ϕ(t)
ϕ ′ (t + ) ,
)
dt =1+4
∫ 1
0
ϕ(t)
ϕ ′ (t) dt,
where ϕ ′ (t + ) is replaced by ϕ ′ (t) in the denominator of the integral because concave
functions are differentiable almost everywhere.
Example 5.6. Consider the Gumbel family with generator ϕ(t) =(− ln t) θ ,for
θ ≥ 1. Then
ϕ(t)
ϕ ′ (t) = t ln t
θ .
Using Theorem 5.4 we can calculate Kendall’s tau for the Gumbel family.
∫ 1
t ln t
τ θ = 1+4 dt
0 θ
= 1+ 4 ( [t 2 ] 1
∫ 1
)
θ 2 ln t t
−
0 0 2 dt
= 1+ 4 (0 − 1/4) = 1 − 1/θ.
θ
Example 5.7. Consider the Clayton family with generator ϕ(t) =(t −θ − 1)/θ, for
θ ∈ [−1, ∞)\{0}. Then
ϕ(t)
ϕ ′ (t) = tθ+1 − t
.
θ
Using Theorem 5.4 we can calculate Kendall’s tau for the Clayton family.
∫ 1
t θ+1 − t
τ θ = 1+4
dt
0 θ
= 1+ 4 ( 1
θ θ +2 − 1 )
2
= 1+ 4 −θ
θ 2(θ +2) = θ
θ +2 .
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