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Extreme-Value Copulas

Gordon Gudendorf and Johan Segers

Universite Catholique de Louvain

February, 5 , 2010

Foundations Parametric models Dependence coefficients Estimation

Let Xi = (Xi1, . . . ,Xid){1,...,n} i.i.d. random vectors withcontinuous distribution function F and marginal distributionfunctions F1, . . . ,Fd and copula CF .

F (x1, . . . , xd) = CF (F1(x1), . . . ,Fd(xd)) (1)

Vector of componentwise maxima:

Mn = (Mn,1, . . . ,Mn,d), where Mn,j =n∨

i=1

Xij , (2)

with ‘∨

’ denoting maximum, F(n) the distribution function ofMn and F(n)1, . . . ,F(n)d the marginals associated toMn,1, . . . ,Mn,d

Gordon Gudendorf and Johan Segers Extreme-Value Copulas


The copula C(n) associated to Mn is given by

C(n)(u1, . . . , ud) = CF (u1/n1 , . . . , u

1/nd )n, (u1, . . . , ud) ∈ [0, 1]d .

Proof.

C(n)(F(n)1(x1), . . . ,F(n)d(x1)) = P(Mn,1 6 x1, . . . ,Mn,d 6 xd)

= P(X11 6 x1, . . . ,X1d 6 xd)n

= CF (F1(x1), . . . ,Fd(xd))n

= CF (F(n)1(x1)1/n, . . . ,F(n)d(xd)1/n)n



Definition 1

A copula C is called an extreme-value copula if there exists acopula CF such that

CF (u1/n1 , . . . , u

1/nd )n → C (u1, . . . , ud) (n→∞) (3)

for all (u1, . . . , ud) ∈ [0, 1]d . The copula CF is said to be in thedomain of attraction of C .

Definition 2

A d-variate copula C is max-stable if it satisfies the relationship

C (u1, . . . , ud) = C (u1/m1 , . . . , u

1/md )m (4)

for every integer m > 1 and all (u1, . . . , ud) ∈ [0, 1]d .



Theorem 3

A copula is an extreme-value copula if and only if it is max-stable.

Proof.

⇐ : Trivial

⇒ :for fixed integer m > 1 and for n = mk , write

C(n)(u1, . . . , ud) = CF (u1/n1 , . . . , u

1/nd )n =

CF (u1/mk1 , . . . , u

1/mkd )mk = CF ((u

1/m1 )1/k , . . . , (u

1/md )1/k)mk =

C(k)(u1/m1 , . . . , u

1/md )m



Theorem 4

A d-variate copula C is an extreme-value copula if and only if there exists afinite Borel measure H on ∆d−1, called spectral measure, such that(u1, . . . , ud) ∈ (0, 1]d ,

C(u1, . . . , ud) = exp(−`(− log u1, . . . ,− log ud)

),

where the tail dependence function ` : [0,∞)d → [0,∞) is given by

`(x1, . . . , xd) =

∫∆d−1

d∨j=1

(wjxj) dH(w1, . . . ,wd), (5)

with (x1, . . . , xd) ∈ [0,∞)d . The spectral measure H is arbitrary except forthe d moment constraints∫

∆d−1

wj dH(w1, . . . ,wd) = 1, j ∈ {1, . . . , d}. (6)



For (x1, . . . , xd) ∈ [0,∞)d :

limt↓0

t−1(1− CF (1− tx1, . . . , 1− txd)

)= `(x1, . . . , xd),

see for instanceDrees & Huang (1998).

s→ 0

F←2 (1− sx2)

F←1 (1− sx1)

x2

x1



Definition 5 (Pickands (1981))

The restriction of ` to the unit simplex

∆d = {(w1, . . . ,wd) :d∑

i=1

wi = 1,wj ≥ 0, ∀j = 1, . . . , d}

is called Pickands dependence function A.

C (u) = exp{−`(− log u1, . . . ,− log ud)}

= exp

d∑

j=1

log uj

A

(log u1∑pj=1 log uj

, . . . ,log ud−1∑dj=1 log uj

)for 0 < uj 6 1.



