Multidimensional extremal dependence coefficients

0.17 MB·

by  Helena Ferreira

Multidimensional extremal dependence coefficients
Helena Ferreira
Universidade da Beira Interior, Centro de Matemática e Aplicações (CMA-UBI),
Avenida Marquês d’Avila e Bolama, 6200-001 Covilhã, Portugal
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1 [email protected]
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2

n
Marta Ferreira
a
J

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1 Center of Mathematics of Minho University

] Center for Computational and Stochastic Mathematics of University of Lisbon
T
S
Center of Statistics and Applications of University of Lisbon, Portugal
.
h
at [email protected]
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[

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v Abstract
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4 Extreme values modeling has attracting the attention of researchers in diverse areas such as
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1 the environment, engineering, or finance. Multivariate extreme value distributions are particularly
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suitable to model the tails of multidimensional phenomena. The analysis of the dependence among
.
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0 multivariate maxima is useful to evaluate risk. Here we present new multivariate extreme value
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1 models, as well as, coefficients toassess multivariate extremal dependence.
:
v
i
X keywords: multivariateextremevaluemodels,taildependence,extremalcoefficients,randomfields
r
a
AMS 2000 Subject Classification: 60G70
1 Introduction
LetX={X(x),x∈Rm}bearandomfield. ForafixedsetoflocationsL={x ,..., x }⊂Rm and
1 d
some partition L = {x ,..., x }, L = {x ,..., x }, ..., L = {x ,..., x }, with 1 ≤
1 1 i1 2 i1+1 i2 p ip−1+1 d
p≤d,considertherandomvectorsX =(X(x ),..., X(x )),...,X =(X(x ),..., X(x )).
L1 1 i1 Lp ip−1+1 d
1
Wearegoingtoevaluatethedependencebetweenthevectorsthroughcoefficients,thatis,thedepen-
dencebetweenthemarginals ofXoverdisjoint regions L ,..., L . Examplesofapplications within
1 p
this context can be found in Naveau et al. ([12] 2009) and Guillou et al. ([9] 2014) for d = p = 2,
i.e.,twolocations,inFonsecaetal. ([8]2015)ford>2andp=2,i.e.,twogroupofseverallocations
and Ferreira and Pereira ([6] 2015) for d=p>2, i.e., several isolated locations.
In the applications, in order to study the dependence between sub-vectors of X we can form an
auxiliary vector (Y ,...,Y ) where each variable Y somehow summarizes the information of X ,
1 p i Li
i = 1,...,p, and study the dependence between the variables Y . This is the approach followed
i
by some authors (Naveau et al. [12] 2009; Marcon et al. [11] 2016). In our proposal to infer the
dependence between clusters of variables, we deal directly with the vectors X , i = 1,...,p. On
Li
the other hand, if the random field is vectorial, that is, for each location x , X(x ) is a vector
i i
(X1(x ),...,Xs(x )), whenever we think of the dependence between X(x ), ..., X(x ) we have
i i 1 d
dependencybetween vectors.
The dependencebetween therandom vectors X , X , ..., X can be characterized through
L1 L2 Lp
theexponentmeasure
ℓx1,...,xd(t1,..., td)=−lnF(X(x1),...,X(xd))(t1,..., td),
whereF(X(x1),...,X(xd)) denotesthedistributionfunction(df)ofXL=(X(x1),..., X(xd)). IfXisa
max-stablerandomfieldwithunitFréchetmarginals,thenℓx1,...,xd ishomogeneousoforder−1and
thepolar transformation used in thePickands representation allows us to see it as a moment-based
tail dependencetool (see, e.g., Finkenstädt and Rootzén [7] 2003 or Beirlant et al. [1] 2004).
Ourproposalalsoaddressesℓx1,...,xd asafunctionofmomentsoftransformationsofXL. Specif-
ically, themoments
p
e(λ ,..., λ )=E Fλj (X(x )) , (λ ,..., λ )∈(0,∞)p,
1 p  X(xi) i  1 p
j_=1xi_∈Lj
 
where a∨b = max(a,b). If p = d = 2, 1e(λ,1−λ) equals the λ-madogram of Naveau et al. ([12]
2
2009), unless the addition of constant 1(E(Uλ)+E(U1−λ)) where U is standard uniform. When
2
p=d≥2,e(λ−1,..., λ−1)with d λ =1equalsthegeneralizedmadogramconsideredinMarcon
1 d j=1 j
et al. ([11] 2016), unless theaddPition of constant 1 d E Uλ−j1 .
d j=1
(cid:16) (cid:17)
P
Herewe also consider a shifted e(λ ,..., λ ) by subtracting theconstant
1 p
p
1 E Fλj (X(x )) .
p  X(xi) i 
Xi=1 xi_∈Lj
 
