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Arthur CHARPENTIER - tails of Archimedean copulas
Tails of Archimedean Copulas
tail dependence in risk management
Arthur Charpentier
CREM-Universite Rennes 1
(joint work with Johan Segers, UCLN)
http ://perso.univ-rennes1.fr/arthur.charpentier/
Colloque Evaluation et couverture des risques extremes
Universite Paris-Dauphine & Chaire AXA de la Fondation du Risque, Juin 2008
1
Arthur CHARPENTIER - tails of Archimedean copulas
Tail behavior and risk management
In reinsurance (XS) pricing, use of Pickands-Balkema-de Haan’s theorem
Theorem 1. F ∈MDA (Gξ) if and only if
limu→xF
sup0<x<xF
{∣∣Pr (X − u ≤ x|X > u)−Hξ,σ(u) (≤ x)∣∣} = 0,
for some positive function σ (·), where Hξ,σ (x) =
1− (1 + ξx/σ)−1/ξ , ξ 6= 0
1− exp (−x/σ) , ξ = 0.
1− F (x) ≈ (1− F (u))[1−Hξ,σ(u) (x− u)
], for all x > u.
So, if u = Xk:n, then
1− F (x) ≈ (1− F (Xk:n))︸ ︷︷ ︸≈1−Fn(Xk:n)=k/n
[1−Hξ,σ(Xk:n) (x−Xk:n)
], for all x > Xk:n,
2
Arthur CHARPENTIER - tails of Archimedean copulas
Pure premium of XS contract
Recall that πd = E((X − d)+) with d large, thus,
πd =1
P(X > d)
∫ ∞d
1− F (x)dx
≈ k
n
σ
1− ξ
(1 + ξ
d−Xn−k:n
σ
)1− 1ξ
,
i.e.
πd =k
n
σk
1− ξk
(1 + ξk
d−Xn−k:n
σk
)1− 1ξk
(see e.g. Beirlant et al. (2005).
Possible to derive explicit formulas for any tail risk measure (VaR, TVaR...).
3
Arthur CHARPENTIER - tails of Archimedean copulas
Extending extreme value theory in higher dimension
univariate case bivariate case
limiting distribution dependence structure of
of Xn:n (G.E.V.) componentwise maximum
when n→∞, i.e. Hξ (Xn:n, Yn:n)
(Fisher-Tippet)
dependence structure of
limiting distribution (X,Y ) |X > x, Y > y
of X|X > x (G.P.D.) when x, y →∞when x→∞, i.e. Gξ,σ dependence structure of
(Balkema-de Haan-Pickands) (X,Y ) |X > x
when x→∞
4
Arthur CHARPENTIER - tails of Archimedean copulas
Tail dependence in risk management
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1e+01 1e+03 1e+05
1e
+0
11
e+
02
1e
+0
31
e+
04
1e
+0
5
Loss (log scale)
Allo
cate
d E
xpe
nse
s
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1 10 100 1000 100001
e+
00
1e
+0
21
e+
04
1e
+0
6
Car claims (log scale)
Ho
use
ho
ld c
laim
s
Fig. 1 – Multiple risks issues.
5
Arthur CHARPENTIER - tails of Archimedean copulas
Motivations : dependence and copulas
Definition 2. A copula C is a joint distribution function on [0, 1]d, withuniform margins on [0, 1].
Theorem 3. (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.
6
Arthur CHARPENTIER - tails of Archimedean copulas
XY
Z
Fonction de répartition à marges uniformes
Fig. 2 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).
7
Arthur CHARPENTIER - tails of Archimedean copulas
xx
z
Densité d’une loi à marges uniformes
Fig. 3 – Density of a copula, c(u, v) =∂2C(u, v)∂u∂v
.
8
Arthur CHARPENTIER - tails of Archimedean copulas
Strong tail dependence
Joe (1993) defined, in the bivariate case a tail dependence measure.