For dimension d = 3:

∆3 = {(w1,w2,w3) :3∑

i=1

wi = 1,wj ≥ 0, ∀j = 1, . . . , 3}

e1 = (1, 0, 0)

e2 = (0, 1, 0)

e3 = (0, 0, 1)∆p

e1 = (1, 0)

e2 = (0, 1)

∆p



A satisfies the following properties:

1 A is convex;

2 max(w1, . . . ,wd) 6 A(w) 6 1.

⇒ A(ej) = 1, for ej = (0, . . . , 1, . . . , 0) being the d vertices.

Remark

Functions satisfying the previous conditions do not characterize theclass of multivariate extreme value copulas except in dimensiond = 2.



Theorem 6

A bivariate copula C is an extreme-value copula if and only if

C(u, v) = (uv)A(log(v)/ log(uv)), (u, v) ∈ (0, 1]2 \ {(1, 1)}, (7)

where A : [0, 1]→ [1/2, 1] is convex and satisfies t ∨ (1− t) 6 A(t) 6 1 for allt ∈ [0, 1].

Pickands dependencefunction A together withthe regiont ∨ (1− t) 6 A(t) 6 1.

A(t) = 1: Independence

A(t)

t

A(t)

A(t)

t

A(t)

A(t)

t



Archimedean copula

Archimedean copula with generator φ : [0, 1]→ [0,∞].

Cφ(u1, . . . , ud) = φ←(φ(u1)+· · ·+φ(ud)

), (u1, . . . , ud) ∈ [0, 1]d

(8)

If the following limit exists,

θ = − lims↓0

φ(1− s)

s φ′(1− s)∈ [0, 1] (9)

thenCφ(u

1/n1 , . . . , u

1/np )n → Cθ(u1, . . . , ud)

with

Cθ(u1, . . . , ud) = exp{−((− log u1)1/θ+ · · ·+(− log ud)1/θ

)θ},Gordon Gudendorf and Johan Segers Extreme-Value Copulas


Archimedean copula

Gumbel–Hougaard or logistic copula

Cθ(u1, . . . , ud) = exp{−((− log u1)1/θ+ · · ·+(− log ud)1/θ

)θ},θ: measures the degree of dependence

θ = 1 : independenceθ = 0 : complete dependence.

`(x1, . . . , xd) =

{(x

1/θ1 + · · ·+ x

1/θd )θ if 0 < θ 6 1,

x1 ∨ · · · ∨ xd if θ = 0,(10)



Example: Logistic or Gumbel copula

Gumbel (1961): A(t) = [t1/θ + (1− t)1/θ]θ



Example: Asymmetric logistic copula

Tawn (1988):

A(t) = (1− ψ1)(1− t) + (1− ψ2)t + [(ψ1t)1/θ + {ψ2(1− t)}1/θ]θ



Example: Archimedean survival copulas

Survival copula of an Archimedean copula:

Cφ(u1, u2) = u1 + u2 − 1 + φ←(φ(1− u1) + φ(1− u2))

If

θ = − lims↓0

φ(s)

sφ′(s)∈ [0,∞]

then Cφ(u1/n1 , u

1/n2 )n → C (u1, u2) where

A(t) = 1− {t−1/θ + (1− t)−1/θ}−θ

C is the negative logistic or Galambos copula



Example: Negative logistic (Galambos) copula

Galambos (1975):

A(t) = 1− {t−1/θ + (1− t)−1/θ}−θ



Example: Asym. negative logistic copula

Joe (1990)

A(t) = 1− [{ψ1(1− t)}−1/θ + (ψ2t)−1/θ]−θ



C admits the following properties:

Positive quadrant dependent, A 6 1 implies that C (u, v) > uvfor all (u, v) ∈ [0, 1]2.

Monotone regression dependent, that is, the conditionaldistribution of U given V = v is stochastically increasing in vand vice versa; see Garralda-Guillem (2000).

