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Arthur CHARPENTIER - Dependence between extremal events

Dependence between extremal events

Arthur Charpentier

Hong Kong University, February 2007Seminar of the department of Statistics and Actuarial Science

1


• Lower tail dependence for Archimedean copulas:characterizations and pitfalls, (2006), to appear, InsuranceMathematics and Economics, with J. Segers,(http://www.crest.fr/.../charpentier-segers-ime.pdf)

• Limiting dependence structures for tail events, with applicationsto credit derivatives , (2006), Journal of Applied Probability, 43, 563 -586, with A. Juri, (http://projecteuclid.org/.../pdf)

• Convergence of Archimedean Copulas, (2006), to appear, Probabilityand 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)

2

http://www.crest.fr/pageperso/charpent/charpentier-segers-ime.pdf

http://projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.jap/1152413742/body/pdf

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=900113

http://www.crest.fr/pageperso/charpent/Charpentier-Segers-JMA.pdf


“Everybody who opens any journal on stochastic processes, probability theory,statistics, econometrics, risk management, finance, insurance, etc., observesthat there is a fast growing industry on copulas [...] The InternationalActuarial Association in its hefty paper on Solvency II recommends usingcopulas for modeling dependence in insurance portfolios [...] Since Basle IIcopulas are now standard tools in credit risk management”.

“Are copulas suitable for modeling multivariate extremes? Copulas generateany multivariate distribution. If one wants to make an honest analysis ofmultivariate extremes the distributions used should be related to extreme valuetheory in some way .” Mikosch (2005).

3


“We are thus generally sympathetic to the primary objective pursued by Dr.Mikosch, which is to caution optimism about what copulas can and cannotachieve as a dependence modeling tool ”.

“Although copula theory has only recently emerged as a distinct field ofinvestigation, its roots go back at least to the 1940s, with the seminal work ofHoeőding on margin-free measures of association [...] It was possiblyDeheuvels who, in a series of papers published around 1980, revealed the fullpotential of the fecund link between multivariate analysis and rank-basedstatistical techniques[...] However, the generalized use of copulas for modelbuilding (and Archimedean copulas in particular) seems to have been largelyfuelled at the end of the 1980s by the publication of significant papers byMarshall and Olkin (1988) and by Oakes (1989) in the influential Journal ofthe American Statistical Association”.

“The work of Pickands (1981) and Deheuvels (1982) also led several authorsto adhere to the copula point of view in studying multivariate extremes”.Genest & Rémillard (2006).

4


Definition 1. A 2-dimensional copula is a 2-dimensional cumulativedistribution 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) = ∂2C(u, v)/∂u∂v.

5


Theorem 2. (Sklar) Let C be a copula, and FX and FY two marginaldistributions, then F (x, y) = C(FX(x), FY (y)) is a bivariate distributionfunction, with F ∈ F(FX , FY ).

Conversely, if F ∈ F(FX , FY ), there exists C such thatF (x, y) = C(FX(x), FY (y). Further, if FX and FY are continuous, then C isunique, and given by

C(u, v) = F (F−1X (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 ).

6


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 ).

7


0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Scatterplot (U,V) from copula C

First component, U

Seco

nd co

mpon

ent, V

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Scatterplot (1−U,1−V) from survival copula C*

First component, 1−U

Seco

nd co

mpon

ent, 1

−V−3 −2 −1 0 1 2 3

−3−2

−10

12

3

Scatterplot (X,Y) from copula C

First component, X

Seco

nd co

mpon

ent, Y

−3 −2 −1 0 1 2 3

−3−2

−10

12

3

Scatterplot (−X,−Y) from survival copula C*

First component, −X

Seco

nd co

mpon

ent, −

Y

Figure 2: Scatterplot of C (pair U, V ) and C∗ (pair 1− U, 1− V ).

8


In dimension 2, consider the following family of copulae

Definition 3. Let ψ denote a convex decreasing function [0, 1] → [0,∞] suchthat 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 < ∞.

ThenC(u, v) = ψ←(ψ(u) + ψ(v)), u, v ∈ [0, 1],

is a copula, called an Archimedean copula, with generator ψ.

