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1. Arthur CHARPENTIER - Dependence between extremal events
Dependence between extremal events
Arthur Charpentier
Hong Kong University, February 2007
Seminar of the department of Statistics and Actuarial Science
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2. Arthur CHARPENTIER - Dependence between extremal events
• Lower tail dependence for Archimedean copulas:
characterizations and pitfalls, (2006), to appear, Insurance
Mathematics and Economics, with J. Segers,
(http://www.crest.fr/.../charpentier-segers-ime.pdf)
• Limiting dependence structures for tail events, with applications
to credit derivatives , (2006), Journal of Applied Probability, 43, 563 -
586, with A. Juri, (http://projecteuclid.org/.../pdf)
• Convergence of Archimedean Copulas, (2006), to appear, Probability
and Statistical Letters, with J. Segers, (http://papers.ssrn.com/...900113)
• Tails of Archimedean Copulas, (2006), submitted, with J. Segers,
(http://www.crest.fr.../Charpentier-Segers-JMA.pdf)
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3. Arthur CHARPENTIER - Dependence between extremal events
“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.” Mikosch (2005).
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“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”.
“The work of Pickands (1981) and Deheuvels (1982) also led several authors
to adhere to the copula point of view in studying multivariate extremes”.
Genest & Rémillard (2006).
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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
Figure 1: Copula C(u, v) and its density c(u, v) = ∂2
C(u, v)/∂u∂v.
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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 ).
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Note that if (X, Y ) has copula C,
P(X ≤ x, Y ≤ y) = C(P(X ≤ x), P(Y ≤ y))
for all (u, v) ∈ [0, 1] × [0, 1], and equivalently
P(X > x, Y > y) = C∗
(P(X > x), P(Y > y))
for all (u, v) ∈ [0, 1] × [0, 1].
C∗
is a copula, called the survival copula of pair (X, Y ), and it satisfies
C∗
(u, v) = u + v − 1 + C(1 − u, 1 − v) for all (u, v) ∈ [0, 1] × [0, 1].
Note that if (U, V ) has distribution C, then C∗
is the distribution function of
(1 − U, 1 − V ).
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot (U,V) from copula C
First component, U
Secondcomponent,V
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot (1−U,1−V) from survival copula C*
First component, 1−U
Secondcomponent,1−V
−3 −2 −1 0 1 2 3
−3−2−10123
Scatterplot (X,Y) from copula C
First component, X
Secondcomponent,Y
−3 −2 −1 0 1 2 3
−3−2−10123
Scatterplot (−X,−Y) from survival copula C*
First component, −X
Secondcomponent,−Y
Figure 2: Scatterplot of C (pair U, V ) and C∗
(pair 1 − U, 1 − V ).
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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(u, v) = ψ←
(ψ(u) + ψ(v)), u, v ∈ [0, 1],
is a copula, called an Archimedean copula, with generator ψ.
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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 7: (Independent) Archimedean copula (C = C⊥
, ψ(t) = − log t).
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Clayton’s copula (Figure 8), 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
.
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Copula density
0.0 0.4 0.8
0.00.51.01.52.0
Archimedean generator
0 1 2 3 4 5 6
0.00.20.40.60.81.0
Laplace Transform
Level curves of the copula
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 8: Clayton’s copula.
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Gumbel’s copula (Figure 9), 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
.
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Copula density
0.0 0.4 0.8
0.00.51.01.52.0
Archimedean generator
0 1 2 3 4 5 6
0.00.20.40.60.81.0
Laplace Transform
Level curves of the copula
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 9: Gumbel’s copula.
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Modeling joint extremal events
“The extension of univariate results is not entirely immediate : the obvious
problem is the lack of natural order in higher dimension.” (Tawn (1988)).
Consider (Xi) an i.i.d. sequence of random variables, with common
distribution function FX . Define, for all n ∈ N∗
the associated statistic order
(Xi:n) and Xn the average, i.e.