Definition 4. Let (X,Y ) denote a random pair, the upper and lower taildependence parameters are defined, if the limit exist, as
λL = limu→0
P(X ≤ F−1
X (u) |Y ≤ F−1Y (u)
),
= limu→0
P (U ≤ u|V ≤ u) = limu→0
C(u, u)u
,
and
λU = limu→1
P(X > F−1
X (u) |Y > F−1Y (u)
)= lim
u→0P (U > 1− u|V ≤ 1− u) = lim
u→0
C?(u, u)u
.
9
Arthur CHARPENTIER - tails of Archimedean copulas
Gaussian copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
GAUSSIAN
●
●
Fig. 4 – L and R cumulative curves.
10
Arthur CHARPENTIER - tails of Archimedean copulas
Gumbel copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
GUMBEL
●
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Fig. 5 – L and R cumulative curves.
11
Arthur CHARPENTIER - tails of Archimedean copulas
Clayton copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.0 0.2 0.4 0.6 0.8 1.0
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1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
CLAYTON
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Fig. 6 – L and R cumulative curves.
12
Arthur CHARPENTIER - tails of Archimedean copulas
Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
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L and R concentration functions
L function (lower tails) R function (upper tails)
STUDENT (df=5)
●
●
Fig. 7 – L and R cumulative curves.
13
Arthur CHARPENTIER - tails of Archimedean copulas
Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
STUDENT (df=3)
●
●
Fig. 8 – L and R cumulative curves.
14
Arthur CHARPENTIER - tails of Archimedean copulas
Weak tail dependence
If X and Y are independent (in tails), for u large enough
P(X > F−1X (u), Y > F−1
Y (u)) = P(X > F−1X (u)) · P(Y > F−1
Y (u)) = (1− u)2,
or equivalently, log P(X > F−1X (u), Y > F−1
Y (u)) = 2 · log(1− u). Further, if Xand Y are comonotonic (in tails), for u large enough
P(X > F−1X (u), Y > F−1
Y (u)) = P(X > F−1X (u)) = (1− u)1,
or equivalently, log P(X > F−1X (u), Y > F−1
Y (u)) = 1 · log(1− u).
=⇒ limit of the ratiolog(1− u)
log P(Z1 > F−11 (u), Z2 > F−1
2 (u)).
15
Arthur CHARPENTIER - tails of Archimedean copulas
Weak tail dependence
Coles, Heffernan & Tawn (1999) defined
Definition 5. Let (X,Y ) denote a random pair, the upper and lower taildependence parameters are defined, if the limit exist, as
ηL = limu→0
log(u)log P(Z1 ≤ F−1
1 (u), Z2 ≤ F−12 (u))
= limu→0
log(u)logC(u, u)
,
and
ηU = limu→1
log(1− u)log P(Z1 > F−1
1 (u), Z2 > F−12 (u))
= limu→0
log(u)logC?(u, u)
.
16
Arthur CHARPENTIER - tails of Archimedean copulas
Gaussian copula
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
Chi dependence functions
lower tails upper tails
GAUSSIAN
●●
Fig. 9 – χ functions.
17
Arthur CHARPENTIER - tails of Archimedean copulas
Gumbel copula
0.0 0.2 0.4 0.6 0.8 1.0
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Chi dependence functions
lower tails upper tails
GUMBEL
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Fig. 10 – χ functions.
18
Arthur CHARPENTIER - tails of Archimedean copulas
Clayton copula
0.0 0.2 0.4 0.6 0.8 1.0
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Chi dependence functions
lower tails upper tails
CLAYTON
●
●
Fig. 11 – χ functions.
19
Arthur CHARPENTIER - tails of Archimedean copulas
Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
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1.0
Chi dependence functions
lower tails upper tails
STUDENT (df=3)
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Fig. 12 – χ functions.
20
Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : Loss-ALAE
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0.0 0.2 0.4 0.6 0.8 1.0
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Allo
cate
d E
xpe
nse
s
Fig. 13 – Losses and allocated expenses.