9


• 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,

• Clayton’s copula, ψ(t) = t−θ − 1, C(u, v) = (uθ + vθ − 1)−1/θ,

• Gumbel’s copula, ψ(t) = (− log t)−θ,C(u, v) = exp

(− [

(− log u)θ + (− log v)θ]1/θ

),

• 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

0.0

0.5

1.0

1.5

2.0

The lower Fréchet bound

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

The independent copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

Gumbel’s copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

Clayton’s copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

Nelsen’s copula

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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(−[αx + βy]θ)] π(θ)dθ,

where ψ(t) = E(exp−tΘ) =∫

exp(−tθ)π(θ)dθ is the Laplace transform of Θ.

Hence P(X > x, Y > y) = φ(αx + βy). Similarly,

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].

11


0 5 10 15

05

10

15

20

Conditional independence, two classes

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

Conditional independence, two classes

Figure 3: Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).

12


0 5 10 15 20 25 30

01

02

03

04

0

Conditional independence, three classes

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

Conditional independence, three classes

Figure 4: Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)),Φ−1(FY (Yi))).

13


0 20 40 60 80 100

02

04

06

08

01

00

Conditional independence, continuous risk factor

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23


Figure 5: Continuous classes, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).

14


0 20 40 60 80 100

02

04

06

08

01

00


−3 −2 −1 0 1 2 3

−3

−2

−1

01

23


Figure 6: Continuous classes, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).

15


Copula density

0.0 0.4 0.8

0.0

0.5

1.0

1.5

2.0

Archimedean generator

0 1 2 3 4 5 6

0.0

0.4

0.8

Laplace Transform

Level curves of the copula

0.0 0.4 0.8

−0.4

−0.2

0.0

Lambda function

0.0 0.4 0.8

0.0

0.4

0.8

Kendall cdf

Figure 7: (Independent) Archimedean copula (C = C⊥, ψ(t) = − log t).

16


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.

17


Copula density

0.0 0.4 0.8

0.0

0.5

1.0

1.5

2.0


0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

Laplace Transform


0.0 0.4 0.8

−0.4

−0.3

−0.2

−0.1

0.0

Lambda function

0.0 0.4 0.8

0.0

0.2

0.4

0.6

0.8

1.0

Kendall cdf

Figure 8: Clayton’s copula.

18


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) = − 1α

log(

1 +(e−αu − 1) (e−αv − 1)

e−α − 1

),

if 1 ≤ α < ∞, for any (u, v) ∈ (0, 1]2.

19


Copula density

0.0 0.4 0.8

0.0

0.5

1.0

1.5

2.0


0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

Laplace Transform


0.0 0.4 0.8

−0.4

−0.3

−0.2

−0.1

0.0

Lambda function

0.0 0.4 0.8

0.0

0.2

0.4

0.6

0.8

1.0

Kendall cdf

Figure 9: Gumbel’s copula.

20


Modeling joint extremal events

“The extension of univariate results is not entirely immediate : the obviousproblem is the lack of natural order in higher dimension.” (Tawn (1988)).

Consider (Xi) an i.i.d. sequence of random variables, with commondistribution 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) andbn =

√V ar (X) /n,

limn→∞

P(

Xn − an

bn≤ x

)= Φ (x) ,

where Φ denotes the c.d.f. of the standard normal distribution.

21


More generally, if X /∈ L2 or X /∈ L1, analogous results could be obtained

Assume that

limn→∞

P(

Xn − an

bn≤ x

)= G (x) .

The set of nondegenerate function is the set of stable distributions, a subset ofinfinitely divisible distributions (see Feller (1971) or Petrov (1995)).

The limiting distributions can be characterized through their Laplacetransform.

22


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:nP−as→ xF = sup{x ∈ R, FX(x) < 1},

and if X is not bounded Xn:nP−as→ xF = ∞

23


In order to obtain some asymptotic distribution for Xn:n, one should consideran 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 satisfiesthe following functional equation

H(a(t)x + b(t))t = H(x) for all x, t ≥ 0.

24


The so-called Fisher-Tippett theorem (see Fisher and Tippett (1928),Gnedenko (1943)), asserts that if a nondegenerate H exists (i.e. adistribution function which does not put all its mass at a single point), itmust 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.

25


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 scaleparameter 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, denotedFX ∈ MDA (Hξ).

The exponential and the Gaussian distributions have light tails (ξ = 0), andthe Pareto distribution has heavy tails (ξ > 0).

