Tail dependence copulas provide a natural perspective from which one can study the dependence in the tail of a multivariate distribution.For Archimedean copulas with continuously differentiable generators, regular variation of the generator near the origin is known to be closely connected to convergence of the corresponding lower tail dependence copulas to the Clayton copula.In this paper, these characterizations are refined and extended to the case of generators which are not necessarily continuously differentiable.Moreover, a counterexample is constructed showing that even if the generator of a strict Archimedean copula is continuously differentiable and slowly varying at the origin, then the lower tail dependence copulas do not need to converge to the independent copula.Archimedean copula;regular variation;tail dependence;de Haan class

Charpentier, A.

Segers, J.J.J.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2006–29 
 
LOWER TAIL DEPENDENCE FOR ARCHIMEDEAN COPULAS: 
CHARACTERIZATIONS AND PITFALLS 
 
By Arthur Charpentier, Johan Segers 
 
April 2006 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 Lower tail dependence for Archimedean
copulas: characterizations and pitfalls
Arthur Charpentier a
aENSAE-CREST, Timbre J120, 3 avenue Pierre Larousse, 92240 Malakoﬀ,
France
Johan Segers b,1
bTilburg University, P.O. Box 90153, NL-5000 LE Tilburg, the Netherlands
Abstract
Tail dependence copulas provide a natural perspective from which one can study the
dependence in the tail of a multivariate distribution. For Archimedean copulas with
continuously diﬀerentiable generators, regular variation of the generator near the
origin is known to be closely connected to convergence of the corresponding lower tail
dependence copulas to the Clayton copula. In this paper, these characterizations are
reﬁned and extended to the case of generators which are not necessarily continuously
diﬀerentiable. Moreover, a counterexample is constructed showing that even if the
generator of a strict Archimedean copula is continuously diﬀerentiable and slowly
varying at the origin, then the lower tail dependence copulas do not need to converge
to the independent copula.
JEL: C14, C16
Key words: Archimedean copula, Regular variation, Tail dependence, de Haan
class
1 Introduction
In ﬁnancial and actuarial risk management, appropriate models for depen-
dence between risks are of obvious importance (e.g. B¨ auerle and M¨ uller, 1987;
Email addresses: arthur.charpentier@ensae.fr (Arthur Charpentier),
jsegers@uvt.nl (Johan Segers).
1 Supported by a VENI grant of the Netherlands Organization for Scientiﬁc Re-
search (NWO).
Preprint submitted to Elsevier Science 14 April 2006Frees and Valdez, 1998; Klugman and Parsa, 1999; Dhaene et al., 2005). Cop-
ulas form a widely accepted tool for building such dependence models. A
versatile subclass of copulas is the one of Archimedean copulas, introduced
in Kimberling (1974) and studied intensively since (e.g. Genest and MacKay,
1986; Genest and Rivest, 1993; M¨ uller and Scarsini, 2004). The need for ac-
curate modelling of extremal events then requires a better understanding of
the behavior of these and other copulas in the tails.
In Juri and W¨ uthrich (2002), tail dependence for bivariate Archimedean copu-
las is described using the concept of lower tail dependence copulas. The lower
tail dependence copula of a copula C at level 0 < u < 1 is deﬁned as the cop-
ula of the conditional distribution of a random pair (U,V ) with distribution
function C when conditioned to be contained in the square [0,u]2.
The central topic in Juri and W¨ uthrich (2002) is the asymptotic behavior of the
lower tail dependence copula of a strict Archimedean copula as the threshold
u decreases to zero. The main result is that, if the generator is continuously
diﬀerentiable, then regular variation of the generator near zero is equivalent
