This document discusses dynamic dependence ordering for Archimedean copulas. It begins by defining copulas and Archimedean copulas. It then shows how conditioning and ageing affect the copula for Archimedean copulas. Specifically, it demonstrates that conditioning and ageing result in copulas that are also Archimedean, with modified generators. The document also provides methods to order the tails of Archimedean copulas based on properties of the generator's derivative. Finally, it analyzes how specific Archimedean copulas, such as Frank, Clayton, and Gumbel, are affected by conditioning and ageing.

1. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Dynamic dependence ordering
for Archimedean copulas
Arthur Charpentier
http ://perso.univ-rennes1.fr/arthur.charpentier/
International Workshop on Applied Probability, July 2008
1

2. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Motivations : dependence and copulas
Definition 1. A copula C is a joint distribution function on [0, 1]d
, with
uniform margins on [0, 1].
Theorem 2. (Sklar) Let C be a copula, and F1, . . . , Fd be d marginal
distributions, then F(x) = C(F1(x1), . . . , Fd(xd)) is a distribution function, with
F ∈ F(F1, . . . , Fd).
Conversely, if F ∈ F(F1, . . . , Fd), there exists C such that
F(x) = C(F1(x1), . . . , Fd(xd)). Further, if the Fi’s are continuous, then C is
unique, and given by
C(u) = F(F−1
1 (u1), . . . , F−1
d (ud)) for all ui ∈ [0, 1]
We will then define the copula of F, or the copula of X.
2

3. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Motivations : dependence and copulas
Given a random vector X with continuous margins. The copula of C satisfies
P[X1 ≤ x1, · · · , Xd ≤ xd] = C (P[X1 ≤ x1], · · · , P[Xd ≤ xd])
and its survival copula C satisfies
P[X1 > x1, · · · , Xd > xd] = C (P[X1 > x1], · · · , P[Xd > xd])
3

4. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Copula density Level curves of the copula
Fig. 1 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).
4

5. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Copula density Level curves of the copula
Fig. 2 – Density of a copula, c(u, v) =
∂2
C(u, v)
∂u∂v
.
5

6. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Archimedean copulas
Definition 3. A copula C is called Archimedean if it is of the form
C(u1, · · · , ud) = φ−1
(φ(u1) + · · · + φ(ud)) ,
where the generator φ : [0, 1] → [0, ∞] is convex, decreasing and satisfies φ(1) = 0.
A necessary and sufficient condition is that φ−1
is d-monotone.
6

7. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Some examples of Archimedean copulas
φ(t) range θ
(1) 1
θ
(t−θ − 1) [−1, 0) ∪ (0, ∞) Clayton, Clayton (1978)
(2) (1 − t)θ [1, ∞)
(3) log
1−θ(1−t)
t
[−1, 1) Ali-Mikhail-Haq
(4) (− log t)θ [1, ∞) Gumbel, Gumbel (1960), Hougaard (1986)
(5) − log e−θt−1
e−θ−1
(−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen (1987)
(6) − log{1 − (1 − t)θ} [1, ∞) Joe, Frank (1981), Joe (1993)
(7) − log{θt + (1 − θ)} (0, 1]
(8) 1−t
1+(θ−1)t
[1, ∞)
(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)
(10) log(2t−θ − 1) (0, 1]
(11) log(2 − tθ) (0, 1/2]
(12) ( 1
t
− 1)θ [1, ∞)
(13) (1 − log t)θ − 1 (0, ∞)
(14) (t−1/θ − 1)θ [1, ∞)
(15) (1 − t1/θ)θ [1, ∞) Genest & Ghoudi (1994)
(16) ( θ
t
+ 1)(1 − t) [0, ∞)
7

8. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Why Archimedean copulas ?
Assume that X and Y are conditionally independent, given the value of an
heterogeneous component Θ. Assume further that
P(X ≤ x|Θ = θ) = (GX(x))θ
and P(Y ≤ y|Θ = θ) = (GY (y))θ
for some baseline distribution functions GX and GY . Then
F(x, y) = ψ(ψ−1
(FX(x)) + ψ−1
(FY (y))).
where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ
).
8

9. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
0 5 10 15
05101520
Conditional independence, two classes
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, two classes
Fig. 3 – Two classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
9

10. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
0 5 10 15 20 25 30
010203040
Conditional independence, three classes
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, three classes
Fig. 4 – Three classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
10

11. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
0 20 40 60 80 100
020406080100
Conditional independence, continuous risk factor
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, continuous risk factor
Fig. 5 – Continuous classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
11

12. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Conditioning with Archimedean copulas
Proposition 4. If (U, V ) has copula C, with generator φ. Then the copula of
(U, V ) given U, V ≤ u is also Archimedean, with generator
φu(x) = φ(x · C(u, u)) − φ(C(u, u)), for all x ∈ (0, 1].
Ageing with Archimedean copulas
Proposition 5. If (X, Y ) is exchangeable, with survival copula C, with generator
φ. Then the survival copula Ct of (X, Y ) given X, Y > t is also Archimedean,
with generator
φt(x) = φ(x · γt) − φ(γt), for all x ∈ (0, 1], t ∈ [0, ∞),
where γt = C(F(t), F(t)) = P(X > t, Y > t).
12

13. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Ageing with Archimedean copulas
0.0 0.2 0.4 0.6 0.8 1.0
0246810
q
Fig. 6 – Initial Archimedean generator, · → φ(·) on (0, 1].
13

14. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Ageing with Archimedean copulas
0.0 0.2 0.4 0.6 0.8 1.0
0246810
P(X>t,Y>t)
q
Fig. 7 – Initial Archimedean generator, · → φ(·) on (0, 1].
14

15. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Ageing with Archimedean copulas
0.0 0.2 0.4 0.6 0.8 1.0
0246810
q
Fig. 8 – Initial Archimedean generator, · → φ(·) on (0, γt], γt = C(F(t), F(t)).
15

16. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Ageing with Archimedean copulas
0.0 0.2 0.4 0.6 0.8 1.0
0246810
q
Fig. 9 – Initial Archimedean generator, · → φ(·) on (0, γt].
16

17. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Ageing with Archimedean copulas
0.0 0.2 0.4 0.6 0.8 1.0
0246810
q
Fig. 10 – Initial Archimedean generator, · → φ(·γt) on [0, 1].
17

18. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Ageing with Archimedean copulas
0.0 0.2 0.4 0.6 0.8 1.0
0246810
q
Fig. 11 – Initial Archimedean generator, · → φ(·γt) − phi(γt) on [0, 1].
18

19. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Ageing with Frailty-Archimedean copulas
Proposition 6. If (X, Y ) is exchangeable with a frailty representation (where Θ
has Laplace transform ψ). Then (X, Y ) given X, Y > t also has a failty
representation, and the factor has Laplace transform
ψt(x) =
ψ[x + ψ−1
(γt)]
γt
, for all x ∈ [0, ∞), t ∈ [0, ∞).
where γt = C(F(t), F(t)).
D´emonstration. The only technical part is to proof that (X, Y ) given X, Y > t
are conditionally independent.
19

20. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Ordering tails of Archimedean copulas
Proposition 7. Set C1 = C(t1, t1) and C2 = C(t2, t2). Given φ, define



f12(x) = φ C1
C2
φ−1
(x + φ(C2)) − φ(C1)
f21(x) = φ C2
C1
φ−1
(x + φ(C1)) − φ(C2)
Then 


Ct2
Ct1
if and only if f21 is subadditive
Ct2 Ct1 if and only if f12 is subadditive
Lemma 8. If φ is twice differentiable, set ψ(t) = log(−φ (t)).



if ψ is concave on (0, 1), Ct2 Ct1 for all 0 < t1 ≤ t2
if ψ is convex on (0, 1), Ct2
Ct1
for all 0 < t1 ≤ t2
20

21. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Some examples : Frank copula
Frank copula has generator φ(t) = − log
e−αx
− 1
e−α − 1
, α ≥ 0
C(u, v) = −
1
α
log 1 +
(e−αu
− 1)(e−αv
− 1)
e−α − 1
on [0, 1] × [0, 1].
ψ(x) = log α − αx − log[1 − e−αx
] is concave, and thus
C Ct1
Ct2
C⊥
, for all 0 ≤ t1 ≤ t2,
with Ct → C as t → ∞. The random pair is less and less positively dependent.
21

22. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Some examples : Clayton copula
Clayton copula has generator φ(t) = t−α
− 1, α ≥ 0,
C(u, v) = u−α
+ v−α
− 1
−1/α
on [0, 1] × [0, 1].
f12 and f21 are linear (see also Lemma 5.5.8. in Schweizer and Sklar
(1983)), hence
Ct1 = Ct2 = C.
22

23. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Some examples : Ali-Mikhail-Haq copula
Ali-Mikhail-Haq copula has generator φ(t) = ...1, α ∈ [0, 1),
C(u, v) =
uv
1 − α(1 − u
(1 − v) on [0, 1] × [0, 1].
ψ(x) = log
1
x
−
α
1 − α[1 − x]
is concave, and thus
C Ct1
Ct2
C⊥
, for all 0 ≤ t1 ≤ t2,
with Ct → C as t → ∞.
23

24. Arthur CHARPENTIER - Dynamic dependence ordering for Archimedean copulas
Some examples : Gumbel copula
Gumbel copula has generator φ(t) = (− log t)α
, α ≥ 0,
C(u, v) = exp − [(− log u)α
+ (− log v)α
]
1
α
) on [0, 1] × [0, 1].
ψ(x) = log α − log x + (α − 1) log[− log x] is neither concave, nor convex.
But see C − Ct is always positive, i.e.
C Ct C⊥
, for all 0 < t.
24