21
Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : Loss-ALAE
0.0 0.2 0.4 0.6 0.8 1.0
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L and R concentration functions
L function (lower tails) R function (upper tails)
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.0
Chi dependence functions
lower tails upper tails
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Gumbel copula
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Fig. 14 – L and R cumulative curves, and χ functions.
22
Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : car-household
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0.0 0.2 0.4 0.6 0.8 1.0
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Car claims
Ho
use
ho
ld c
laim
s
Fig. 15 – Motor and Household claims.
23
Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : car-household
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L and R concentration functions
L function (lower tails) R function (upper tails)
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0.0 0.2 0.4 0.6 0.8 1.00
.00
.20
.40
.60
.81
.0
Chi dependence functions
lower tails upper tails
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Fig. 16 – L and R cumulative curves, and χ functions.
24
Arthur CHARPENTIER - tails of Archimedean copulas
Archimedean copulas
Definition 6. 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.
25
Arthur CHARPENTIER - tails of Archimedean copulas
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,∞)
26
Arthur CHARPENTIER - tails of Archimedean copulas
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) = P(X ≤ x, Y ≤ y) = E(P(X ≤ x, Y ≤ y|Θ = θ))
= E(P(X ≤ x|Θ = θ)× P(Y ≤ y|Θ = θ))
= E((GX(x))Θ × (GY (y))Θ
)= ψ(− logGX(x)− logGY (y))
where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ). Since
FX(x) = ψ(− logGX(x)) and FY (y) = ψ(− logGY (y))
and thus, the joint distribution of (X,Y ) satisfies
F (x, y) = ψ(ψ−1(FX(x)) + ψ−1(FY (y))).
27
Arthur CHARPENTIER - tails of Archimedean copulas
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. 17 – Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
28
Arthur CHARPENTIER - tails of Archimedean copulas
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. 18 – Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
29
Arthur CHARPENTIER - tails of Archimedean copulas
0 20 40 60 80 100
020
4060
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Conditional independence, continuous risk factor
!3 !2 !1 0 1 2 3
!3
!2
!1
01
23
Conditional independence, continuous risk factor
Fig. 19 – Continuous classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
30
Arthur CHARPENTIER - tails of Archimedean copulas
Properties of Archimedean copulas
• the countercomonotonic copula C− is Archimedean, φ(t) = 1− t,• the independent copula C⊥ is Archimedean, φ(t) = − log(t),• the comonotonic copula is not Archimedean (but can be a limit of
Archimedean copulas).
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Arthur CHARPENTIER - tails of Archimedean copulas
Properties of Archimedean copulas
• Frank copula is the only Archimedean such that (U, V ) L= (1− U, 1− V )(stability by symmetry),
• Gumbel copula is the only Archimedean such that (U, V ) has the same copulaas (max{U1, ..., Un},max{V1, ..., Vn}) for all n ≥ 1 (max-stability),
• Clayton copula is the only Archimedean such that (U, V ) has the same copulaas (U, V ) given (U ≤ u, V ≤ v) (stability by truncature).
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Arthur CHARPENTIER - tails of Archimedean copulas
Lower tails of Archimedean copulas
Study regular variation property of φ at 0,
lims→0
φ(st)φ(s)
= t−θ0 , t ∈ (0,∞)⇐⇒ θ0 = − lims→0
sφ′(s)φ(s)
.
If θ0 > 0 : asymptotic dependence
Proposition 7. If 0 < θ0 <∞, then for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,
lims↓0
Pr[∀i = 1, . . . , d : Ui ≤ syi | ∀i ∈ I : Ui ≤ sxi]
=(∑
i∈Ic y−θ0i +
∑i∈I(xi ∧ yi)−θ0∑
i∈I x−θ0i
)−1/θ0
This is Clayton’s copula.
33
Arthur CHARPENTIER - tails of Archimedean copulas
Lower tails of Archimedean copulas
Study regular variation property of φ at 0,
lims→0
φ(st)φ(s)
= t−θ0 , t ∈ (0,∞)⇐⇒ θ0 = − lims→0
sφ′(s)φ(s)
.