to convergence of the lower tail dependence copula to a Clayton copula, the
parameter of the latter being determined by the index of regular variation of
the generator. The key role of the Clayton copula was also recognized later in
Charpentier and Juri (2004) or Bassan and Spizzichino (2005).
Our aim, then, is twofold: First, we extend the results in Juri and W¨ uthrich
(2002) to the case of generators that are not necessarily continuously diﬀer-
entiable. Here we rely on results in Charpentier and Segers (2006), extend-
ing known characterizations for convergence of Archimedean copulas (Nelsen,
1999, Theorems 4.4.7 and 4.4.8) by removing redundant smoothness assump-
tions. Second, we correct the statement in Theorem 3.5 in Juri and W¨ uthrich
(2002) that slow variation of the generator implies convergence of the lower
tail dependence copula to the independent one. Indeed, we construct a coun-
terexample contradicting the previous claim.
The outline of the paper is as follows: after some preliminaries in section 2,
the main result on convergence of lower tail dependence copulas is stated and
proven in section 3. A counterexample is constructed in section 4, followed by a
technical discussion in section 5 on the relation between the various conditions
imposed on the behaviour of the generator near the origin.
2 Preliminaries
A function C : [0,1]2 → [0,1] is called a bivariate copula if it is the restriction
to [0,1]2 of a bivariate distribution function whose marginals are given by the
2uniform distribution on the interval [0,1]. A function ψ : [0,1] → [0,∞] is
called a strict generator if it is decreasing, convex, ψ(0) = ∞ and ψ(1) = 0.
The inverse function of a strict generator ψ is denoted by ψ−1. A function
C : [0,1]2 → [0,1] is called a strict Archimedean copula if there exists a strict
generator ψ such that
C(u,v) = ψ
−1{ψ(u) + ψ(v)}, (u,v) ∈ [0,1]
2.
Note that the generator is unique up to a multiplicative constant. A strict
Archimedean copula is a copula. See the survey monograph by Nelsen (1999)
and the references therein for more details.
Let C be a copula and let (U,V ) be a random pair with joint distribution
function C. Let 0 < u < 1 be such that C(u,u) > 0. The lower tail depen-
dence copula relative to C at level u is deﬁned as the copula, Cu, of the joint
distribution of (U,V ) conditionally on the event {U ≤ u,V ≤ u}. Formally,
Cu(x,y) =
C(x0,y0)
C(u,u)
where 0 ≤ x0 ≤ u and 0 ≤ y0 ≤ u are the solutions to the equations
C(x0,u)
C(u,u)
= x and
C(u,y0)
C(u,u)
= y;
see Juri and W¨ uthrich (2002, Deﬁnition 3.1) or Juri and W¨ uthrich (2003,
Deﬁnition 2.2). Upper tail dependence copulas are deﬁned in a similar way
(Juri and W¨ uthrich, 2003, Deﬁnition 2.1). Moreover, the deﬁnition can be
extended by allowing the thresholds for the two margins to be diﬀerent, that
is, by conditioning on the event {U ≤ u,V ≤ v}, where (u,v) ∈ (0,1]2 are
such that C(u,v) > 0, see Charpentier and Juri (2004, Deﬁnition 2.5). In this
note, we will be interested only in the diagonal.
If C is a strict bivariate Archimedean copula with generator ψ, then the lower
tail dependence copula relative to C at level u is given by the strict Archime-
dean copula with generator ψu deﬁned by
ψu(t) = ψ(tv) − ψ(v), 0 ≤ t ≤ 1, (1)
where v = v(u) = ψ−1{2ψ(u)} (Juri and W¨ uthrich, 2002, Proposition 3.2).
Since v(u) → 0 as u ↓ 0, the asymptotic behaviour of the lower tail dependence
copula Cu as u ↓ 0 depends on the asymptotic behaviour of ψ near the origin.
A positive, measurable function f deﬁned in a right-neighbourhood of zero is
said to be regularly varying at zero of index τ ∈ R if
lim
u↓0
f(ux)
f(u)
= x
τ, 0 < x < ∞;
3notation: f ∈ Rτ. If τ = 0, then the limit is equal to one for all 0 < x < ∞;
in this case, f is said to be slowly varying at zero. A limiting case is obtained
when τ = −∞: f is said to be rapidly varying at zero of index −∞, notation
f ∈ R−∞, if
lim
u↓0
f(ux)
f(u)
=

