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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

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“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 extremevalue theory in some way .” Mikosh (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 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 model building (andArchimedean copulas in particular) seems to have been largely fuelled at theend of the 1980s by the publication of significant papers by Marshall andOlkin (1988) and by Oakes (1989) in the influential Journal of the AmericanStatistical Association”. Genest & Rémillard (2006).

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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

4


Why using copulas ?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 ).

5


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(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 cumulativedistribution function restricted to [0, 1]d with standard uniform margins.

Sklar’s theorem can be extended in dimension d as followsTheorem 5. (Sklar) Let C be a d-copula, and F1, ..., Fd be marginaldistributions, then F (x1, ..., , xd) = C(F1(x1), ..., Fd(xd)) is a d-dimensionaldistribution function, with F ∈ F(F1, ..., Fd).

Conversely, if F ∈ F(F1, ..., Fd), there exists C such thatF (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−11 (u1), ..., F−1

d (ud)) for all (u1, ..., ud) ∈ [0, 1]d

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Definition 6. Let ψ be an generator of order d, i.e. ψ is decreasing andψ(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 andψ−1 is completely monotone, i.e. (−1)k(ψ−1)(k)(·) ≥ 0 on (0,∞).

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• 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

0.0

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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Some more examples of Archimedean copulas

ψ(t) range θ

(1) 1θ

(t−θ − 1) [−1, 0) ∪ (0,∞) Clayton, Clayton (1978)

(2) (1 − t)θ [1,∞)

(3) log 1−θ(1−t)t

[−1, 1) Ali-Mikhail-Haq

(4) (− log t)θ [1,∞) Gumbel, Gumbel (1960), Hougaard (1986)

(5) − log e−θt−1e−θ−1

(−∞, 0) ∪ (0,∞) Frank, Frank (1979), Nelsen (1987)

(6) − log{1 − (1 − t)θ} [1,∞) Joe, Frank (1981), Joe (1993)

(7) − log{θt + (1 − θ)} (0, 1]

(8) 1−t1+(θ−1)t

[1,∞)

(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)

(10) log(2t−θ − 1) (0, 1]

(11) log(2 − tθ) (0, 1/2]

(12) ( 1t− 1)θ [1,∞)

(13) (1 − log t)θ − 1 (0,∞)

(14) (t−1/θ − 1)θ [1,∞)

(15) (1 − t1/θ)θ [1,∞) Genest & Ghoudi (1994)

(16) ( θt

+ 1)(1 − t) [0,∞)

Table 1: Archimedean copulas, from Nelsen (2006).

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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 acontinuous risk factor θ ∈ (0,∞).

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

12


0 5 10 15 20 25 30

010

20

30

40

Conditional independence, three classes

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

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

020

40

60

80

100

Conditional independence, continuous risk factor

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23


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

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0 20 40 60 80 100

020

40

60

80

100


−3 −2 −1 0 1 2 3

−3

−2

−1

01

23


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

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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.

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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 ofmultivariate 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.

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This approach can be used in motor insurance ratemaking, and in credit risk.

A finite sequence {X1, ..., Xd} of random variables is exchangeable, ord-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 simplyexchangeable) if

(X1, X2, ...)L=

(Xσ(1), Xσ(2), ...

), (2)

for any finite permutation σ of N∗ (that is Card {i, σ (i) 6= i} < ∞).

A d-exchangeable sequence {X1, ..., Xd} is called m-extendible (for somem > d), if (X1, ..., Xd)

L= (Z1, ..., Zd), where {Z1, ..., Zm} is somem-exchangeable sequence.

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Using the formulation of Aldous (1985), de Finetti’s theorem states that“aninfinite exchangeable sequence is a mixture of i.i.d. sequences”: X1, X2, ... ofBernoulli random variables is exchangeable if and only there is a randomvariable Θ, taking values in [0, 1] such that, given Θ = θ the Xi’s areindependent, and Xi ∼ B(θ) (see Schervish (1995) or Chow & Teicher(1997)).Example 8. This result can be easily interpreted in credit risk, wherevariables of interest are dichotomous (default or non-default). Let X1, X2, ...

be an infinite exchangeable sequence of Bernoulli variables, and letSn = X1 + .... + Xn the number of defaults within n companies (for a givenperiod of time). Then, the distribution of Sn is a mixture of binomialdistributions, 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) .

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• The survival distribution approach

Assume that for a random vector (X, Y ), there exists a convex survivaldistribution 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 withgenerator 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 isClayton’s. If S is the survival Weibull distribution, the associated copula isGumbel’s.

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• The serial iterate approach

A “natural ” idea to define d dimension copula from 2 dimensional copulascan 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 forall u ∈ (0, 1) if and only if C is Archimedean.

Hence, the only copulas that can be constructed by serial iteration areArchimedean copulas.Remark 11. This is actually where the word Archimedean comes from.