The referred works consider max-stable random fields with standard Fréchet marginals, except
2
Guillouetal.([9]2014)whereℓ (t ,t )ishomogeneousoforder−1/ηandF (t)=P(X(x )≤
x1,x2 1 2 X(xi) i
t)=exp(−σ(x )t−1/η), i=1,2, η ∈(0,1], corresponding to the bivariate extreme values model ob-
i
tained in Ramos and Ledford ([14] 2011).
We will also consider that F(X(x1),...,X(xd)) is such that ℓx1,...,xd(t1,..., td) is homogeneous of
order −1/η and F (t) = P(X(x) ≤ t) = exp(−σ(x)t−1/η) for some constants σ(x) > 0 and
X(x)
η ∈ (0,1]. Under this hypothesis, which includes all the other mentioned works whenever η = 1
and σ(x)= 1, we define extremal dependence functions that provide us coefficients to measure the
dependenceamong X , ..., X through the dependencebetween M(L ), j =1,..., p and relate
L1 Lp j
theextremalcoefficientswiththeuppertaildependencefunctionintroducedinFerreiraandFerreira
([4] 2012) (Section 2). We compute the extremal coefficients for several choices of F(X(x1),...,X(xd))
inSection3. Finallyweconsideranasymptotictailindependencecoefficienttomeasurean“almost"
independencefor a class of models wider than max-stable ones (Section 4).
In order to simplify notations, we will write X instead of X(x ) and, for any vector a and any
i i
subset of its indexesS, we will write a to denote thesub-vectorof a with indexesin S.
S
2 Model and coefficients of multivariate extremal depen-
dence
Let I ={1,..., d} and I ={α(I )=1,..., ω(I )}, I ={α(I )=ω(I )+1,..., ω(I )}, ..., I =
1 1 1 2 2 1 2 p
{α(I ) =ω(I )+1,..., ω(I ) = d} be a partition of I, 1 ≤p ≤d. Consider X =(X ,..., X )
p p−1 p I 1 d
hasdf F and univariate marginals F such that
XI i
(i) F (t)=exp −σ t−1/η , i=1,..., d
i i
(cid:16) (cid:17)
(ii) ℓ (t ,...,t )=−lnF (t ,...,t ) is homogeneous of order −1/η,
XI 1 d XI 1 d
for some constants σ >0 and η∈(0,1]. Thus, thecopula C of F is max-stable, i.e.
i XI XI
C (us,...,us)=Cs (u ,..., u ), s>0. (1)
XI 1 d XI 1 d
In thefollowing we use notation M(I)= F (X ).
i∈I i i
W
Lemma 2.1. If X = (X ,..., X ) satisfies conditions (i) and (ii) then, for all (u ,...,u ) ∈
I 1 d 1 p
(0,1)p,
p σ η p σ η
P(M(I )≤u ,..., M(I )≤u )=exp −ℓ − 1 δ (I ),..., − d δ (I ) .
1 1 p p ( XI j=1(cid:18) lnuj(cid:19) 1 j j=1(cid:18) lnuj(cid:19) d j !)
X X
3
Proof. Wehavesuccessively
P(M(I )≤u ,..., M(I )≤u )
1 1 p p
p p
= C u δ (I ),..., u δ (I )
XI j 1 j j d j !
j=1 j=1
X X
p p
= exp −ℓ F−1 u δ (I ) ,..., F−1 u δ (I ) .
( XI 1 j 1 j ! d j d j !!)
j=1 j=1
X X
Analogously, we obtain, for 1≤j <j′ ≤p,
P(M(Ij)≤uj,M(Ij′)≤uj′)
σ η σ η
= exp −ℓ − α(Ij∪Ij′) δ (I ),..., − ω(Ij∪Ij′) δ (I ) ,
 XIj∪Ij′ i∈X{j,j′}(cid:18) lnui (cid:19) α(Ij∪Ij′) i i∈X{j,j′}(cid:18) lnui (cid:19) ω(Ij∪Ij′) i 
 
