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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

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Motivations : dependence and copulas

Definition 1. A copula C is a joint distribution function on [0, 1]d, withuniform margins on [0, 1].

Theorem 2. (Sklar) Let C be a copula, and F1, . . . , Fd be d marginaldistributions, then F (x) = C(F1(x1), . . . , Fd(xd)) is a distribution function, withF ∈ F(F1, . . . , Fd).

Conversely, if F ∈ F(F1, . . . , Fd), there exists C such thatF (x) = C(F1(x1), . . . , Fd(xd)). Further, if the Fi’s are continuous, then C isunique, and given by

C(u) = F (F−11 (u1), . . . , F−1

d (ud)) for all ui ∈ [0, 1]

We will then define the copula of F , or the copula of X.

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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])

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Copula density Level curves of the copula

Fig. 1 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).

4


Copula density Level curves of the copula

Fig. 2 – Density of a copula, c(u, v) =∂2C(u, v)∂u∂v

.

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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.

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Some examples of Archimedean copulas

φ(t) range θ

(1) 1θ

(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−1e−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−t1+(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) ( 1t− 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) ( θt

+ 1)(1 − t) [0,∞)

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Why Archimedean copulas ?

Assume that X and Y are conditionally independent, given the value of anheterogeneous 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).

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0 5 10 15

05

1015

20

Conditional independence, two classes

!3 !2 !1 0 1 2 3

!3

!2

!1

01

23

Conditional independence, two classes

Fig. 3 – Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).

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0 5 10 15 20 25 30

010

2030

40

Conditional independence, three classes

!3 !2 !1 0 1 2 3

!3

!2

!1

01

23

Conditional independence, three classes

Fig. 4 – Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).

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0 20 40 60 80 100

020

4060

80100

Conditional independence, continuous risk factor

!3 !2 !1 0 1 2 3

!3

!2

!1

01

23

Conditional independence, continuous risk factor

Fig. 5 – Continuous classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).

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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 generatorφ. 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).

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0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

●

Fig. 6 – Initial Archimedean generator, · 7→ φ(·) on (0, 1].

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0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

P(X>t,Y>t)

●

Fig. 7 – Initial Archimedean generator, · 7→ φ(·) on (0, 1].

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0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

●

Fig. 8 – Initial Archimedean generator, · 7→ φ(·) on (0, γt], γt = C(F (t), F (t)).

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0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

●

Fig. 9 – Initial Archimedean generator, · 7→ φ(·) on (0, γt].

16



0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

●

Fig. 10 – Initial Archimedean generator, · 7→ φ(·γt) on [0, 1].

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0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

●

Fig. 11 – Initial Archimedean generator, · 7→ φ(·γt)− phi(γt) on [0, 1].

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Ageing with Frailty-Archimedean copulas

Proposition 6. If (X,Y ) is exchangeable with a frailty representation (where Θhas Laplace transform ψ). Then (X,Y ) given X,Y > t also has a failtyrepresentation, 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)).

Demonstration. The only technical part is to proof that (X,Y ) given X,Y > t

are conditionally independent.

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Ordering tails of Archimedean copulas

Proposition 7. Set C1 = C(t1, t1) and C2 = C(t2, t2). Given φ, define f12(x) = φ(C1C2φ−1(x+ φ(C2))

)− φ(C1)

f21(x) = φ(C2C1φ−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 ≤ t2if ψ is convex on (0, 1), Ct2 � Ct1 for all 0 < t1 ≤ t2

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Some examples : Frank copula

Frank copula has generator φ(t) = − loge−αx − 1e−α − 1

, α ≥ 0

C(u, v) = − 1α

log(

1 +(e−αu − 1)(e−αv − 1)

e−α − 1

)on [0, 1]× [0, 1].

ψ(x) = logα− 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.

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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)), henceCt1 = Ct2 = C.

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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(

1x− α

1− α[1− x]

)is concave, and thus

C � Ct1 � Ct2 � C⊥, for all 0 ≤ t1 ≤ t2,

with Ct → C as t→∞.

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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]

1α

)) on [0, 1]× [0, 1].

ψ(x) = logα− 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.

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