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Arthur CHARPENTIER - Archimedean copulas.
Les copules Archimédiennes,
quelques motivations et applications
Arthur Charpentier
Katholieke Universiteit Leuven, ENSAE/CREST
Institut de Mathématiques Appliquées, Angers, Novembre 2006
1

“Everybody who opens any journal on stochastic processes, probability theory,
statistics, econometrics, risk management, ﬁnance, insurance, etc., observes
that there is a fast growing industry on copulas [...] The International
Actuarial Association in its hefty paper on Solvency II recommends using
copulas for modeling dependence in insurance portfolios [...] Since Basle II
copulas are now standard tools in credit risk management”.
“Are copulas suitable for modeling multivariate extremes? Copulas generate
any multivariate distribution. If one wants to make an honest analysis of
multivariate extremes the distributions used should be related to extreme
value theory in some way.” Mikosh (2005).
2

“We are thus generally sympathetic to the primary objective pursued by Dr.
Mikosch, which is to caution optimism about what copulas can and cannot
achieve as a dependence modeling tool”.
“Although copula theory has only recently emerged as a distinct field of
investigation, its roots go back at least to the 1940s, with the seminal work of
Hoeőding on margin-free measures of association [...] “It was possibly
Deheuvels who, in a series of papers published around 1980, revealed the full
potential of the fecund link between multivariate analysis and rank-based
statistical techniques”.
“However, the generalized use of copulas for model building (and
Archimedean copulas in particular) seems to have been largely fuelled at the
end of the 1980s by the publication of significant papers by Marshall and
Olkin (1988) and by Oakes (1989) in the influential Journal of the American
Statistical Association”. Genest & Rémillard (2006).
3

Deﬁnition 1. A 2-dimensional copula is a 2-dimensional cumulative
distribution function restricted to [0, 1]2
with standard uniform margins.
Copula (cumulative distribution function) Level curves of the copula
Copula density Level curves of the copula
4

Why using copulas ?
Theorem 2. (Sklar) Let C be a copula, and FX and FY two marginal
distributions, then F(x, y) = C(FX (x), FY (y)) is a bivariate distribution
function, with F ∈ F(FX, FY ).
Conversely, if F ∈ F(FX , FY ), there exists C such that
F(x, y) = C(FX(x), FY (y). Further, if FX and FY are continuous, then C is
unique, and given by
C(u, v) = F(F−1
X (u), F−1
Y (v)) for all (u, v) ∈ [0, 1] × [0, 1]
We will then deﬁne the copula of F, or the copula of (X, Y ).
5

In dimension 2, consider the following family of copulae
Deﬁnition 3. Let ψ denote a convex decreasing function [0, 1] → [0, ∞] such
that ψ(1) = 0. Deﬁne the inverse (or quasi-inverse if ψ(0) < ∞) as
ψ←
(t) =



ψ−1
(t) for 0 ≤ t ≤ ψ(0)
0 for ψ(0) < t < ∞.
Then
C(u1, u2) = ψ←
(ψ(u1) + ψ(u2)), u1, u2 ∈ [0, 1],
is a copula, called an Archimedean copula, with generator ψ.
Note that ψ←
◦ ψ(t) = t on [0, 1]. ψ is said to be strict if ψ(0) = ∞.
The generator is unique up to a multiplicative positive constant.
6

In higher dimension, most of the notions and results can be extended.
Deﬁnition 4. A d-dimensional copula is a d-dimensional cumulative
distribution function restricted to [0, 1]d
with standard uniform margins.
Sklar’s theorem can be extended in dimension d as follows
Theorem 5. (Sklar) Let C be a d-copula, and F1, ..., Fd be marginal
distributions, then F(x1, ..., , xd) = C(F1(x1), ..., Fd(xd)) is a d-dimensional
distribution function, with F ∈ F(F1, ..., Fd).
Conversely, if F ∈ F(F1, ..., Fd), there exists C such that
F(x1, ..., , xd) = C(F1(x1), ..., Fd(xd)). Further, if F1, ..., Fd are continuous,
then C is unique, and given by
C(u1, ..., ud) = F(F−1
1 (u1), ..., F−1
d (ud)) for all (u1, ..., ud) ∈ [0, 1]d
7