X1:n ≤ X2:n ≤ ... ≤ Xn:n and Xn =
X1 + ... + Xn
n
.
Assume that V ar (X) < ∞, from the central limit theorem, if an = E (X) and
bn = V ar (X) /n,
lim
n→∞
P
Xn − an
bn
≤ x = Φ (x) ,
where Φ denotes the c.d.f. of the standard normal distribution.
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More generally, if X /∈ L2
or X /∈ L1
, analogous results could be obtained
Assume that
lim
n→∞
P
Xn − an
bn
≤ x = G (x) .
The set of nondegenerate function is the set of stable distributions, a subset of
infinitely divisible distributions (see Feller (1971) or Petrov (1995)).
The limiting distributions can be characterized through their Laplace
transform.
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In the case of the maxima, consider an i.i.d. sequence of random variables,
X1, X2, ..., with common distribution function FX, F(x) = P{Xi ≤ x}. Then
P{Xn:n ≤ x} = FX (x)n
.
This result simply says that for any fixed x for which F(x) < 1,
P{Xn:n ≤ x} → 0. Hence,
Xn:n
P−as
→ xF = sup{x ∈ R, FX(x) < 1},
and if X is not bounded Xn:n
P−as
→ xF = ∞
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In order to obtain some asymptotic distribution for Xn:n, one should consider
an affine transformation, i.e. find an > 0, bn such that
P
Xn:n − bn
an
≤ x = F(anx + bn)n
→ H(x),
for some nondegenerated function H.
The limiting distibution necessarily satisfies some stability condition, i.e.
H(anx + bn)n
= H(x) for some an > 0, bn, for any n ∈ N. Hence, H satisfies
the following functional equation
H(a(t)x + b(t))t
= H(x) for all x, t ≥ 0.
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The so-called Fisher-Tippett theorem (see Fisher and Tippett (1928),
Gnedenko (1943)), asserts that if a nondegenerate H exists (i.e. a
distribution function which does not put all its mass at a single point), it
must be one of three types:
• H (x) = exp (−x−γ
) if x > 0, α > 0, the Fréchet distribution,
• H (x) = exp (− exp (−x)), the Gumbel distribution,
• H (x) = exp − (−x)
−γ
if x < 0 ,α > 0 , the Weibull distribution.
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The three types may be combined into a single Generalised Extreme Value
(GEV) distribution:
Hξ,µ,σ(x) = exp − 1 + ξ
x − µ
σ
−1/ξ
+
, (2.6)
(where y+ = max(y, 0)) where µ is a location parameter, σ > 0 is a scale
parameter and ξ is a shape parameter.
• the limit ξ → 0 corresponds to the Gumbel distribution,
• ξ > 0 to the Fréchet distribution with γ = 1/ξ,
• ξ < 0 to the Weibull distribution with γ = −1/ξ.
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Furthermore, note that
• µ and σ depend on the affine transformation, an and bn,
• ξ depends on the distribution F.
Definition 4. If there are an and bn such that a non-degenerate limit exists,
FX will be said to be in the max-domain of attraction of Hξ, denoted
FX ∈ MDA (Hξ).
The exponential and the Gaussian distributions have light tails (ξ = 0), and
the Pareto distribution has heavy tails (ξ > 0).
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In order to characterize distributions in some max-domain of attraction, let us
introduce the following concept of regular variation.
Definition 5. A measurable function f : (0, ∞) → (0, ∞) is said to be
regularly varying with index α at infinity, denoted f ∈ Rα (∞) if
lim
u→∞
f (ux)
f (u)
= xα
.
If α = 0, the function will be said to be slowly varying. Notice that f ∈ Rα if
and only if there is L slowly varying such that f (x) = xα
L (x).
Proposition 6. If FX ∈ Rα (∞), α < 0, then the limiting distribution is
Fréchet with index −α, i.e. H−1/α. Analogous properties could be obtained if
ξ ≤ 0 .
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Consider the distribution of X conditionally on exceeding some high threshold
u,
Fu(y) = P{X − u ≤ y | X > u} =
F(u + y) − F(u)
1 − F(u)
.