If θ0 = 0 : asymptotic independence (dependence in independence) for strictgenerators (φ(0) =∞)Proposition 8. If θ0 = 0 and φ(0) =∞, for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,
lims↓0
Pr[∀i ∈ I : Ui ≤ syi;∀i ∈ Ic : Ui ≤ χs(yi) | ∀i ∈ I : Ui ≤ sxi]
=∏i∈I
(yjxj∧ 1)|I|−κ ∏
i∈Icexp
(−|I|−κy−1
i
),
where χs(·) = φ−1 (−sφ′(s)/·), and κ is the index of regular variation of ψ, withψ(·) = −φ−1(·)φ′(φ−1(·)).
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Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
Study regular variation property of φ at 1,
lims→0
φ(1− st)φ(1− s)
= tθ1 , t ∈ (1,∞)⇐⇒ θ1 = − lims→0
sφ′(1− s)φ(1− s)
.
If θ1 > 1 : asymptotic dependence
Proposition 9. If 1 < θ0 <∞, then for every ∅ 6= I ⊂ {1, . . . , d}, every(xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,
lims↓0
Pr[∀i = 1, . . . , d : Ui ≥ 1− syi | ∀i ∈ I : Ui ≥ 1− sxi] =rd(z1, . . . , zd; θ1)r|I|((xi)i∈I ; θ1)
where zi = xi ∧ yi for i ∈ I and zi = yi for i ∈ Ic and
rk(u1, . . . , uk; θ1) =∑
∅ 6=J⊂{1,...,k}
(−1)|J|−1(∑i∈J
uθ1j)1/θ1
for integer k ≥ 1 and (u1, . . . , uk) ∈ (0,∞)k.
35
Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
Study regular variation property of φ at 1,
lims→0
φ(1− st)φ(1− s)
= tθ1 , t ∈ (1,∞)⇐⇒ θ1 = − lims→0
sφ′(1− s)φ(1− s)
.
If θ1 > 1 and φ′(1) < 0 : asymptotic independence, or near independence
Proposition 10. If 1 < θ1 = 1 and φ′(1) < 0, then for all (xi)i∈I ∈ (0,∞)|I| and(y1, . . . , yd) ∈ (0, 1]d ,
lims↓0
Pr[∀i ∈ I : Ui ≥ 1− syi;∀i ∈ Ic : Ui ≤ yi | ∀i ∈ I : Ui ≥ 1− sxi]
=∏i∈I
yj ·(−D)|I|φ−1(
∑i∈Ic φ(yi))
(−D)|I|φ−1(0).
36
Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
If θ > 1 and φ′(1) = 0 : asymptotic independence, dependence in independence
Proposition 11. If 1 < θ1 = 1 and φ′(1) = 0, if I ⊂ {1, . . . , d} and |I| ≥ 2, thenfor every (xi)i∈I ∈ (0,∞)|I| and every (y1, . . . , yd) ∈ (0,∞)d,
lims↓0
Pr[∀i = 1, . . . , d : Ui ≥ 1− syi | ∀i ∈ I : Ui ≥ 1− sxi] =rd(z1, . . . , zd)r|I|((xi)i∈I)
where zi = xi ∧ yi for i ∈ I and zi = yi for i ∈ Ic and
rk(u1, . . . , uk) :=∑
∅ 6=J⊂{1,...,k}
(−1)|J|(∑J
uj) log(∑J
uj)
= (k − 2)!∫ u1
0
· · ·∫ uk
0
(t1 + · · ·+ tk)−(k−1)dt1 · · · dtk
for integer k ≥ 2 and (u1, . . . , uk) ∈ (0,∞)k.