Deﬁnition 6. Let ψ be an generator of order d, i.e. ψ is decreasing and
ψ(1) = 0, the inverse ψ−1
is d − 2 times continuously diﬀerentiable on
(0, ∞), and (ψ−1
)(d−2)
is convex. Then
C(u1, ..., ud) = ψ←
(ψ(u1) + ... + ψ(ud)), u1, ..., ud ∈ [0, 1],
is a copula, called an Archimedean copula, with generator ψ.
Note that ψ is a generator in any dimension d if and only if ψ(1) = 0 and
ψ−1
is completely monotone, i.e. (−1)k
(ψ−1
)(k)
(·) ≥ 0 on (0, ∞).
8

• the lower Fréchet bound, ψ(t) = 1 − t, C−
(u, v) = min{u + v − 1, 0},
• the independent copula, ψ(t) = − log t, C⊥
(u, v) = uv,
• the Clayton’s copula, ψ(t) = t−θ
− 1, C(u, v) = (uθ
+ vθ
− 1)−1/θ
,
• the Gumbel’s copula, ψ(t) = (− log t)−θ
,
C(u, v) = exp − (− log u)θ
+ (− log v)θ 1/θ
,
• the Nelsen’s copula, ψ(t) = (1 − t)/t, C(u, v) = uv/(u + v − uv),
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
The lower Fréchet bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
The independent copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
Gumbel’s copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
Clayton’s copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
Nelsen’s copula
9

Some more examples of Archimedean copulas
ψ(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, ∞)
Table 1: Archimedean copulas, from Nelsen (2006).
10

Where do these copulas come from ?
• The conditional independence and frailty approach
Consider two risks, X and Y , such that
X|Θ = θG ∼ E(θG) and Y |Θ = θG ∼ E(θG) are independent,
X|Θ = θB ∼ E(θB) and Y |Θ = θB ∼ E(θB) are independent,
(unobservable good (G) and bad (B) risks).
The following ﬁgures start from 2 classes of risks, then 3, and then a
continuous risk factor θ ∈ (0, ∞).
11

0 5 10 15
05101520
Conditional independence, two classes
−3 −2 −1 0 1 2 3
−3−2−10123
Conditional independence, two classes
Figure 1: Two classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
12

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
Figure 2: Three classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
13

0 20 40 60 80 100
020406080100
Conditional independence, continuous risk factor
−3 −2 −1 0 1 2 3
−3−2−10123
Figure 3: Continuous classes, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
14

0 20 40 60 80 100
020406080100
−3 −2 −1 0 1 2 3
−3−2−10123
Figure 4: Continuous classes, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
15

Assume that, given Θ, X|Θ ∼ E(αΘ) and Y |Θ ∼ E(βΘ) are independent,
P(X > x, Y > y) =
∞
0
P(X > x, Y > y|Θ = θ)π(θ)dθ
=
∞
0
P(X > x|Θ = θ)P(Y > y|Θ = θ)π(θ)dθ
=
∞
0
exp(−αθx) exp(−βθy)π(θ)dθ
=
∞
0
[exp(−[αx + βy]θ)] π(θ)dθ,
where ψ(t) = E(exp −tΘ) = exp(−tθ)π(θ)dθ is the Laplace transform of Θ.
16

Hence P(X > x, Y > y) = φ(αx + βy). Or,
P(X > x) =
∞
0
P(X > x|Θ = θ)π(θ)dθ
=
∞
0
exp(−αθx)π(θ)dθ
= φ(αx),
and thus αx = φ−1
(P(X > x)) (similarly for βy). And therefore,
P(X > x, Y > y) = φ(φ−1
(P(X > x)) + φ−1
(P(Y > y)))
= C(P(X > x), P(Y > y)),
setting C(u, v) = φ(φ−1
(u) + φ−1
(v)) for any (u, v) ∈ [0, 1] × [0, 1].
Using any Laplace transforms, one can generate several families of
multivariate distributions.
Example 7. If Θ is Gamma distributed, the associated copula is Clayton’s.
If Θ has an α-stable distributed, the associated copula is Gumbel’s.
17