27

http://www.crest.fr/pageperso/charpent/loi-max-exponentielle-comp.avi

http://www.crest.fr/pageperso/charpent/loi-max-gaussienne-comp.avi

http://www.crest.fr/pageperso/charpent/loi-max-pareto-comp.avi


In order to characterize distributions in some max-domain of attraction, let usintroduce the following concept of regular variation.Definition 5. A measurable function f : (0,∞) → (0,∞) is said to beregularly varying with index α at infinity, denoted f ∈ Rα (∞) if

limu→∞

f (ux)f (u)

= xα.

If α = 0, the function will be said to be slowly varying. Notice that f ∈ Rα ifand only if there is L slowly varying such that f (x) = xαL (x).Proposition 6. If FX ∈ Rα (∞), α < 0, then the limiting distribution isFréchet with index −α, i.e. H−1/α. Analogous properties could be obtained ifξ ≤ 0 .

28


Consider the distribution of X conditionally on exceeding some high thresholdu,

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 limitingdistribution 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

limn→∞

P (Xn:n ≤ anx + bn) = Hξ (x) , x ∈ R.

2. There exists a positive, measurable function a (·) such that for 1 + ξx > 0,

limu→∞

F (u + xa (u))F (u)

= limu→∞

P(

X − u

a (u)> x |X > u

)

=

(1 + ξx)−1/ξ if ξ 6= 0,

exp (−x) if ξ = 0.

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The general structure for such bivariate extreme value distributions has beenknown since the end of the 50’s, due to Tiago de Olivera (1958),Geoffroy (1958) or Sibuya (1960). Those three papers obtained equivalentrepresentations (in dimension 2 or higher).

Most of the results on multivariate extremes have been obtained consideringcomponentwise ordering, i.e. considering possible limiting distributions for(Xn:n, Yn:n). As pointed out in Tawn (1988) “A difficulty with this approachis that in some applications it may be impossible for (Xn:n, Yn:n) to occur as avector observation”. Despite this problem, this is the approach most widelyused in bivariate extreme value analysis.

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−4 −2 0 2 4

−3

−2

−1

01

23

Maximum componentwise

First component

Se

con

d c

om

po

ne

nt

−4 −2 0 2 4

−3

−2

−1

01

23

Joint exceedance approach

First componentS

eco

nd

co

mp

on

en

t

Figure 10: Modeling joint extremal events.

32


Suppose that there are sequences of normalizing constant αX,n, αY,n > 0 andβX,n, βY,n such that

P(

Xn:n − βX,n

αX,n≤ x,

Yn:n − βY,n

αY,n≤ y

)

= FnX,Y (αX,nx + βX,n, αY,ny + βY,n) → G (x, y) ,

as n →∞, where G is a proper distribution function, non-degenerated in eachmargin.

Bivariate extreme value distributions are obtained as limiting distributions of

limn→∞

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.

33


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 isMarshall and Olkin copula.

34


0.0 0.2 0.4 0.6 0.8 1.0

0.5

0.6

0.7

0.8

0.9

1.0

Pickands dependence function A

0.0 0.2 0.4 0.6 0.8 1.0

0.5

0.6

0.7

0.8

0.9

1.0

Pickands dependence function A

Figure 11: Gumbel, and Marshall & Olkin’s dependence function A(ω).

35


Proposition 9. Consider (X1, Y1), ..., (Xn, Yn), ... sequence of i.i.d. versionsof (X,Y ), with c.d.f. (X,Y ). Assume that there are normalizing sequencesαX,n, αY,n, α′X,n, α′Y,n > 0 and βX,n, βY,n, β′X,n, β′Y,n such that

FnX,Y (αX,nx + βX,n, αY,ny + βY,n) → G (x, y)

FnX,Y

(α′X,nx + β′X,n, α′Y,ny + β′Y,n

) → G′ (x, y) ,

as n →∞, for two non-degenerated distributions G and G′. Then marginaldistributions of G and G′ are unique up to an affine transformation, i.e. thereare αX , αY , βX , βY such that

GX(x) = G′X(αXx + βX) and GY (y) = G′Y (αY y + βY ).

Further, the dependence structures of G and G′ are equal, i.e. the copulae areequal, CG = CG′ .

Frank copula has independence in tails (A = A⊥) and the survival Claytoncopula has dependence in tails (A 6= A⊥). The associated limited copula isGumbel.

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http://www.crest.fr/pageperso/charpent/copule-maximum-surv-clayton.avi


Joe (1993) defined, in the bivariate case a tail dependence measure.Definition 10. 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)

),

andλU = lim

u→1P

(X > F−1

X (u) |Y > F−1Y (u)

).