     
     
0 if 1 < x < ∞,
1 if x = 1,
∞ if 0 < x < 1.
Classically, regular (and rapid) variation are considered at inﬁnity rather than
at zero. However, it is typically straightforward to translate results from regu-
lar variation at inﬁnity to regular variation at zero by considering the function
y 7→ f(1/y) (see for instance Bingham et al., 1987, p. 18).
The Clayton copula with parameter α ∈ [0,∞) is the Archimedean copula
with strict generator given by
ψ(x;α) =
Z 1
x
t
−α−1dt =

  
  
x−α − 1
α
if 0 < α < ∞,
−log(x) if α = 0
for 0 < x ≤ 1; the corresponding copula is
C(x,y;α) =

 
 
(x−α + y−α − 1)−1/α if 0 < α < ∞,
xy if α = 0
for (x,y) ∈ (0,1]2. Note that limα↓0 ψ(t;α) = ψ(t;0) and limα↓0 C(x,y;α) =
C(x,y;0). The comonotone copula, which is itself not an Archimedean copula,
arises as the limit of the Clayton copula as α → ∞, that is,
C(x,y;∞) = lim
α→∞C(x,y;α) = min(x,y)
for (x,y) ∈ [0,1]2. The Clayton copula has the special property that at every
level 0 < u < 1, its lower tail dependence copula is again a Clayton copula
and with the same parameter; see also Charpentier and Juri (2004, Proposi-
tion 4.15).
3 Main result
Our main result, Theorem 1, can be seen as an extension of Theorems 3.3, 3.5
and 3.6 of Juri and W¨ uthrich (2002). The asymptotic behavior of lower tail
dependence copulas for general symmetric bivariate copulas is studied in Juri
4and W¨ uthrich (2003), and for nonsymmetric bivariate copulas in Charpentier
and Juri (2004).
Theorem 1 Let C be a strict Archimedean copula with generator ψ, whose
right-hand derivative is denoted by ψ0. Let 0 ≤ α ≤ ∞. Consider the following
four statements:
(i) limu↓0 Cu(x,y) = C(x,y;α) for all (x,y) ∈ [0,1]2;
(ii) −ψ0 ∈ R−α−1.
(iii) ψ ∈ R−α.
(iv) limu↓0 uψ0(u)/ψ(u) = −α.
If α = 0 (tail independence),
(i) ⇐⇒ (ii) =⇒ (iii) ⇐⇒ (iv),
and if α ∈ (0,∞] (tail dependence),
(i) ⇐⇒ (ii) ⇐⇒ (iii) ⇐⇒ (iv).
As we will see in Theorem 2, if α = 0, then (iii) does not imply (i) or (ii), in
contradiction to Lemma 3.4 and Theorem 3.5 of Juri and W¨ uthrich (2002).
Proof. Recall from (1) that the lower tail dependence copula of C at 0 < u < 1
is the Archimedean copula with generator ψu(x) = ψ(xv)−ψ(v) for 0 ≤ x ≤ 1,
where v ≡ v(u) = ψ−1{2ψ(u)}. Note that v(u) is continuous in u and decreases
to 0 as u decreases to zero. The (right-hand) derivative of ψu at 0 < y < 1 is
equal to ψ0
u(y) = vψ0(vy).
• The case α ∈ (0,∞) (tail dependence)
(i) implies (ii). By Charpentier and Segers (2006, Proposition 2) [extending
Nelsen (1999, Theorem 4.4.7) and Genest and MacKay (1986, Proposition 4.2)
to the case of generators which are not twice continuously diﬀerentiable], for
0 < x ≤ 1,
lim
u↓0
ψu(x)
ψ0
u(1/2)
=
ψ(x;α)
ψ0(1/2;α)
= −2
αψ(x;α).
Deﬁne g(v) = −ψ0
u(1/2) = −vψ0(v/2). By the previous display, for 0 < x ≤ 1,
lim
v↓0
ψ(vx) − ψ(v)
g(v)
= −2
αψ(x;α).
For 0 < x ≤ 1, we get
5g(vx)
g(v)
=
 
ψ(vx2) − ψ(v)
g(v)
−
ψ(vx) − ψ(v)
g(v)
! ,
ψ(vx2) − ψ(vx)
g(vx)
→{ψ(x
2;α) − ψ(x;α)}/ψ(x;α) = x
−α, as v ↓ 0.
Hence, g ∈ R−α, and thus −ψ0 ∈ R−α−1.
(ii) implies (i). If −ψ0 ∈ R−α−1, then for all 0 < x < 1 and 0 < y < 1,
ψu(x)
ψ0
u(y)
=
ψ(vx) − ψ(v)
vψ0(vy)
= −
Z v
vx
ψ0(t)
vψ0(vy)
dt = −
Z 1
x
ψ0(vt)
ψ0(vy)
dt
→−
Z 1
x
 