Kendall’s τ and Spearman’s ρS

τ = 4

∫∫[0,1]2

C (u, v) dC (u, v)− 1 =

∫ 1

0

t(1− t)

A(t)dA′(t),

ρS = 12

∫∫[0,1]2

uv dC (u, v)− 3 = 12

∫ 1

0

1

(1 + A(t))2dt − 3.



Coefficient of upper tail dependence: For a bivariate copula CF inthe domain of attraction of an extreme-value copula with taildependence function ` and Pickands dependence function A, wefind

λU = limu↑1

P(U > u | V > u)

= limt↓0

t−1(2t − 1 + CF (1− t, 1− t))

= 2− `(1, 1) = 2(1− A(1/2)

)∈ [0, 1].

λL = limu↓0

P(U 6 u | V 6 u)

= limu↓0

u(2A(1/2)−1)

A(t)

t

(1−A(1/2))

A(t)

t



Estimation

Parametric estimation: not treated here

Nonparametric estimation

Known margins: apply probability integral transform (i.i.d.sample from C )

(Ui ,1, . . . ,Ui ,d) := (F1(Xi ,1), . . . ,Fd(Xi ,d))

Unknown margins: use ranks (empirical copula)

(Ui , . . . , Ui ,d) :=

(Ri ,1

n + 1, . . . ,

Ri ,d

n + 1

), where Ri ,1 =

∑nk=1 I (Xk,1 6 Xi ,1).



Pickands estimator

For t ∈ [0, 1], define

ξi (t) = min

(log Ui ,1

1− t,

log Ui ,2

t

),

with the obvious conventions for division by zero.

P[ξi (t) > x ] = P[Ui < e−(1−t)x ,Vi < e−tx ]

= C (e−(1−t)x , e−tx) = e−x A(t). (11)

Pickands (1981) proposed the estimator:

1

AP(t)=

1

n

n∑i=1

ξi (t). (12)



Deheuvels estimator:

Deheuvels (1991):

1

AD(t)=

1

n

n∑i=1

ξi (t)− t1

n

n∑i=1

ξi (1)− (1− t)1

n

n∑i=1

ξi (0) + 1.

(13)

Endpoint constraints verified: AD(0) = AD(1) = 1

Weights (1− t) and t rather pragmatic choices:



Deheuvels estimator:

Segers (2007):

1

AD(t)=

1

n

n∑i=1

ξi (t)− β1(t)1

n

n∑i=1

ξi (1)− β2(t)1

n

n∑i=1

ξi (0) + 1.

(14)Variance-minimizing weight functions via a linear regression:

ξi (t) = β0(t) + β1(t) {ξi (0)− 1}+ β2(t) {ξi (1)− 1}+ εi (t).

β0(t): minimum-variance estimator for 1/A(t) in the class ofDeheuvels estimators.



CFG-estimator

Caperaa, Fougeres and Genest (1997):

log ACFG (t) = −1

n

∑i=1

log ξi (t)− (1− t)1

n

n∑i=1

log ξi (0)− t1

n

n∑i=1

log ξi (1) (15)

for t ∈ [0, 1], using

E[− log ξi (t)] = log A(t) + γ, t ∈ [0, 1],

with the Euler–Mascheroni constant γ = 0.5772 . . .

log ACFG (t) = −1

n

∑i=1

log ξi (t) + β1(t)1

n

n∑i=1

log ξi (0) + β2(t)1

n

n∑i=1

log ξi (1).



Estimation

Hall-Tajvidi (2000)

Optimize weight functions β1(w), . . . , βd(w); see Segers(2007) and Gudendorf & Segers (2009).

Estimation using unknown margins; see Genest & Segers(2009).

Other estimators . . .



Shape constraints

Even after endpoint correction, the estimators still do notsatisfy the shape constraints

A is convexmax(t, 1− t) ≤ A(t) ≤ 1

Possible solutions:

Convex minorant to {A(t) ∨ t ∨ (1− t)} ∧ 1 (Deheuvels1991)Spline smoothing (Hall & Tajvidi 2000)L2 projection (Fils-Villetard, Guillou & Segers 2008)



Reading

References can be found in:

G. Gudendorf & J. Segers (2009)“Extreme-Value Copulas”IS - DP 2009-26


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