Proposition 11. Let (X, Y ) denote a random pair with copula C, the upperand lower tail dependence parameters are defined, if the limit exist, as

λL = limu→0

C(u, u)u

and λU = limu→1

C∗(u, u)1− u

.

Example 12. If (X, Y ) has a Gaussian copula with parameter θ < 1, thenλ = 0.

37


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Marges uniformes

Co

pu

le d

e G

um

be

l

−2 0 2 4

−2

02

4Marges gaussiennes

Figure 12: Simulations of Gumbel’s copula θ = 1.2.

38


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Marges uniformes

Co

pu

le G

au

ssie

nn

e

−2 0 2 4

−2

02

4Marges gaussiennes

Figure 13: Simulations of the Gaussian copula (θ = 0.95).

39


Example 13. Consider the case of Archimedean copulas, then

λU = 2− limx→0

1− φ−1(2x)1− φ−1(x)

and λL = limx→0

φ−1(2φ(x))x

= limx→∞

φ−1(2x)φ−1(x)

.

Ledford and Tawn (1996) propose the following model to study taildependence. Consider standardized marginal variables, with unit Fréchetdistributions, 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 calledcoefficient of tail dependence,

• η describes the kind of limiting dependence,

• L describes the relative strength, given η.

40


More precisely,

• η = 1, perfect positive dependence (tail comontonicity),

• 1/2 < η < 1 and L → c > 0, more dependent than independence, butasymptotically 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).

41


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.

42


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)

43


Conditional copulae

Let U = (U1, ..., Un) be a random vector with uniform margins, anddistribution 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)

44


A bivariate regular variation property

In the univariate case, h is regularly varying if there

limt→0

h(tx)h(t)

= λ(x), for all x > 0.

For all x, y > 0, limt→0

h(txy)h(t)

= λ(xy), and

limt→0

h(txy)h(t)

= limt→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 andthus, necessarily, λ(x) = xθ (power function) for some θ ∈ R.

45


In the (standard) bivariate case (see Resnick (1981)), h is regularly varying ifthere

limt→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 thatλ(tx, ty) = tθλ(x, y) (homogeneous function).

46


A general extention is to consider (see Meerschaert & Scheffer (2001)) isto assume that there is a sequence (At) of operators, regularly varying withindex E such that

limt→0

h

A−1

t

x

y

· g(t)−1 = λ(x, y).

Then there is θ ∈ R such that λ(tE

x

y

) = tθλ(x, y) (generalized

homogeneous function).

47


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate regular variation, ray convergence

First component, X

Se

co

nd

co

mp

on

en

t, Y

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Bivariate regular variation, directional convergence

First component, X

Se

co

nd

co

mp

on

en

t, Y

Figure 14: Two concepts of regular variation in R2.

48


A subdefinition has been proposed by de Haan, Omey & Resnick (1984), adirectional convergence: given r, s : [0, 1] → [0, 1] such that r(t), s(t) → 0 ast → 0, both regularly varying (with index α and β respectively), then

limt→0

h(r(t)x, s(t)y) · g(t)−1 = λ(x, y),

then there is θ ∈ R such that

λ(tαx, tβy) = tθλ(x, y),

for all x, y, t > 0, i.e. λ is a (generalized homogeneous function).

49


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 ands ∈ Rβ , so that s(t), r(t) → 0 when t → 0, so that

limt→0

C(r(t)x, s(t)yC(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) = tθφ(x, y) forsome θ > 0. Hence, φ is a so-called generalized homogeneous function (seeAczél (1966)), which has an explicit general solution (in dimension 2). tThemost general solution is given by

φ(x, y) =

xθ/αh(yx−β/α) if x 6= 0

cyθ/β if x = 0 and y 6= 0

0 if x = y = 0

, (5)

where c is a constant and h is function of one variable.

50


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Marshall and Olkin’s copula Level curves of the copula

DISCONTINUITY

Marshall and Olkin’s copula

Figure 15: Marshall and Olkin’s copula.

51


2 4 6 8 10 12 14

46

81

01

2Scatterplot, LOSS−ALAE

Losses amounts (log)

Allo

ca

ted

exp

en

se

s (

log

)

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Copula density, LOSS−ALAE

Figure 16: On statistical inference for tail events.