t
y
!−α−1
dt =
ψ(x;α)
ψ0(y;α)
, u ↓ 0.
By Charpentier and Segers (2006, Proposition 2), extending Nelsen (1999,
Theorem 4.4.7), we ﬁnd that (i) must hold.
(iii) implies (iv). This follows from the Monotone Density Theorem (Bingham
et al., 1987, Theorem 1.7.2) applied to the function x 7→ ψ(1/x).
(iv) implies (iii). This follows from the Representation Theorem for regularly
varying functions (Bingham et al., 1987, equation (1.5.2)).
So far, we have established the equivalences (i) ⇐⇒ (ii) and (iii) ⇐⇒ (iv).
(ii) implies (iii). This follows from Karamata’s Theorem (Bingham et al.,
1987, Proposition 1.5.8) applied to the function x 7→ ψ(1/x).
(iii) and (iv) imply (ii). This is immediate, since −ψ0(x) ∼ αx−1ψ(x) as x ↓ 0
and ψ ∈ R−α.
• The case α = 0 (tail independence)
The proofs of all the implications, except for the last one, also hold when
α = 0.
• The case α = ∞ (tail comonotonicity)
By Charpentier and Segers (2006, Proposition 3), extending Nelsen (1999,
Theorem 4.4.8) and Genest and MacKay (1986, Proposition 4.3), (i) is equiv-
alent to
lim
u↓0
ψu(x)
ψ0
u(x)
= 0, 0 < x ≤ 1.
Combine the above three displays to ﬁnd that (i) is equivalent to
lim
v↓0
ψ(vx) − ψ(v)
vψ0(vx)
= 0, 0 < x ≤ 1. (2)
6We show ﬁrst the circle of implications (i) =⇒ (ii) =⇒ (iv) =⇒ (i) and then
the equivalence (iii) ⇐⇒ (iv).
(i) implies (ii). Since ψ is decreasing and convex,
0 ≤ (x − 1)vψ
0(v) ≤ ψ(vx) − ψ(v), 0 < x ≤ 1;0 < v ≤ 1.
Since (i) is equivalent to (2), the above inequality implies
lim
v↓0
ψ0(v)
ψ0(vx)
= 0, 0 < x < 1.
Hence ψ0 ∈ R−∞.
(ii) implies (iv). Let 1 < x < ∞. There exists 0 < u0 ≤ 1/x such that
ψ0(ux)
ψ0(u)
≤
1
2x
, 0 < u ≤ u0.
Let 0 < u ≤ u0 and let k = 0,1,2,... be such that uxk < u0 ≤ uxk+1. Since ψ
is decreasing and convex,
ψ(u)=
k X
j=0
{ψ(ux
j) − ψ(ux
j+1)} + ψ(ux
k+1)
≤
k X
j=0
ux
j(1 − x)ψ
0(ux
j) + ψ(u0)
≤uψ
0(u)(1 − x)
k X
j=0
x
j 1
(2x)j + ψ(u0)
≤2(1 − x)uψ
0(u) + ψ(u0).
Since ψ is strict, there exists 0 < u1 < u0 such that ψ(u) ≥ 2ψ(u0) for all
0 < u ≤ u1. Hence, by the previous display,
ψ(u) ≤ 4(1 − x)uψ
0(u), 0 < u ≤ u1.
Let u decrease to zero to ﬁnd
limsup
u↓0
ψ(u)
−uψ0(u)
≤ 4(x − 1).
Since x was an arbitrary element in (1,∞), we arrive at (iv).
(iv) implies (i). Since (i) is equivalent to (2), it is suﬃcient to show that (iv)
implies (2). Let 0 < v ≤ 1 and 0 < x < 1. We have

 


ψ(vx) − ψ(v)
vψ0(vx)

 