This approach can be used in motor insurance ratemaking, and in credit risk.
A finite sequence {X1, ..., Xd} of random variables is exchangeable, or
d-exchangeable, if
(X1, ..., Xd)
L
= Xσ(1), ..., Xσ(d) , (1)
for any permutation σ of {1, ..., d}. More generally, an infinite sequence
{X1, X2...} of random variables is infinitely exchangeable (or simply
exchangeable) if
(X1, X2, ...)
L
= Xσ(1), Xσ(2), ... , (2)
for any finite permutation σ of N∗
(that is Card {i, σ (i) = i} < ∞).
A d-exchangeable sequence {X1, ..., Xd} is called m-extendible (for some
m > d), if (X1, ..., Xd)
L
= (Z1, ..., Zd), where {Z1, ..., Zm} is some
m-exchangeable sequence.
18

Using the formulation of Aldous (1985), de Finetti’s theorem states that“an
inﬁnite exchangeable sequence is a mixture of i.i.d. sequences”: X1, X2, ... of
Bernoulli random variables is exchangeable if and only there is a random
variable Θ, taking values in [0, 1] such that, given Θ = θ the Xi’s are
independent, and Xi ∼ B(θ) (see Schervish (1995) or Chow & Teicher
(1997)).
Example 8. This result can be easily interpreted in credit risk, where
variables of interest are dichotomous (default or non-default). Let X1, X2, ...
be an inﬁnite exchangeable sequence of Bernoulli variables, and let
Sn = X1 + .... + Xn the number of defaults within n companies (for a given
period of time). Then, the distribution of Sn is a mixture of binomial
distributions, i.e. there is a distribution function H on [0, 1] such that
P (Sn = k) =
1
0
n
k
ωk
(1 − ω)
n−k
dH (θ) .
19

• The survival distribution approach
Assume that for a random vector (X, Y ), there exists a convex survival
distribution S, such that S(0) = 1 and
P(X > x, Y > y) = S(x + y),
then the joint survival copula of (X, Y ), such that
P(X > x, Y > y) = C(P(X > x), P(Y > y),
is C(u, v) = S(S−1
(u) + S−1
(v)), which is an Archimedean copula with
generator ψ = S−1
.
This is the notion of Schur-constant survival distribution of random pair
(X, Y ).
Example 9. If S is the survival Pareto distribution, the associated copula is
Clayton’s. If S is the survival Weibull distribution, the associated copula is
Gumbel’s.
20

• The serial iterate approach
A “natural” idea to deﬁne d dimension copula from 2 dimensional copulas
can be the serial iterate approach. Given a 2-copula C2, deﬁne recursively
Cn(u1, . . . , un−1, un) = C2(Cn−1(u1, . . . , un−1), un).
Proposition 10. C is an associative copula, i.e.
C(u, C(v, w) = C(C(u, v), w) for all u, v, w ∈ [0, 1], such that C(u, u) < u for
all u ∈ (0, 1) if and only if C is Archimedean.
Hence, the only copulas that can be constructed by serial iteration are
Archimedean copulas.
Remark 11. This is actually where the word Archimedean comes from.
21

• The distorted copula approach
Deﬁnition 12. A distortion function is a function h : [0, 1] → [0, 1] strictly
increasing such that h(0) = 0 and h(1) = 1.
The set of distortion function will be denoted H.
Note that h ∈ H if and only if h−1
∈ H. Given a copula C, deﬁne
Ch(u, v) = h−1
(C(h(u), h(v))).
If h is convex, then Ch is a copula, called distorted copula.
Example 13. if h(x) = x1/n
, the distorted copula is
Ch(u, v) = Cn
(u1/n
, v1/n
), for all n ∈ N, (u, v) ∈ [0, 1]2
.
if the survival copula of the (Xi, Yi)’s is C, then the survival copula of
(Xn:n, Yn:n) = (max{X1, ..., Xn}, max{Y1, ..., Yn}) is Ch.
Example 14. if C(u, v) = uv = C⊥
(u, v) (the independent copula), and
φ(·) = log h(·), then
Ch(u, v) = h−1
(h(u)h(v)) = φ−1
(φ(u) + φ(v)).
22

