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  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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