 
where α(Ij∪Ij′) and ω(Ij∪Ij′) denote thefirst and last point of Ij∪Ij′, respectively.
Lemma 2.2. If X = (X ,..., X ) satisfies conditions (i) and (ii) then, for all (λ ,..., λ ) ∈
I 1 d 1 p
(0,∞)p,
p p
ℓ ση ληδ (I ),..., ση ληδ (I )
p XI 1 j 1 j d j d j !
E M(I )λj = Xj=1 Xj=1 . (2)
j p p
!
j_=1 1+ℓXI σ1η ληjδ1(Ij),..., σdη ληjδd(Ij)!
j=1 j=1
X X
Proof. From Lemma 2.1 and byapplying thehomogeneity of order−1/η of ℓX , we have
I
p
P M(Ij)≤uλ−j1 =uℓXI(σ1η pj=1ληjδ1(Ij),...,σdη pj=1ληjδd(Ij))
! P P
j=1
_
and
E j=p1M(Ij)λj!=Z01uℓXI(σ1ηPpj=1ληjδ1(Ij),...,σdηPpj=1ληjδd(Ij))ℓXI σ1ηj=p1ληjδ1(Ij),..., σdηj=p1ληjδd(Ij)!du,
_ X X
which leads tothe result.
The naturalextension of the madogram toour context is thefunction
p
1
ν (λ ,..., λ )=e(λ ,..., λ )− E M(I )λj , (λ ,..., λ )∈(0,∞)p.
XI1,...,XIp 1 p 1 p p j 1 p
Xi=1 (cid:16) (cid:17)
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Motivated by therelation between E pj=1M(Ij)λj and ℓXI presented in Lemma 2.2,we first
propose thefollowing definition for theex(cid:16)tWremal depend(cid:17)encefunction between X ,..., X .
I1 Ip
Definition2.1. IfX =(X ,..., X ) satisfies conditions (i)and (ii)then the extremal dependence
I 1 d
function ε (λ ,..., λ ) among X ,..., X is defined by
XI1,...,XIp 1 p I1 Ip
ε (λ ,..., λ )= E pj=1M(Ij)λj , (λ ,..., λ )∈(0,∞)p.
XI1,...,XIp 1 p 1−E(cid:16)W pj=1M(Ij)λ(cid:17)j 1 p
(cid:16) (cid:17)
W
AsaconsequenceofLema2.2andDefinition2.1whichcomparesthedistancesofE pj=1M(Ij)λj ∈
(cid:16) (cid:17)
(0,1)tozeroandone,wehavethefollowingpropertythatdisclosesε (λ ,...,Wλ )asamea-
XI1,...,XIp 1 p
sureof thedependencebetween X ,..., X .
I1 Ip
Proposition 2.3. If X =(X ,..., X ) satisfies conditions (i) and (ii) then, for all(λ ,..., λ )∈
I 1 d 1 p
(0,∞)p,
p p
ε (λ ,..., λ )=ℓ ση ληδ (I ),..., ση ληδ (I ) .
XI1,...,XIp 1 p XI 1 j 1 j d j d j !
j=1 j=1
X X
Therefore, the extremal dependence function among X ,..., X at the point (λ ,..., λ ) co-
I1 Ip 1 p
incides with thetail dependencefunction of X at thepoint
I
((σ λ )η,..., (σ λ )η, (σ λ )η,..., (σ λ )η,..., (σ λ )η,..., (σ λ )η).
1 1 ω(I1) 1 α(I2) 2 ω(I2) 2 α(Ip) p ω(Ip) p
In thecontext of thevalidity of conditions (i) and (ii), by Proposition 2.3, we have
ε (1,...,1)=ℓ (ση,..., ση), (3)
XI1,...,XIp XI 1 d
ε (1,1)=ℓ ση ,..., ση , ση ,..., ση , 1≤j <j′ ≤p
XIj,XIj′ XIj∪Ij′ α(Ij) ω(Ij) α(Ij′) ω(Ij′)
(cid:16) (cid:17)
and
ε (1)=ℓ ση ,..., ση , 1≤j ≤p.
XIj XIj α(Ij) ω(Ij)
(cid:16) (cid:17)
Note that, when η =1= σ , i=1,..., d, ε (1,..., 1) coincides with the usual concept of