• Kendall’s distribution approach
Archimedean copulas can also be characterized through Kendall’s cdf, K,
K(t) = P(C(U, V ) ≤ t), t ∈ [0, 1].
where (U, V ) has cdf C.
Note that K(t) = t − λ(t) where λ(t) = ψ(t)/ψ (t). And conversely ψ is
ψ(u) = ψ(u0) exp
u
u0
1
λ(t)
dt for all 0 < u0 < 1.
23

• Characterizations of some archimedean copulas
1. Frank copula is the only Archimedean such that (U, V )
L
= (1 − U, 1 − V )
(stability by symmetry),
2. Clayton copula is the only Archimedean such that (U, V ) has the same
copula as (U, V ) given (U ≤ u, V ≤ v), for all u, v ∈ (0, 1] (stability by
truncature),
3. Gumbel copula is the only Archimedean such that (U, V ) has the same
copula as (max{U1, ..., Un}, max{V1, ..., Vn}) for all n ≥ 1 (max-stability),
24

Copula density
0.0 0.4 0.8
0.00.51.01.52.0
Archimedean generator
0 1 2 3 4 5 6
0.00.40.8
Laplace Transform
Level curves of the copula
0.0 0.4 0.8
−0.4−0.20.0
Lambda function
0.0 0.4 0.8
0.00.40.8
Kendall cdf
Figure 5: (Independent) Archimedean copula (C = C⊥
, ψ(t) = − log t).
25

Some “famous” Archimedean copulas
Clayton’s copula (Figure 6), with parameter α ∈ [0, ∞) has generator
ψ(x; α) =
x−α
− 1
α
if 0 < α < ∞, with the limiting case ψ(x; 0) = − log(x), for any 0 < x ≤ 1.
The inverse function is the Laplace transform of a Gamma distribution.
The associated copula is
C(u, v; α) = (u−α
+ v−α
− 1)−1/α
if 0 < α < ∞, with the limiting case C(u, v; 0) = C⊥
(u, v), for any
(u, v) ∈ (0, 1]2
.
26

Copula density
0.0 0.4 0.8
0.00.51.01.52.0
0 1 2 3 4 5 6
0.00.20.40.60.81.0
Laplace Transform
0.0 0.4 0.8
−0.4−0.3−0.2−0.10.0
Lambda function
0.0 0.4 0.8
0.00.20.40.60.81.0
Kendall cdf
Figure 6: Clayton’s copula.
27

Gumbel’s copula (Figure 7), with parameter α ∈ [1, ∞) has generator
ψ(x; α) = (− log x)α
if 1 ≤ α < ∞, with the limiting case ψ(x; 0) = − log(x), for any 0 < x ≤ 1.
The inverse function is the Laplace transform of a 1/α-stable distribution.
The associated copula is
C(u, v; α) = −
1
α
log 1 +
(e−αu
− 1) (e−αv
− 1)
e−α − 1
,
if 1 ≤ α < ∞, for any (u, v) ∈ (0, 1]2
.
28

Copula density
0.0 0.4 0.8
0.00.51.01.52.0
0 1 2 3 4 5 6
0.00.20.40.60.81.0
Laplace Transform
0.0 0.4 0.8
−0.4−0.3−0.2−0.10.0
Lambda function
0.0 0.4 0.8
0.00.20.40.60.81.0
Kendall cdf
Figure 7: Gumbel’s copula.
29

How to deﬁne more general parametric families ?
Given an Archimedean generator ψ, deﬁne ψα and ψβ as follows,
ψα,1(x) = ψ(xα
) and ψ1,β(x) = ψ(x)β
,
where β ≥ 1 and α ∈ (0, 1]. Note that a composite family can also be
considered, ψα,β(x) = [ψ(xα
)]β
.
30

Distorted Archimedean copula, from Frank
β = 1
α = 1
β = 3 2
α = 1
β = 1
α = 1 2
β = 3 2
α = 1 2
Figure 8: Distorted Archimedean copula (φα,β), from Frank copula.
31

