Multivariate Archimax copulas

Anne-Laure FougeresInstitut Camille Jordan, Universite Lyon 1

joint work with

A. Charpentier, Ch. Genest and J.G. Neslehova

November 3, 2014

Workshop “Extreme Value Theory, Spatial and Temporal Aspects”

Besancon

I. Motivation

II. Cψ,L is a copula

III. Stochastic representations for Archimax copulas

IV. Simulation algorithms

V. Extremal behavior of Archimax copulas

VI. Conclusion - Perspectives

I. Motivation

Multivariate risks often deal with extremes of dependent variables:

I Alimentary risks: Global exposition to the contamination riskvia a set of aliments.

I Insurance risks: ruin probabilities, when several types ofcontracts are concerned (natural disaster).

I Coastal flooding: electrical infrastructures, dikes.

Multivariate extreme-value theory provides a usefulmathematical framework to handle such risks.

Consider a d-variate sample X1, . . . ,Xn, with Xi = (X i1, . . . ,X

id),

for each i = 1, . . . , n. Define

P(Xi ≤ x) = F (x) = C (F1(x1), . . . ,Fd(xd)) ,

so that F1, . . . ,Fd are the marginal cdfs (assume them continuous),F is the joint cdf, and C is the associated copula.

Assumption: existence of a multivariate domain of attraction

There exist (an), (bn),G such that, when n→∞,

F n(an,1 x1 + bn,1, . . . , an,d xd + bn,d) = F n(an x + bn)→ G (x),

where the attractor G is a d-variate cdf with non degeneratemargins G1, . . . ,Gd , and x is any continuity point of G .

This means equivalently that:

. the marginal cdfs Fj are “in the univariate domain of attraction”of the Gj ’s (j = 1, . . . , d).

. there exists a d-variate copula C ? such that for any u ∈ [0, 1]d ,

limn→∞

C (u1/n1 , . . . , u

1/nd )n = C ?(u1, . . . , ud) , (1)

and the limiting cdfs are related via G (x) = C?(G1(x1), . . . ,Gd(xd)).

Notation: F ∈ D(G ) or C ∈ D(C ?).

Equation (1) is equivalent to

n[1− C

(1− x1

n, . . . , 1− xd

n

)]−→ − log C ?(e−x1 , . . . , e−xd ) = L?(x) .

L? = stable tail dependence function (Huang, 1992)

L?(x) = limn→∞

n[1− C

(1− x1

n, . . . , 1− xd

n

)]= lim

n→∞n P[F1(X1) > 1− x1

nor . . . or Fd(Xd) > 1− xd

n

].

Tail regions of interest for L?:

at least one of the components X1, . . . ,Xd becomes large.

Some properties of the stable tail dependence function L

I LM(x) := max(x1, . . . , xd) ≤ L(x) ≤ LΠ(x) := x1 + · · ·+ xd

comonotonicity case independence case

I margins are standardized: L(0, . . . , 0, xj , 0, . . . , 0) = xj

I L is homogeneous of order 1

L(αx) = lims→∞

s

[1− C

(1− x1

s/α, . . . , 1− xd

s/α

)]= lim

t→∞αt[1− C

(1− x1

t, . . . , 1− xd

t

)]= αL(x) .

I L is convex, i.e. for each λ ∈ [0, 1],

Lλx + (1− λ)y ≤ λL(x) + (1− λ)L(y) .

Question: fix an attractor C ? (equivalently L?); which kind ofdistribution F does belong to its domain of attraction?

I theoretical descriptive interest.

I practical modeling interest: To get large families with aflexible structure in a specific domain of attraction.

I numerical interest: Risk evaluation requires estimation of C ?.Several estimators exist. How to compare them? For a smallsample simulation study, designs of experiments involve to

. fix several attractors C?;. simulate, for each attractor, from several distributionsC ∈ D(C?).

Attractor of classical multivariate distributions

I Multivariate normal d.f. −→ Independence

I Archimedean copulas

Cψ(u1, . . . , ud) = ψψ−1(u1) + · · ·+ ψ−1(ud)

where the generator ψ : R+ → [0, 1] satisfies specific conditions.