i XI1,...,XIp
extremal coefficient ε of X. Under this framework, the family of possible extremal coefficients of
X
all sub-vectorsof X is characterized in Strokorb and Schlather ([15] 2012).
Moreover,sinceF isamultivariateextremevalues(MEV)model,wehave,fort=(t ,...,t ),
XI 1 d
p p
ℓ (t )≤ℓ (t)≤ ℓ (t ),
XIj Ij XI XIj Ij
j=1 j=1
^ X
5
which,alongwithProposition2.3,alowustoboundtheextremaldependencefunctionofX ,...,X .
I1 Ip
Proposition 2.4. If X =(X ,..., X ) satisfies conditions (i) and (ii) then, for all(λ ,..., λ )∈
I 1 d 1 p
(0,∞)p, we have
p p
λ−1ε (1)≤ε (λ ,...,λ )≤ λ−1ε (1),
j XIj XI1,...,XIp 1 p j XIj
j=1 j=1
^ X
withtheupperboundcorrespondingtoindependentrandomvectorsX ,..., X andthelowerbound
I1 Ip
to totally dependent margins X ,...,X .
1 d
Observe that, if X ,..., X are totally dependent vectors, then the copula of X is the minimum
I1 Ip
copula (Nelsen [13] 2006).
Now we analyze how εXIi,XIj′(λj,λj′) relates with the dependence within the tails of XIi and
X , 1 ≤ j < j′ ≤ p. Analogously to Ferreira and Ferreira ([4] 2012), we are going to consider an
Ij′
uppertail dependencefunction of vector(X ,X ) given by thecommon valueof
Ij Ij′
tl→im∞P(M(Ij)>1−λj/t|M(Ij′)>1−λj′/t)λj′εXIj′(1) (4)
and
tl→im∞P(M(Ij′)>1−λj′/t|M(Ij)>1−λj/t)λjεXIj(1). (5)
Considering thefirst limit, observe that
lim P(M(Ij)>1−λj/t|M(Ij′)>1−λj′/t)
t→∞
(6)
= lim 1+ 1−P(M(Ij)≤1−λj/t) − 1−P(M(Ij)≤1−λj/t,M(Ij′)≤1−λj′/t)
t→∞(cid:18) 1−P(M(Ij′)≤1−λj′/t) 1−P(M(Ij′)≤1−λj′/t) (cid:19)
and that
lim tP(M(Ij)≤1−λj/t,M(Ij′)≤1−λj′/t)
t→∞
= −lnC (e−λj,...,e−λj,e−λj′,...,e−λj′),
XIj,XIj′
since C is max-stable. By Lemma 2.1, we obtain
XIj,XIj′
−lnC (e−λj,...,e−λj,e−λj′,...,e−λj′)
XIj,XIj′
= ℓ σα(Ij) η,..., σω(Ij) η, σα(Ij′) η,..., σω(Ij′) η .
XIj∪Ij′ λj λj λj′ λj′
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)
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By thehomogeneity of order −1/η of ℓ, thelimit in (6) becomes
1+ λjεXIj(1) − εXIj,XIj′(λ−j1,λ−j′1)
λj′εXIj′(1) λj′εXIj′(1)
Switchingtherolesofj andj′ intheconditionalprobabilities, wecanseethatbothfunctionsin(4)
and (5) are equaland its common valueis given in the following definition.
Definition 2.2. For X = (X ,...,X ) under conditions (i) and (ii) and 1≤ j <j′ ≤ p, the tail
I 1 d
dependence function χXIj,XIj′(λj,λj′) for (XIj,XIj′) is defined by
χXIj,XXIj′ (λj,λj′)=λjεXIj(1)+λj′εXIj′(1)−εXIj,XIj′(λ−j1,λ−j′1)
and the value χ (1,1)≡χ isdenoted by coefficient of tail dependence for (X ,X ).
XIj,XIj′ XIj,XIj′ Ij Ij′