A short word on the zero-set area
Define the zero-set boundary curve φ(u) + φ(v) = φ(0). If φ(0) = ∞, the
zero-set boundary curve has a null measure (e.g. Clayton’s copula). If
φ(0) < ∞, zero-set boundary curve is −φ(0)/φ (0) (e.g. Copula 4.2.2 in
Nelsen (2006)).
Example 15. Clayton’s copula can be defined for θ ∈ [−1, 0) ∪ (0, +∞), as
C(u, v) = max{ u−θ
+ v−θ
− 1
−1/θ
, 0}.
If θ < 0, this copula has a zero-set, below the curve y = (1 − x−θ
)−1/θ
Example 16. Consider Copula 4.2.2 in Nelsen (2006), defined for
θ ∈ [1, +∞), as
C(u, v) = max{1 − (1 − u)θ
+ (1 − v)θ 1/θ
, 0},
with generator (1 − t)θ
.
32

0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Generating Clayton’s copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Generating Clayton’s copula
Figure 9: Clayton’s copula, θ = 2 and θ = −1/2.
33

0.0 0.2 0.4 0.6 0.8 1.0
0.20.40.60.81.0
Generating copula 4.2.2
0.0 0.2 0.4 0.6 0.8 1.0
0.20.40.60.81.0
Generating copula 4.2.2
Figure 10: Copula 4.2.2, θ = 2 and θ = 5.
34

Sequences of Archimedean copulas
Extension of results due to Genest & Rivest (1986),
Proposition The ﬁve following statements are equivalent,
(i) lim
n→∞
Cn(u, v) = C(u, v) for all (u, v) ∈ [0, 1]2
,
(ii) lim
n→∞
ψn(x)/ψn(y) = ψ(x)/ψ (y) for all x ∈ (0, 1] and y ∈ (0, 1) such that
ψ such that is continuous in y,
(iii) lim
n→∞
λn(x) = λ(x) for all x ∈ (0, 1) such that λ is continuous in x,
(iv) there exists positive constants κn such that limn→∞ κnψn(x) = ψ(x) for
all x ∈ [0, 1],
(v) lim
n→∞
Kn(x) = K(x) for all x ∈ (0, 1) such that K is continuous in x.
35

Proposition 17. The four following statements are equivalent
(i) lim
n→∞
Cn(u, v) = C+
(u, v) = min(u, v) for all (u, v) ∈ [0, 1]2
,
(ii) lim
n→∞
λn(x) = 0 for all x ∈ (0, 1),
(iii) lim
n→∞
ψn(y)/ψn(x) = 0 for all 0 ≤ x < y ≤ 1,
(iv) lim
n→∞
Kn(x) = x for all x ∈ (0, 1).
Note that one can get non Archimedean limits,
0.0 0.4 0.8
051015
0.0 0.4 0.8
0.00.20.40.60.81.0
Sequence of generators and Kendall cdf’s
36

Statistical inference for Archimedean copulas
Recall that the Archimedean generator can be expressed as
ψ(u) = ψ(u0) exp
u
u0
1
λ(t)
dt for all 0 < u0 < 1,
where λ(t) = t − K(t), K begin Kendall’s distribution function, i.e.
K(t) = P(F(X, Y ) ≤ t), t ∈ [0, 1].
where F(x, y) = P(X ≤ x, Y ≤ y).
Note that K can be written
K(t) = Pr[F(X, Y ) ≤ t] = E[1{F(X, Y ) ≤ t}]
=
∞
0
∞
0
1[F(x, y) ≤ t]dF(x, y).
37

Thus, a natural estimator can be deﬁned as
K(t) =
∞
0
∞
0
1[F(x, y) ≤ t]dF(x, y)
where F is a nonparametric estimate of the joint cdf F.
Hence, a natural estimate for φ is then
φ(u) = exp
u
u0
1
λ(t)
dt = exp
u
u0
1
t − K(t)
dt
which leads to Cφ (see Genest & Rivest (1993)).
38

−2 −1 0 1 2
−2−1012
Scatterplot
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Kendall’s distribution function
0.0 0.2 0.4 0.6 0.8 1.0
−0.25−0.20−0.15−0.10−0.050.00
Lambda function
0.0 0.2 0.4 0.6 0.8 1.0
0123456
Figure 11: Estimation of the Archimedean generator, n = 100.
39

−2 −1 0 1 2
−2−1012
Scatterplot
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Kendall’s distribution function
0.0 0.2 0.4 0.6 0.8 1.0
−0.25−0.20−0.15−0.10−0.050.00
Lambda function
0.0 0.2 0.4 0.6 0.8 1.001234567
Figure 12: Estimation of the Archimedean generator, n = 100.
40