 ≤
ψ(vx)
v|ψ0(vx)|
≤
ψ(vx)
vx|ψ0(vx)|
.
7By (iv), the right-hand side of this equation tends to zero as v ↓ 0, whence
(2), as required.
(iii) implies (iv). Let 0 < u < 1 and 1 < x < 1/u. Since ψ is convex,
ψ(u) − ψ(ux) ≤ (1 − x)uψ
0(u).
By (iii), limu↓0 ψ(ux)/ψ(u) = 0 for every 1 < x < ∞. Divide both sides of the
inequality in the previous display by ψ(u) and let u decrease to zero to ﬁnd
liminf
u↓0
−uψ0(u)
ψ(u)
≥
1
x − 1
, 1 < x < ∞.
The right-hand side in the previous display becomes arbitrarily large as x ↓ 1,
whence (iv).
(iv) implies (iii). Let 0 < x < 1. Since ψ is convex, we have for 0 < u ≤ 1,
ψ(ux) − ψ(u) ≥ (x − 1)uψ
0(u),
whence
ψ(ux)
ψ(u)
≥ (x − 1)
uψ0(u)
ψ(u)
+ 1.
By (iv), the right-hand side side of this inequality tends to inﬁnity as u ↓ 0,
yielding (iii). 2
4 Counterexample
We claim in Theorem 1 that for general α ∈ [0,∞], statements (i) and (ii)
imply statements (iii) and (iv). If α > 0, the converse is also true. However, if
α = 0, then the converse does not hold, as shown by the following counterex-
ample, contradicting Juri and W¨ uthrich (2002, Theorem 3.5).
Theorem 2 There exists a strict Archimedean copula C whose generator ψ
is continuously diﬀerentiable and slowly varying at the origin, but such that
the lower tail dependence copula of C at level u does not converge to the
independence copula as u ↓ 0.
Proof. Let f : (0,1] → R be the piece-wise linear function with knots
f(2
−k) = 2
k, k = 0,1,2,....
That is, f is the linear interpolation of the function (0,1] 3 x 7→ x−1 at the
points {2−k | k = 0,1,2,...}. Deﬁne the function ψ : [0,1] → [0,∞] by
ψ(s) =
Z 1
s
f(x)dx, s ∈ [0,1].
8By construction, the function ψ is continuously diﬀerentiable with derivative
ψ0 = −f. Since f is decreasing, ψ0 is increasing, whence ψ is convex. Hence,
ψ is a strict generator.
As s−1 ≤ f(s) ≤ 2s−1 for all s ∈ (0,1], we have ψ(s) ≥ log(1/s) and thus
0 ≤
sf(s)
ψ(s)
≤
2
log(1/s)
→ 0, as s ↓ 0.
Hence, as ψ is convex, for every 1 < x < ∞,
0 ≤ 1 −
ψ(sx)
ψ(s)
≤
s(1 − x)f(s)
ψ(s)
→ 0, as s ↓ 0.
Therefore, ψ is slowly varying at the origin.
Let C be the Archimedean copula with generator ψ and let Cu be the tail
dependence copula relative to C at level 0 < u < 1. We will show that C2−k =
C for every positive integer k. Hence, Cu cannot converge to the independence
copula as u ↓ 0.
By the deﬁnition of the function f,
ψ(2
−k−1) − ψ(2
−k) =
Z 2−k
2−k−1 f(x)dx =
3
4
for all nonnegative integer k. Since also ψ(1) = 0, we get ψ(2−k) = 3
4k and thus
ψ−1{2ψ(2−k)} = 2−2k for all nonnegative integer k. By (1), the tail dependence
copula of C at level u = 2−k is therefore Archimedean with generator
ψ2−k(t) = ψ(2
−2kt) − ψ(2
−2k) =
Z 2−2k
2−2kt
f(x)dx =
Z 1
t
2
−2kf(2
−2kx)dx
for t ∈ [0,1]. The function (0,1] 3 x 7→ fk(x) = 2−2kf(2−2kx) is piece-wise
linear with knots fk(2−j) = 2j for all nonnegative integer j. Hence, fk must
coincide with f. But then, ψ2−k coincides with ψ, and thus C2−k coincides with
C for all nonnegative integer k, as required 2
5 Discussion: Asymptotic independence
The problem with Theorem 3.5 in Juri and W¨ uthrich (2002) comes from the
auxiliary Lemma 3.4 in the same paper. In this Lemma, it is claimed that if ψ
is a strict generator, diﬀerentiable and slowly varying at the origin, then there
9exists a positive function g on (0,1) such that
lim
u↓0
ψ(ux) − ψ(u)
g(u)
= −log(x) (3)
for every 0 < x < ∞. However, the generator ψ appearing in the proof of
Theorem 2 satisﬁes ψ(ux)−ψ(u) = ψ(x) for every u = 2−k with k = 0,1,2,...,
contradicting the claim.
The condition (3) states that the function ψ belongs to the de Haan class Π
with auxiliary function g, notation ψ ∈ Πg (e.g. Bingham et al., 1987, chap-
ter 3). [Here, we conveniently shift from asymptotics at inﬁnity to asymptotics
at zero by considering the function y 7→ ψ(1/y) for y ≥ 1.] By the Monotone
Density Theorem (Bingham et al., 1987, Theorem 3.6.8), equation (3) is equiv-
alent to
−ψ
0 ∈ R−1 (4)
and in this case, g(s) ∼ −sψ0(s) as s ↓ 0. Moreover, by Karamata’s theorem
(Bingham et al., 1987, Proposition 1.5.9a), (4) implies ψ ∈ R0. The converse
is not true however, as demonstrated by our counterexample.
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Lower Tail Dependence for Archimedean Copulas: Characterizations and Pitfalls

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