As u → xF = sup{x : F(x) < 1}, we often find a limit
Fu(y) ∼ G(y; σu, ξ),
where G is Generalised Pareto Distribution (GPD) defined as
G(y; σ, ξ) = 1 − 1 + ξ
y
σ
−1/ξ
+
. (2.8)
The Gaussian distribution has light tails (ξ = 0). The associated limiting
distribution is the exponential distribution.
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Theorem 7. For ξ ∈ R, the following assertions are equivalent,
1. F ∈ MDA (Hξ), i.e. there are (an) and (bn) such that
lim
n→∞
P (Xn:n ≤ anx + bn) = Hξ (x) , x ∈ R.
2. There exists a positive, measurable function a (·) such that for 1 + ξx > 0,
lim
u→∞
F (u + xa (u))
F (u)
= lim
u→∞
P
X − u
a (u)
> x |X > u
=
(1 + ξx)
−1/ξ
if ξ = 0,
exp (−x) if ξ = 0.
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The general structure for such bivariate extreme value distributions has been
known since the end of the 50’s, due to Tiago de Olivera (1958),
Geoffroy (1958) or Sibuya (1960). Those three papers obtained equivalent
representations (in dimension 2 or higher).
Most of the results on multivariate extremes have been obtained considering
componentwise ordering, i.e. considering possible limiting distributions for
(Xn:n, Yn:n). As pointed out in Tawn (1988) “A difficulty with this approach
is that in some applications it may be impossible for (Xn:n, Yn:n) to occur as a
vector observation”. Despite this problem, this is the approach most widely
used in bivariate extreme value analysis.
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−4 −2 0 2 4
−3−2−10123
Maximum componentwise
First component
Secondcomponent
−4 −2 0 2 4
−3−2−10123
Joint exceedance approach
First componentSecondcomponent
Figure 10: Modeling joint extremal events.
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Suppose that there are sequences of normalizing constant αX,n, αY,n > 0 and
βX,n, βY,n such that
P
Xn:n − βX,n
αX,n
≤ x,
Yn:n − βY,n
αY,n
≤ y
= Fn
X,Y (αX,nx + βX,n, αY,ny + βY,n) → G (x, y) ,
as n → ∞, where G is a proper distribution function, non-degenerated in each
margin.
Bivariate extreme value distributions are obtained as limiting distributions of
lim
n→∞
P
Xn:n − an
bn
≤ x,
Yn:n − cn
dn
≤ y = C (HξX (x) , HξY (y)) .
i.e. the normalized distribution of the vector of componentwise maxima.
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C is called an extreme value copula,
C (u, v) = exp (log u + log v) A
log u
log u + log v
, (1)
where 0 < u, v < 1, and A is a convex function on [0, 1] such that
A+
(t) = max {t, 1 − t} ≤ A (t) ≤ 1 = A⊥
(t).
(see Capéraà, Fougères and Genest (1997), based on Pickands (1981)).
Example 8. If A(ω) = exp (1 − ω)θ
+ ωθ 1/θ
, then C is Gumbel copula.
Further, if A (ω) = max {1 − αω, 1 − β (1 − ω)}, where 0 ≤ α, β ≤ 1, then C is
Marshall and Olkin copula.
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0.0 0.2 0.4 0.6 0.8 1.0
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Pickands dependence function A
0.0 0.2 0.4 0.6 0.8 1.0
0.50.60.70.80.91.0
Pickands dependence function A
Figure 11: Gumbel, and Marshall & Olkin’s dependence function A(ω).
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Proposition 9. Consider (X1, Y1), ..., (Xn, Yn), ... sequence of i.i.d. versions
of (X, Y ), with c.d.f. (X, Y ). Assume that there are normalizing sequences
αX,n, αY,n, αX,n, αY,n > 0 and βX,n, βY,n, βX,n, βY,n such that
Fn
X,Y (αX,nx + βX,n, αY,ny + βY,n) → G (x, y)
Fn
X,Y αX,nx + βX,n, αY,ny + βY,n → G (x, y) ,
as n → ∞, for two non-degenerated distributions G and G . Then marginal
distributions of G and G are unique up to an affine transformation, i.e. there
are αX, αY , βX, βY such that
GX (x) = GX (αXx + βX) and GY (y) = GY (αY y + βY ).