Archimedean copulas −→ Multivariate logistic EV

L?(x1, . . . , xd) =∑d

j=1 x1/rj

r

(with r ≥ 1).

But in fact Independence case (r = 1) for

. Clayton’s family ψα(t) = (1 + αt)−1/α, α > 0

. Frank’s family ψα(t) = − log 1− e−t(1− e−α) /α

I Elliptical distributions

X = µ+ RAU

where µ location parameter, R random radial component,A d × d-matrix invertible such that AAT positive definite, andU random d-vector uniformly distributed on Sd−1.

Elliptical distribution −→ Independencewith R rapidly varying

I Extreme value d.f. CA −→ itself CA

Objective: Construct a family of multivariate copulas

I which can have any extreme value distribution as itsmaximum attractor

I which is easy to simulate.

Caperaa, Fougeres and Genest (2000) : bivariate Archimax copulas

Cψ,A(u1, u2) = ψ

[ψ−1(u1) + ψ−1(u2)A

ψ−1(u1)

ψ−1(u1) + ψ−1(u2)

],

(2)where A : [0, 1]→ [1/2, 1] and ψ : [0,∞)→ [0, 1] such that

(i) A is convex and, for all t ∈ [0, 1], max(t, 1− t) ≤ A(t) ≤ 1;

(ii) ψ : (0, 1]→ [0,∞) is convex, decreasing, such that ψ(0) = 1 andlimx→∞ ψ(x) = 0.

“Archimax” because... two important particular cases

I if A ≡ 1, Cφ,A reduces to an Archimedean copula,

Cψ(u1, u2) = ψψ−1(u1) + ψ−1(u2)

I if ψ(t) = e−t , Cψ,A is an extreme-value copula,

CA(u1, u2) = exp

[ln(u1u2)A

ln(u1)

ln(u1u2)

].

Result: Archimax copulas are in the domain of attraction of anEV copula CA? where, for all t ∈ (0, 1),

A?(t) = t1/α + (1− t)1/ααAα

t1/α

t1/α + (1− t)1/α

whenever t 7→ ψ−1(1− 1/t) is regularly varying of degree−1/α with α ∈ (0, 1];

Additional references

I Application in hydrology: see Basigal, Jagr and Mesiar (2011);

I Applications in finance: see Zivot and Wang (2006), Jaworski,Durante and Hardle (2013), Mai and Scherer (2014);

I R package: acopula (Basigal);

I Mesiar and Jagr (2013): conjecture that a suitable extensionto arbitrary dimension should be

Cψ,L(u1, . . . , ud) = ψ Lψ−1(u1), . . . , ψ−1(ud). (3)

Open problem 4.1 (Mesiar and Jagr, 2013) : Cψ,L is a copula assoon as L is a stable tail dependence function and ψ is anArchimedean generator.[sounds reasonable, since for d = 2, A(t) = L(t, 1− t), so that (3) is (2).]

Purpose of our work

I prove that Cψ,L is a copulasolve Open problem of Mesiar and Jagr, 2013

I study the d-variate Archimax family

. in terms of attractor;

. in terms of simulation issues.

Refer to Charpentier, Fougeres, Genest and Neslehova (2014),JMVA 126, pp. 118-136.

Cψ,L is a copula: main ingredients

For all u1, . . . , ud ∈ (0, 1), consider

Cψ(u1, . . . , ud) = ψψ−1(u1) + · · ·+ ψ−1(ud).

1. Characterization of an Archimedean generator

Then Cψ is a copula if and only if ψ : [0,∞)→ [0, 1] satisfies

. ψ(0) = 1,

. limx→∞ ψ(x) = 0

. ψ is d-monotone, i.e. ψ has d − 2 derivatives on (0,∞),(−1)jψ(j) ≥ 0 (for all j ∈ 0, . . . , d − 2), and (−1)d−2ψ(d−2)

non-increasing and convex on (0,∞).

McNeil and Neslehova (2009)

Cψ,L is a copula: main ingredients (cont.)