Generating Archimedean copulas
In dimension 2, recall that if (U, V ) has joint distribution C.
P (V ≤ v|U = u) = lim
h→0
P (V ≤ v|u ≤ U < u + h)
= lim
h→0
P (U < u + h, V ≤ v) − P (U < u, V ≤ v)
P (U < u + h) − P (U < u)
= lim
h→0
C(u + h, v) − C(u, v)
(u + h) − u
= lim
h→0
C(u + h, v) − C(u, v)
(u + h) − u
=
∂C
∂u (u,v)
.
The general algorithm is then
U ← Random, and V ←
∂C(U, ·)
∂u
−1
(Random),
Genest & MacKay (1986), Genest (1987) and Lee (1993) proposed the
following algorithm to generate random vectors (X, Y ) with Archimedean
41

copula.
U ⇐= Random,
V is the solution of Random =
(φ−1
) (V ))
(φ−1) (0)
.
Note that other algorithms can be used, to generate pairs (U, V )
U ⇐=Random, T ⇐=Random,
W ⇐= (φ )−1
(φ (U)/T)
V ⇐= φ−1
(φ(W) − φ(U))
Or equivalently
W ⇐= K−1
(Random), S ⇐=Random,
U ⇐= φ−1
(Sφ(W))
V ⇐= φ−1
((1 − S)φ(W))
42

Histogram, first component
0.0 0.2 0.4 0.6 0.8 1.0
020406080100
−2 0 2 4
−3−2−1012
Gaussian distribution N(0,1)
GaussiandistributionN(0,1)0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Uniform distribution on [0,1]
Uniformdistributionon[0,1]
Histogram, second component
0.0 0.2 0.4 0.6 0.8 1.0
020406080100
Figure 13: Simulations using Clayton’s copula.
43

Tails for Archimedean copulas (Fisher-Tippett)
Standard approach, introduce λL and λU (lower and upper tail indices),
λL = lim
u→0
P X ≤ F−1
X (u) |Y ≤ F−1
Y (u) ,
λU = lim
u→1
P X > F−1
X (u) |Y > F−1
Y (u) .
Those measures are copula based, i.e. λL = lim
u→0
C(u, u)
u
For Archimedean copulas, note that
λU = 2 − lim
x→0
1 − φ−1
(2x)
1 − φ−1(x)
and λL = lim
x→0
φ−1
(2φ(x))
x
= lim
x→∞
φ−1
(2x)
φ−1(x)
.
44

ψ(t) range θ λL λU
(1) 1
θ
(t−θ − 1) [−1, 0) ∪ (0, ∞) max(2−1/θ, 0) 0
(2) (1 − t)θ [1, ∞) 0 2 − 21/θ
(3) log
1−θ(1−t)
t
[−1, 1) 0 0
(4) (− log t)θ [1, ∞) 0 2 − 21/θ
(5) − log e−θt−1
e−θ−1
(−∞, 0) ∪ (0, ∞) 0 0
(6) − log{1 − (1 − t)θ} [1, ∞) 0 2 − 21/θ
(7) (θ − 1) log{θt + (1 − θ)} (0, 1] 0 0
(8) 1−t
1+(θ−1)t
[1, ∞) 0 0
(9) log(1 − θ log t) (0, 1] 0 0
(10) log(2t−θ − 1) (0, 1] 0 0
(11) log(2 − tθ) (0, 1/2] 0 0
(12) ( 1
t
− 1)θ [1, ∞) 2−1/θ 2 − 21/θ
(13) (1 − log t)θ − 1 (0, ∞) 0 0
(14) (t−1/θ − 1)θ [1, ∞) 1/2 2 − 21/θ
(15) (1 − t1/θ)θ [1, ∞) 0 2 − 21/θ
(16) ( θ
t
+ 1)(1 − t) [0, ∞) 1/2 0
(17) − log
(1+t)−θ−1
2−θ−1
(−∞, 0) ∪ (0, ∞) 0 0
(18) eθ/(t−1) [2, ∞) 0 1
(19) eθ/t − eθ (0, ∞) 1 0
(20) e−tθ
− e (0, ∞) 1 0
(21) 1 − {1 − (1 − t)θ}1/θ [1, ∞) 0 2 − 21/θ
(22) arcsin(1 − tθ) (0, 1] 0 0
45