Further, the dependence structures of G and G are equal, i.e. the copulae are
equal, CG = CG .
Frank copula has independence in tails (A = A⊥
) and the survival Clayton
copula has dependence in tails (A = A⊥
). The associated limited copula is
Gumbel.
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Joe (1993) defined, in the bivariate case a tail dependence measure.
Definition 10. Let (X, Y ) denote a random pair, the upper and lower tail
dependence parameters are defined, if the limit exist, as
λL = lim
u→0
P X ≤ F−1
X (u) |Y ≤ F−1
Y (u) ,
and
λU = lim
u→1
P X > F−1
X (u) |Y > F−1
Y (u) .
Proposition 11. Let (X, Y ) denote a random pair with copula C, the upper
and lower tail dependence parameters are defined, if the limit exist, as
λL = lim
u→0
C(u, u)
u
and λU = lim
u→1
C∗
(u, u)
1 − u
.
Example 12. If (X, Y ) has a Gaussian copula with parameter θ < 1, then
λ = 0.
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0.00.20.40.60.81.0
Marges uniformes
CopuleGaussienne
−2 0 2 4
−2024
Marges gaussiennes
Figure 13: Simulations of the Gaussian copula (θ = 0.95).
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Example 13. Consider the case of Archimedean copulas, then
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)
.
Ledford and Tawn (1996) propose the following model to study tail
dependence. Consider standardized marginal variables, with unit Fréchet
distributions, such that
P(X > t, Y > t) ∼ L(t) · [P(X > t)]1/η
, t → ∞,
where L denotes some slowly varying functions, and η ∈ (0, 1] will be called
coefficient of tail dependence,
• η describes the kind of limiting dependence,
• L describes the relative strength, given η.
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More precisely,
• η = 1, perfect positive dependence (tail comontonicity),
• 1/2 < η < 1 and L → c > 0, more dependent than independence, but
asymptotically independent,
• η = 1/2, tail independence
• 0 < η < 1/2 less dependent than independence.
Example 14. : distribution with Gumbel copula,
P(X ≤ x, Y ≤ y) = exp(−(x−α
+ y−α
)1/α
), α ≥ 0
then η = 1 and (t) → (2 − 21/θ
).
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A short word on tail parameter estimation
For the estimation of η, define
T =
1
1 − FX (X)
∧
1
1 − FY (Y )
,
then FT , is regularly varying with parameter η: Hill’s estimator can be used.
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univariate case bivariate case
limiting distribution dependence structure of
of Xn:n (G.E.V.) componentwise maximum
when n → ∞ (Xn:n, Yn:n)
(Fisher-Tippet)
limiting distribution dependence structure of
of X|X > x (G.P.D.) (X, Y ) |X > x, Y > y
when x → ∞ when x, y → ∞
(Balkema-de Haan-Pickands)
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Conditional copulae
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, (2)
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 is
Cr(u) =
C(F←
1|r(u1), ..., F←
d|r(ud))
C(r1, ..., rd)
. (3)
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A bivariate regular variation property
In the univariate case, h is regularly varying if there
lim
t→0
h(tx)
h(t)
= λ(x), for all x > 0.
For all x, y > 0, lim
t→0
h(txy)
h(t)
= λ(xy), and
lim
t→0
h(txy)
h(t)
= lim
t→0
h(txy)
h(tx)
×
h(tx)
h(t)
= λ(y) × λ(x).
Thus, necessarily λ(xy) = λ(x) × λ(y). It is Cauchy functional equation and
thus, necessarily, λ(x) = xθ
(power function) for some θ ∈ R.