2. Characterization of a stable tail dependence function [stdf]

L : [0,∞)d → [0,∞) is a d-variate stdf if and only if

(a) L is homogeneous of degree 1;

(b) L(e1) = · · · = L(ed) = 1;

(c) L is fully d-max decreasing, i.e., for any J ⊆ 1, . . . , d ofarbitrary size |J| = k and all x1, . . . , xd , h1, . . . , hd ∈ [0,∞),∑ι1,...,ιk∈0,1

(−1)ι1+···+ιk L(x1+ι1h111∈J , . . . , xd+ιdhd1d∈J) ≤ 0.

Ressel (2013)

Remark: (c) is equivalent to f : (−∞, 0]d → (−∞, 0] defined by

f (y1, . . . , yd) = −L(−y1, . . . ,−yd) (4)

is totally increasing as defined in Morillas (2005), which states∑ι1,...,ιk∈0,1

(−1)k−ι1−···−ιk f (y1 + ι1h111∈J , . . . , yd + ιdhd1d∈J) ≥ 0.

First result

TheoremLet L be a d-variate stdf and ψ be the generator of a d-variateArchimedean copula. There exists a vector (X1, . . . ,Xd) of strictlypositive random variables such that, for all x1, . . . , xd ∈ [0,∞),

Pr(X1 > x1, . . . ,Xd > xd) = ψ L(x1, . . . , xd).

In particular, Pr(Xj > xj) = ψ(xj) for xj ∈ [0,∞) and j ∈ 1, . . . , d.

Sketch of proof:

I Morillas (2005) - McNeil and Neslehova (2009) :ψ Archimedean generator ⇔ ψ† absolutely monotone of order dwhere ψ† : t ∈ (−∞, 0] 7→ ψ(−t) ∈ [0, 1].

I Morillas (2005) + (c) ⇒ ψ† f totally increasing.

I L satisfies (b)⇒ ψ† f (y1, 0, . . . , 0) = ψ†[−L(−y1, 0, . . . , 0)] = ψ†(y1).

I ψ continuous ⇒ ψ† f continuous.

Consequence: ψ† f is a cdf on (−∞, 0]d . This meansequivalently that ψ† f (−x1, . . . ,−xd) = ψ L(x1, . . . , xd) is asurvival function on [0,∞)d .

Corollary

Let L be a d-variate stable tail dependence function and ψ be thegenerator of a d-variate Archimedean copula. Then

Cψ,L(u1, . . . , ud) = ψ Lψ−1(u1), . . . , ψ−1(ud)

is a copula, as conjectured by Mesiar and Jagr (2013).

Some examples

I Recall that LΠ(x) := x1 + · · ·+ xd . Then Cψ,LΠis the

d-variate Archimedean copula Cψ.

I If ψ(t) = e−t , Cψ,L is the extreme-value copula with stdf L

Cψ,L(u1, . . . , ud) = exp[−L− ln(u1), . . . ,− ln(ud)] .

I Let θ ≥ 1 and consider the stdf of the d-variate logisticextreme-value copula

Lθ(x1, . . . , ud) = (xθ1 + · · ·+ xθd )1/θ.

Then for any generator ψ,

Cψ,θ(u1, . . . , ud) = ψ

[[ψ−1(u1)θ + · · ·+ ψ−1(ud)θ

]1/θ]

is an Archimedean copula with generator ψθ(t) = ψ(t1/θ).

III. Stochastic representations for Archimax copulas

1. ψ is a Laplace transform. Suppose that ψ is the Laplacetransform of a strictly positive r.v. Θ with cdf G , so that

ψ(t) =

∫ ∞0

e−tθ dG (θ).

Bernstein’s Theorem (Widder, 1941) ⇒ ψ is completely monotone,i.e., it is differentiable of any order and for all k ∈ N, (−1)kψ(k) ≥ 0.

Let L be a d-variate stdf. Let (T1, . . . ,Td) be a random vectorwith survival function

Pr(T1 > t1, . . . ,Td > td) = exp−L(t1, . . . , td). (5)

This means T1, . . . ,Td ∼ E(1) with survival copula theextreme-value copula with stable tail dependence function L.

Stochastic representations for Archimax copulas (cont.)