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• The distorted copula approachDefinition 12. A distortion function is a function h : [0, 1] → [0, 1] strictlyincreasing 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), andφ(·) = log h(·), then

Ch(u, v) = h−1(h(u)h(v)) = φ−1(φ(u) + φ(v)).

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• 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

1λ(t)

dt

)for all 0 < u0 < 1.

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• 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 samecopula as (U, V ) given (U ≤ u, V ≤ v), for all u, v ∈ (0, 1] (stability bytruncature),

3. Gumbel copula is the only Archimedean such that (U, V ) has the samecopula as (max{U1, ..., Un}, max{V1, ..., Vn}) for all n ≥ 1 (max-stability),

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

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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.

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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 6: Clayton’s copula.

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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) = − 1α

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

0.0

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 7: Gumbel’s copula.

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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 beconsidered, ψα,β(x) = [ψ(xα)]β .

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Distorted Archimedean copula, from Frank

β = 1α = 1


β = 3 2α = 1


β = 1α = 1 2


β = 3 2α = 1 2

Figure 8: Distorted Archimedean copula (φα,β), from Frank copula.

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A short word on the zero-set area

Define the zero-set boundary curve φ(u) + φ(v) = φ(0). If φ(0) = ∞, thezero-set boundary curve has a null measure (e.g. Clayton’s copula). Ifφ(0) < ∞, zero-set boundary curve is −φ(0)/φ′(0) (e.g. Copula 4.2.2 inNelsen (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 forθ ∈ [1, +∞), as

C(u, v) = max{1− ((1− u)θ + (1− v)θ

)1/θ, 0},

with generator (1− t)θ.

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0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Generating Clayton’s copula

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Generating Clayton’s copula

Figure 9: Clayton’s copula, θ = 2 and θ = −1/2.

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0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Generating copula 4.2.2

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Generating copula 4.2.2

Figure 10: Copula 4.2.2, θ = 2 and θ = 5.

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Sequences of Archimedean copulas

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.

35


Proposition 17. 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

36


Statistical inference for Archimedean copulas

Recall that the Archimedean generator can be expressed as

ψ(u) = ψ(u0) exp(∫ u

u0

1λ(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

K̂(t) =∫ ∞

0

∫ ∞

0

1[F̂ (x, y) ≤ t]dF̂ (x, y)

where F̂ is a nonparametric estimate of the joint cdf F .

Hence, a natural estimate for φ is then

φ̂(u) = exp

(∫ u

u0

1

λ̂(t)dt

)= exp

(∫ u

u0

1

t− K̂(t)dt

)

which leads to Cbφ (see Genest & Rivest (1993)).

38


−2 −1 0 1 2

−2−1

01

2

Scatterplot

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Kendall’s distribution function

0.0 0.2 0.4 0.6 0.8 1.0

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

Lambda function

0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

6


Figure 11: Estimation of the Archimedean generator, n = 100.

39


−2 −1 0 1 2

−2−1

01

2

Scatterplot

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Kendall’s distribution function

0.0 0.2 0.4 0.6 0.8 1.0

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

Lambda function

0.0 0.2 0.4 0.6 0.8 1.00

12

34

56

7


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) = limh→0

P (V ≤ v|u ≤ U < u + h)

= limh→0

P (U < u + h, V ≤ v)− P (U < u, V ≤ v)P (U < u + h)− P (U < u)

= limh→0

C(u + h, v)− C(u, v)(u + h)− u

= limh→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 thefollowing 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

020

4060

8010

0

−2 0 2 4

−3−2

−10

12

Gaussian distribution N(0,1)

Gaus

sian d

istrib

ution

N(0,

1)0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Uniform distribution on [0,1]

Unifo

rm di

stribu

tion o

n [0,1

]

Histogram, second component

0.0 0.2 0.4 0.6 0.8 1.0

020

4060

8010

0

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 = limu→0

P(X ≤ F−1

X (u) |Y ≤ F−1Y (u)

),

λU = limu→1

P(X > F−1

X (u) |Y > F−1Y (u)

).

Those measures are copula based, i.e. λL = limu→0

C(u, u)u

For Archimedean copulas, note that

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)

.

44


ψ(t) range θ λL λU

(1) 1θ

(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−1e−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−t1+(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) ( 1t− 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) ( θt

+ 1)(1 − t) [0,∞) 1/2 0

(17) − log (1+t)−θ−12−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 copulasDefinition 18. Let U = (U1, ..., Un) be a random vector with uniformmargins, and distribution function C. Let Cr denote the copula of randomvector

(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 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

3.0Generators 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, 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).

48


Proposition 21. 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)).

49


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

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 indexα, then necessarily α ∈ [1,∞). If if (−D)ψ(1) > 0, then α = 1 (but theconverse is 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


51


Proposition 22. Let C be an Archimedean copula with generator ψ.Assume that f : s 7→ ψ(1− s) is regularly varying with index α ∈ [1,∞) andthat −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) 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

53


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

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 definecopulas 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 donesimply in the case of Archimedean copulas. Consider φ, ψ and λ threeArchimedean 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

Business

Slides ima