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In the (standard) bivariate case (see Resnick (1981)), h is regularly varying if
there
lim
t→0
h(tx, ty)
g(t, t)
= λ(x, y), for all x, y > 0.
This will be called ray-convergence. Then, there is θ ∈ R such that
λ(tx, ty) = tθ
λ(x, y) (homogeneous function).
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A general extention is to consider (see Meerschaert & Scheffer (2001)) is
to assume that there is a sequence (At) of operators, regularly varying with
index E such that
lim
t→0
h
A−1
t
x
y
· g(t)−1
= λ(x, y).
Then there is θ ∈ R such that λ(tE
x
y
) = tθ
λ(x, y) (generalized
homogeneous function).
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0.00.20.40.60.81.0
Bivariate regular variation, ray convergence
First component, X
Secondcomponent,Y
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Bivariate regular variation, directional convergence
First component, X
Secondcomponent,Y
Figure 14: Two concepts of regular variation in R2
.
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A subdefinition has been proposed by de Haan, Omey & Resnick (1984), a
directional convergence: given r, s : [0, 1] → [0, 1] such that r(t), s(t) → 0 as
t → 0, both regularly varying (with index α and β respectively), then
lim
t→0
h(r(t)x, s(t)y) · g(t)−1
= λ(x, y),
then there is θ ∈ R such that
λ(tα
x, tβ
y) = tθ
λ(x, y),
for all x, y, t > 0, i.e. λ is a (generalized homogeneous function).
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Let C denote a copula, such that C(u, v) > 0 for all u, v > 0. Furthermore,
consider r and s two continuous functions, regularly varying at 0, r ∈ Rα and
s ∈ Rβ , so that s(t), r(t) → 0 when t → 0, so that
lim
t→0
C(r(t)x, s(t)y
C(r(t), s(t))
= φ(x, y), (4)
where φ is a positive measurable function.
Then φ satisfies the following functional equation φ(tα
, tβ
) = tθ
φ(x, y) for
some θ > 0. Hence, φ is a so-called generalized homogeneous function (see
Aczél (1966)), which has an explicit general solution (in dimension 2). tThe
most general solution is given by
φ(x, y) =
xθ/α
h(yx−β/α
) if x = 0
cyθ/β
if x = 0 and y = 0
0 if x = y = 0
, (5)
where c is a constant and h is function of one variable.
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51. Arthur CHARPENTIER - Dependence between extremal events
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Marshall and Olkin’s copula Level curves of the copula
DISCONTINUITY
Marshall and Olkin’s copula
Figure 15: Marshall and Olkin’s copula.
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55. Arthur CHARPENTIER - Dependence between extremal events
Conditional dependence for Archimedean copulae
Proposition 15. 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).
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Generators of conditional Archimedean copulae
(1) (2)
(3)
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56. Arthur CHARPENTIER - Dependence between extremal events
Archimedean copulae in lower tails
Proposition 16. 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).
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57. Arthur CHARPENTIER - Dependence between extremal events
Proposition 17. 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)).
Frank copula has independence in tails (C = C⊥
) and the 4-2-14 copula has
dependence in tails (C = C⊥
). The associated limited copula is Clayton.
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59. Arthur CHARPENTIER - Dependence between extremal events
Archimedean copulae in upper tails
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).
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Archimedean copula at 1
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Archimedean copula at 1
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0.000.020.040.060.080.100.12
Archimedean copula at 1
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−0.020.000.020.040.060.080.10
Archimedean copula at 1
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60. Arthur CHARPENTIER - Dependence between extremal events
Proposition 18. 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.
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62. Arthur CHARPENTIER - Dependence between extremal events
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0246810
Archimedean copula density on the diagonal
Dependence
Dependence in independence
Independence in independence
Copula density
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63. Arthur CHARPENTIER - Dependence between extremal events
On sequences of Archimedean copulae
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.
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64. Arthur CHARPENTIER - Dependence between extremal events
Proposition 19. 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,
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Sequence of generators and Kendall cdf’s
64