TheoremThe copula Cψ,L is Archimax with d-variate stdf L and completelymonotone Archimedean generator ψ if and only if it is the survivalcopula of the random vector

(X1, . . . ,Xd) = (T1/Θ, . . . ,Td/Θ) ,

where Θ has Laplace transform ψ and is independent of therandom vector (T1, . . . ,Td) defined in (5).

Sketch of proof:

Pr(X1 > x1, . . . ,Xd > xd) =

∫ ∞0

Pr(T1 > θx1, . . . ,Td > θxd) dG (θ)

=

∫ ∞0

exp−θL(x1, . . . , xd) dG (θ)

= ψ L(x1, . . . , xd).

Stochastic representations for Archimax copulas (cont.)

2. ψ d-monotone. Consider ψ0(t) = max(0, 1− t)d−1 (t ≥ 0).

ψ0 is d-monotone ⇒ there exists (S1, . . . ,Sd) such that,

Pr(S1 > s1, . . . ,Sd > sd) = [max0, 1− L(s1, . . . , sd)]d−1.

ThenI support of this joint survival function:

Ωd(`) = (s1, . . . , sd) ∈ [0, 1]d : L(s1, . . . , sd) ≤ 1

I S1, . . . ,Sd are (dependent) Beta r.v. B(1, d − 1).

Now let R be a strictly positive r.v. with cdf F , independent of(S1, . . . ,Sd) and consider

(X1, . . . ,Xd) = (RS1, . . . ,RSd). (6)

Stochastic representations for Archimax copulas (cont.)

Theorem

(i) If (X1, . . . ,Xd) has form (6), then its survival copula is theArchimax copula Cψ,L, where ψ is the Williamson d-transformof R, i.e., for all t ∈ [0,∞),

ψ(t) =

∫ ∞t

(1− t

r

)d−1

dF (r).

(ii) Let L be a d-variate stdf and ψ be a generator of a d-variateArchimedean copula. Then Cψ,L is the survival copula of arandom vector (X1, . . . ,Xd) of the form (6), where the cdf Fof R is the inverse Williamson d-transform of ψ,

F (r) = 1−d−2∑k=0

(−1)k rkψ(k)(r)

k!−

(−1)d−1rd−1ψ(d−1)+ (r)

(d − 1)!,

where ψ(d−1)+ denotes the right-hand derivative of ψ(d−2).

Corollary

A function L : [0,∞)d → [0,∞) is a d-variate stdf if and only if

(a) L is homogeneous of degree 1;

(b) The function given, for all x1, . . . , xd ∈ [0,∞), by

G`(x1, . . . , xd) = [max0, 1− L(x1, . . . , xd)]d−1 (7)

defines a d-variate survival function with B(1, d − 1) margins.

Remark: The distribution in (7) is related to the multivariategeneralized Pareto distribution of Falk and Reiss (2005). See alsoHofmann (2009).

Reminder: Purpose of our work

I prove that Cψ,L is a copulasolve Open problem of Mesiar and Jagr, 2013

I study the d-variate Archimax family

. in terms of simulation issues;

. in terms of attractor.

IV. Simulation algorithms

Algorithm 1Let L be the d-variate stdf associated to an extreme-value copulaD, and let ψ be a d-variate Archimedean copula generator.Suppose that ψ is the Laplace transform of a r.v. Θ.

To simulate an observation (U1, . . . ,Ud) from a d-variateArchimax copula Cψ,L, proceed as follows:

1.1 Generate an observation (V1, . . . ,Vd) from copula D.

1.2 Set T1 = − ln(V1), . . . ,Td = − ln(Vd).

1.3 Generate an observation Θ.

1.4 Set U1 = ψ(T1/Θ), . . . ,Ud = ψ(Td/Θ).

Algorithm 2Let L be a d-variate stdf, and let ψ be a d-variate Archimedeancopula generator.

To simulate an observation (U1, . . . ,Ud) from Cψ,L:

2.1 Generate an observation (S1, . . . ,Sd) from the joint survivalfunction defined, for all s1, . . . , sd ∈ [0,∞), by

G`(s1, . . . , sd) = [max0, 1− L(s1, . . . , sd)]d−1.