52


2 4 6 8 10 12 14

46

810

12Scatterplot, LOSS−ALAE, maximum componentwise


Alloc

ated e

xpen

ses (

log)

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Copula density, LOSS−ALAE, n=50

2 4 6 8 10 12 14

46

810

12

Scatterplot, LOSS−ALAE, maximum componentwise


Alloc

ated e

xpen

ses (

log)

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Copula density, LOSS−ALAE, n=50


53


2 4 6 8 10 12 14

46

810

12

Scatterplot, LOSS−ALAE, joint−exceedences


Alloc

ated e

xpen

ses (

log)

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Copula density, LOSS−ALAE, u=85%

2 4 6 8 10 12 14

46

810

12

Scatterplot, LOSS−ALAE, joint−exceedences


Alloc

ated e

xpen

ses (

log)

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Copula density, LOSS−ALAE, u=95%


54


Conditional dependence for Archimedean copulaeProposition 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 anyr ∈ (0, 1]d, will also have an Archimedean generator, with generatorψr(t) = ψ(tc)− ψ(c) where c = C(r1, ..., rd).

0.0 0.2 0.4 0.6 0.8 1.0

0.00.5

1.01.5

2.02.5

3.0

Generators of conditional Archimedean copulae

(1) (2)

(3)

55


Archimedean copulae in lower tailsProposition 16. Let C be an Archimedean copula with generator ψ, and0 ≤ 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).

56


Proposition 17. There exists Archimedean copulae, with generators havingcontinuous derivatives, slowly varying such that the conditional copula doesnot 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 = limu↓0

C(u, u)u

= 2−1/α,

with proper interpretations for α equal to zero or infinity (see e.g. Theorem3.9 of Juri and Wüthrich (2003)).

Frank copula has independence in tails (C = C⊥) and the 4-2-14 copula hasdependence in tails (C 6= C⊥). The associated limited copula is Clayton.

57

http://www.crest.fr/pageperso/charpent/conditional-copula-Frank.avi

http://www.crest.fr/pageperso/charpent/conditional-copula-4-2-14.avi


ψ(t) range θ α

(1) 1θ

(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−1e−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−t1+(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) ( 1t− 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) ( θt

+ 1)(1 − t) [0,∞) 1

(17) − log (1+t)−θ−12−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

58


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 converseis not true).

0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Archimedean copula at 1

0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6


0.5 0.6 0.7 0.8 0.9 1.0

0.00

0.02

0.04

0.06

0.08

0.10

0.12


0.5 0.6 0.7 0.8 0.9 1.0

−0.0

20.

000.

020.

040.

060.

080.

10


59


Proposition 18. Let C be an Archimedean copula with generator ψ. Assumethat f : s 7→ 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.

60


ψ(t) range θ α κ

(1) 1θ

(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−1e−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−t1+(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) ( 1t− 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) ( θt

+ 1)(1 − t) [0,∞) 1 θ + 1

(17) − log (1+t)−θ−12−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

61


0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

Archimedean copula density on the diagonal

DependenceDependence in independenceIndependence in independence

Copula density

62


On sequences of Archimedean copulae

Extension of results due to Genest & Rivest (1986),

Proposition The five following statements are equivalent,

(i) limn→∞

Cn(u, v) = C(u, v) for all (u, v) ∈ [0, 1]2,

(ii) limn→∞

ψn(x)/ψ′n(y) = ψ(x)/ψ′(y) for all x ∈ (0, 1] and y ∈ (0, 1) such that

ψ′ such that is continuous in y,

(iii) limn→∞

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) forall x ∈ [0, 1],

(v) limn→∞

Kn(x) = K(x) for all x ∈ (0, 1) such that K is continuous in x.

63


Proposition 19. The four following statements are equivalent

(i) limn→∞

Cn(u, v) = C+(u, v) = min(u, v) for all (u, v) ∈ [0, 1]2,

(ii) limn→∞

λn(x) = 0 for all x ∈ (0, 1),

(iii) limn→∞

ψn(y)/ψn(x) = 0 for all 0 ≤ x < y ≤ 1,

(iv) limn→∞

Kn(x) = x for all x ∈ (0, 1).

Note that one can get non Archimedean limits,

0.0 0.4 0.8

05

1015

0.0 0.4 0.8

0.00.2

0.40.6

0.81.0

Sequence of generators and Kendall cdf’s

64

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