Truncature of Archimedean copulas
Deﬁnition 18. Let U = (U1, ..., Un) be a random vector with uniform
margins, and distribution function C. Let Cr denote the copula of random
vector
(U1, ..., Un)|U1 ≤ r1, ..., Ud ≤ rd, (3)
where r1, ..., rd ∈ (0, 1].
If Fi|r(·) denotes the (marginal) distribution function of Ui given
{U1 ≤ r1, ..., Ui ≤ ri, ..., Ud ≤ rd} = {U ≤ r},
Fi|r(xi) =
C(r1, ..., ri−1, xi, ri+1, ..., rd)
C(r1, ..., ri−1, ri, ri+1, ..., rd)
,
and therefore, the conditional copula (or truncated copula) is
Cr(u) =
C(F←
1|r(u1), ..., F←
d|r(ud))
C(r1, ..., rd)
. (4)
46

Proposition 19. The class of Archimedean copulae is stable by truncature.
More precisely, if U has cdf C, with generator ψ, U given {U ≤ r}, for any
r ∈ (0, 1]d
, will also have an Archimedean generator, with generator
ψr(t) = ψ(tc) − ψ(c) where c = C(r1, ..., rd).
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.02.53.0 Generators of conditional Archimedean copulae
(1) (2)
(3)
47

Tails for Archimedean copulas (Pickands-Balkema-de Haan)
Proposition 20. Let C be an Archimedean copula with generator ψ, and
0 ≤ α ≤ ∞. If C(·, ·; α) denote Clayton’s copula with parameter α.
(i) limu→0 Cu(x, y) = C(x, y; α) for all (x, y) ∈ [0, 1]2
;
(ii) −ψ ∈ R−α−1.
(iii) ψ ∈ R−α.
(iv) limu→0 uψ (u)/ψ(u) = −α.
If α = 0 (tail independence),
(i) ⇐⇒ (ii)=⇒(iii) ⇐⇒ (iv),
and if α ∈ (0, ∞] (tail dependence),
(i) ⇐⇒ (ii) ⇐⇒ (iii) ⇐⇒ (iv).
48

Proposition 21. There exists Archimedean copulae, with generators having
continuous derivatives, slowly varying such that the conditional copula does
not convergence to the independence.
Generator ψ integration of a function piecewise linear, with knots 1/2k
,
If −ψ ∈ R−1, then ψ ∈ Πg (de Haan class), and not ψ /∈ R0.
This generator is slowly varying, with the limiting copula is not C⊥
.
Note that lower tail index is
λL = lim
u↓0
C(u, u)
u
= 2−1/α
,
with proper interpretations for α equal to zero or inﬁnity (see e.g. Theorem
3.9 of Juri and Wüthrich (2003)).
49

ψ(t) range θ α
(1) 1
θ
(t−θ − 1) [−1, 0) ∪ (0, ∞) max(θ, 0)
(2) (1 − t)θ [1, ∞) 0
(3) log
1−θ(1−t)
t
[−1, 1) 0
(4) (− log t)θ [1, ∞) 0
(5) − log e−θt−1
e−θ−1
(−∞, 0) ∪ (0, ∞) 0
(6) − log{1 − (1 − t)θ} [1, ∞) 0
(7) (θ − 1) log{θt + (1 − θ)} (0, 1] 0
(8) 1−t
1+(θ−1)t
[1, ∞) 0
(9) log(1 − θ log t) (0, 1] 0
(10) log(2t−θ − 1) (0, 1] 0
(11) log(2 − tθ) (0, 1/2] 0
(12) ( 1
t
− 1)θ [1, ∞) θ
(13) (1 − log t)θ − 1 (0, ∞) 0
(14) (t−1/θ − 1)θ [1, ∞) 1
(15) (1 − t1/θ)θ [1, ∞) 0
(16) ( θ
t
+ 1)(1 − t) [0, ∞) 1
(17) − log
(1+t)−θ−1
2−θ−1
(−∞, 0) ∪ (0, ∞) 0
(18) eθ/(t−1) [2, ∞) 0
(19) eθ/t − eθ (0, ∞) ∞
(20) e−tθ
− e (0, ∞) 0
(21) 1 − {1 − (1 − t)θ}1/θ [1, ∞) 0
(22) arcsin(1 − tθ) (0, 1] 0
50