2.2 Generate R from the cdf defined, for all r ∈ (0,∞), by

F (r) = 1−d−2∑k=0

(−1)k rkψ(k)(r)

k!−

(−1)d−1rd−1ψ(d−1)+ (r)

(d − 1)!.

2.3 Set U1 = ψ(RS1), . . . ,Ud = ψ(RSd).

V. Extremal behavior of Archimax copulas

Let X1,X2, . . . be iid copies of a vector X = (X1, . . . ,Xd) whosedistribution is the Archimax copula Cψ,L, and define for each n ∈ N,

Mn = max(X1, . . . ,Xn),

where vector algebra is meant component-wise.

Objective:find the limiting behavior, as n→∞, of the sequence (Mn).

Reminder for equation (1):

limn→∞

Cψ,L(u1/n1 , . . . , u

1/nd )n = C ?(u1, . . . , ud) .

Extremal behavior of Archimax copulas (cont.)

TheoremSuppose that ψ is the generator of a d-variate Archimedean copulasuch that w 7→ 1− ψ(1/w) is regularly varying of index −α forsome α ∈ (0, 1]. Then the copula Cψ,L belongs to the maximumdomain of attraction of an extreme-value distribution whose uniqueunderlying copula is defined, for all u1, . . . , ud ∈ (0, 1), by

CL?(u1, . . . , ud) = exp[−Lα| ln(u1)|1/α, . . . , | ln(ud)|1/α].

VI. Conclusion - Perspectives

Our purpose has been to

I prove that Cψ,L is a copulaas conjectured by Mesiar and Jagr, 2013

I study the d-variate Archimax family. in terms of simulation issues;

. in terms of attractor.

Some questions remains to be considered:

I computational issues associated with Algorithms 1 and 2.

I dependence structure.

For d = 2, Caperaa, Fougeres and Genest (2000) :

τψ,L = τL + (1− τL)τψ.

How to extend this relation to the multivariate case ?

References (1/2)

. T. Bacigal, V. Jagr, R. Mesiar (2011), Non-exchangeable random variables,Archimax copulas and their fitting to real data, Kybernetika 47, 519-531.

. P. Caperaa, A.-L. Fougeres, C. Genest (2000), Bivariate distributions with givenextreme value attractor, J. Multivariate Anal. 72, 30-49.

. A. Charpentier, A.-L. Fougeres, C. Genest, J.G. Neslehova (2014), MultivariateArchimax copulas, J. Multivariate Anal. 126, 118-136.

. M. Falk, R.-D. Reiss (2005), On Pickands coordinates in arbitrary dimensions, J.Multivariate Anal. 92 426-453.

. A.J. McNeil, J. Neslehova (2009), Multivariate Archimedean copulas,d-monotone functions and `1-norm symmetric distributions, Ann. Statist. 37,3059-3097.

. J. F. Mai, M. Scherer (2014) Financial Engineering with Copulas Explained,Palgrave Macmillan.

. D. Hofmann (2009), Characterization of the D-norm corresponding to amultivariate extreme value distribution, Ph.D.Thesis, BayerischeJulius-Maximilians-Universitat Wurzburg, Germany.

References (2/2)

. X. Huang (1992), Statistics of bivariate extreme values, Ph.D. Thesis,Tinbergen Institute Research Series, The Netherlands.

. P. Jaworski, F. Durante, W. K. Hardle (2013) Copulae in Mathematical andQuantitative Finance, Lecture Notes in Statistics, Vol. 213. Springer.

. R. Mesiar, V. Jagr (2013), d-dimensional dependence functions and Archimaxcopulas, Fuzzy Sets and Systems 228, 78-87.

. P.M. Morillas (2005), A characterization of absolutely monotonic functions of afixed order, Publ. Inst. Math. (Beograd) (N.S.) 78 (92), 93-105.

. P. Ressel (2013), Homogeneous distributions - and a spectral representation ofclassical mean values and stable tail dependence functions, J. Multivariate Anal.117, 246-256.

. D.V. Widder (1941), The Laplace Transform, in: Princeton MathematicalSeries, vol. 6, Princeton University Press, Princeton, NJ.

. E. Zivot, J. Wang (2006) Modeling Financial Time Series with Splus, Springer.