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Arthur CHARPENTIER - tails of Archimedean copulas
Tails of Archimedean copulas
• upper tail : calculate φ′(1) and θ1 = − lims→0
sφ′(1− s)φ(1− s)
,
◦ φ′(1) < 0 : asymptotic independence
◦ φ′(1) = 0 et θ1 = 1 : dependence in independence
◦ φ′(1) = 0 et θ1 > 1 : asymptotic dependence
• lower tail : calculate φ(0) and θ0 = − lims→0
sφ′(s)φ(s)
,
◦ φ(0) <∞ : asymptotic independence
◦ φ(0) =∞ et θ0 = 0 : dependence in independence
◦ φ(0) =∞ et θ0 > 0 : asymptotic dependence
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Arthur CHARPENTIER - tails of Archimedean copulas
upper tail lower tail
φ(t) range θ −φ′(1) θ1 φ(0) θ0 κ
(1) 1θ
(t−θ − 1) [−1,∞) 1 1 1(−θ)∨0 θ ∨ 0 ·
(2) (1 − t)θ [1,∞) 1(θ = 1) θ 1 0 ·
(3) log 1−θ(1−t)t
[−1, 1) 1 − θ 1 ∞ 0 0
(4) (− log t)θ [1,∞) 1(θ = 1) θ ∞ 0 1 − 1θ
(5) − log e−θt−1e−θ−1
θeθ−1
1 ∞ 0 0
(6) − log{1 − (1 − t)θ} [1,∞) 1(θ = 1) θ ∞ 0 0
(7) − log{θt + (1 − θ)} (0, 1] θ 1 − log(1 − θ) 0 ·(8) 1−t
1+(θ−1)t [1,∞) 1θ
1 1 0 ·
(9) log(1 − θ log t) (0, 1] θ 1 ∞ 0 −∞(10) log(2t−θ − 1) (0, 1] 2θ 1 ∞ 0 0
(11) log(2 − tθ) (0, 1/2] θ 1 log 2 0 ·(12) ( 1
t− 1)θ [1,∞) 1(θ = 1) θ ∞ θ ·
(13) (1 − log t)θ − 1 (0,∞) θ 0 ∞ 0 1 − 1θ
(14) (t−1/θ − 1)θ [1,∞) 1(θ = 1) θ ∞ 1 ·(15) (1 − t1/θ)θ [1,∞) 1(θ = 1) θ 1 0 ·(16) ( θ
t+ 1)(1 − t) [0,∞) 1 + θ 1 ∞ 1 ·
(17) − log (1+t)−θ−12−θ−1
θ2(2θ−1)
1 ∞ 0 0
(18) eθ/(t−1) [2,∞) 0 ∞ e−θ 0 ·(19) eθ/t − eθ (0,∞) θeθ 1 ∞ ∞ ·
(20) et−θ− e (0,∞) θe 1 ∞ ∞ ·
(21) 1 − {1 − (1 − t)θ}1/θ [1,∞) 1(θ = 1) θ 1 0 ·(22) arcsin(1 − tθ) (0, 1] θ 1 π/2 0 ·
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Arthur CHARPENTIER - tails of Archimedean copulas
How to extend to more general dependence structures ?
• mixtures of generators, since convex sums of generators defines a generator,• the α− β transformations in Nelsen (1999), i.e.
φα(t) = φ(tα) and φβ(t) = [φ(t)]β , where α ∈ (0, 1) and β ∈ (1,∞).
• other transformations, e.g.◦ exp(αφ(t))− 1, α ∈ (0,∞),◦ φ(1− [1− t]α), α ∈ (1,∞),◦ φ(αt)− φ(α), α ∈ (0, 1),
=⇒ can be related to distortion of Archimedean copulas.
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Arthur CHARPENTIER - tails of Archimedean copulas
upper tail lower tail
φα(t) range α φ′α(1) θ1(α) φα(0) θ0(α) κ(α)
(1) (φ(t))α (1,∞) 0 αθ1 (φ(0))α αθ0κα
+ 1 − 1α
(2) eαφ(t)−1α
(0,∞) αφ′(1) θ1αφ(0)−1
α∗ ∗
(3) φ(tα) (0, 1) αφ′(1) θ1 φ(0) αθ0 κ
(4) φ(1 − (1 − t)α) (1,∞) 0 αθ1 φ(0) θ0 κ
(5) φ(αt) − φ(α) (0, 1) αφ′(α) 1 φ(0) − φ(α) θ0 κ
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