Tails for Archimedean copulas (Pickands-Balkema-de Haan)
Analogy with lower tails.
Recall that ψ(1) = 0, and therefore, using Taylor’s expansion yields
ψ(1 − s) = −sψ (1) + o(s) as s → 0.
And moreover, since ψ is convex, if ψ(1 − ·) is regularly varying with index
α, then necessarily α ∈ [1, ∞). If if (−D)ψ(1) > 0, then α = 1 (but the
converse is not true).
0.5 0.6 0.7 0.8 0.9 1.0
0.00.10.20.30.40.50.60.7
Archimedean copula at 1
0.5 0.6 0.7 0.8 0.9 1.0
0.00.10.20.30.40.50.6
0.5 0.6 0.7 0.8 0.9 1.0
0.000.020.040.060.080.100.12
0.5 0.6 0.7 0.8 0.9 1.0
−0.020.000.020.040.060.080.10
51

Proposition 22. Let C be an Archimedean copula with generator ψ.
Assume that f : s → ψ(1 − s) is regularly varying with index α ∈ [1, ∞) and
that −ψ (1) = κ. Then three cases can be considered
(i) if α ∈ (1, ∞), case of asymptotic dependence,
(ii) if α = 1 and if κ = 0, case of dependence in independence,
(iii) if α = 1 and if κ > 0, case of independence in independence.
52

ψ(t) range θ α κ
(1) 1
θ
(t−θ − 1) [−1, 0) ∪ (0, ∞) 1 1
(2) (1 − t)θ [1, ∞) θ 0
(3) log
1−θ(1−t)
t
[−1, 1) 1 1 − θ
(4) (− log t)θ [1, ∞) θ 0
(5) − log e−θt−1
e−θ−1
(−∞, 0) ∪ (0, ∞) 1 θe−θ
e−θ−1
(6) − log{1 − (1 − t)θ} [1, ∞) θ 0
(7) − log{θt + (1 − θ)} (0, 1] 1 θ
(8) 1−t
1+(θ−1)t
[1, ∞) 1 1/θ
(9) log(1 − θ log t) (0, 1] 1 θ
(10) log(2t−θ − 1) (0, 1] 1 2θ
(11) log(2 − tθ) (0, 1/2] 1 θ
(12) ( 1
t
− 1)θ [1, ∞) θ 0
(13) (1 − log t)θ − 1 (0, ∞) 1 θ
(14) (t−1/θ − 1)θ [1, ∞) θ 0
(15) (1 − t1/θ)θ [1, ∞) θ 0
(16) ( θ
t
+ 1)(1 − t) [0, ∞) 1 θ + 1
(17) − log
(1+t)−θ−1
2−θ−1
(−∞, 0) ∪ (0, ∞) 1 −θ2−θ−1
2−θ−1
(18) eθ/(t−1) [2, ∞) ∞ 0
(19) eθ/t − eθ (0, ∞) 1 θeθ
(20) e−tθ
− e (0, ∞) 1 θe
(21) 1 − {1 − (1 − t)θ}1/θ [1, ∞) θ 0
(22) arcsin(1 − tθ) (0, 1] 1 θ
(·) (1 − t) log(t − 1) 1 0
53

0.0 0.2 0.4 0.6 0.8 1.0
0246810
Archimedean copula density on the diagonal
Dependence
Dependence in independence
Independence in independence
Copula density
54

A short word on hierarchical copulas
As pointed out in Genest, Quesada Molina & Rodríguez Lallena
(1995), or Li, Scarsini & Shaked (1996) it is usually difficult to define
copulas as follows,
C(u1, u2, u3, u4) = C0(C1,2(u1, u2), C3,4(u3, u4))
where only 2-dimensional copulas are considered. But this can be done
simply in the case of Archimedean copulas. Consider φ, ψ and λ three
Archimedean copulas, and set
C(u1, u2, u3, u4) = λ←
(λ ◦ ψ←
(ψ(u1) + ψ(u2)) + λ ◦ φ←
(φ(u3) + φ(u4))).
Under weak conditions such a function defines a 4 dimensional copula.
55

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