In the following we present a property of the generalized madogram coming from the function
ε (λ ,..., λ ).
XI1,...,XIp 1 p
Proposition 2.5. If X =(X ,..., X ) satisfies conditions (i) and (ii) then, for all(λ ,..., λ )∈
I 1 d 1 p
(0,∞)p,
ν (λ ,..., λ )= εXI1,...,XIp(λ1,..., λp) − 1 p εXIj(λj) .
XI1,...,XIp 1 p 1+εXI1,...,XIp(λ1,..., λp) pXj=1 1+εXIj(λj)
Inparticular, considering p=d=2 and λ =λ =1, we recovertheinitial relation between the
1 2
madogram ν and theextremal coefficient ε,given byν = ε−1 (Cooley et al. [2] 2006).
2(ε+1)
3 Examples
Considerr≥1integer,β ,i=1,...,d,j =1,...,r,nonnegativeconstantssuchthat r β =1,
ji j=1 ji
i=1,...,d, and α , j =1,...,r, constants in (0,1]. Consider C , j =1,...,r, max-sPtable copulas
j j
and define
C (u ,...,u )=exp − r −lnC e−(−βj1lnu1)η/αj,...,e−(−βjdlnud)η/αj αj/η , (7)
η 1 d j
( j=1(cid:18) (cid:18) (cid:19)(cid:19) )
X
withη∈(0,1]andsuchthatα /η∈(0,1]. Thisparametricfamilyofcopulascanbeobtainedfroma
j
mixturemodelofvariousMEVdistributions(FerreiraandPereira[5]2011)andencompassesseveral
known copulas such as logistic symmetric and asymmetric and geometric means.
Consider X has marginals in (i) and copula in (7). Then
I
FXI(t1,...,td)=exp(−j=r1(cid:18)−lnCj(cid:18)e−(βj1σ1t−11/η)η/αj,...,e−(βjdσdt−d1/η)η/αj(cid:19)(cid:19)αj/η) .
X
7
The tail dependence function ℓ (t ,...,t ) is homogeneous of order −1/η and thus we are in the
XI 1 d
context of the previous section. We will consider different particular cases in the choice of the
constants and MEV copulas and we determine the respective extremal coefficients and coefficients
of tail dependence.
Example 3.1. Considering r=1, β =1, i=1,...,d, we obtain
1i
FXI(t1,...,td)=exp(−(cid:18)−lnC(cid:18)e−(σ1t−11/η)η/α,...,e−(σdt−d1/η)η/α(cid:19)(cid:19)α/η) .
and if we take C = , we find
Q
α/η
F (t ,...,t )= exp − (σ t−1/η)η/α+...+(σ t−1/η)η/α
XI 1 d 1 1 d d
(cid:26) (cid:16) (cid:17) (cid:27)
α/η
= exp − ση/αt−1/α+...+ση/αt−1/α .
1 1 d d
(cid:26) (cid:16) (cid:17) (cid:27)
We have
εXI1,...,XIp(1,...,1)=dα/η, εXIj(1)=|Ij|α/η, εXIj,XIj′(1,1)=|Ij∪Ij′|α/η,
dα/η 1 p |I |α/η
ν (1,...,1)= − j
XI1,...,XIp 1+dα/η p 1+|Ij|α/η
j=1
X
and
χXIj,XIj′ =|Ij|α/η+|Ij′|α/η−(|Ij|+|Ij′|)α/η,
for all 1 ≤ j < j′ ≤ d, this latter generalizing the known result χ = 2−2α/η for the logistic
Xj,Xj′
model.
Example 3.2. Considering the previous example with positive constants β =β , i=1,...,d, not
1i i
necessarily equal to 1, we have
α/η
F (t ,...,t )= exp − (β σ )η/αt−1/α+...+(β σ )η/αt−1/α .
XI 1 d 1 1 1 d d d
(cid:26) (cid:16) (cid:17) (cid:27)
We obtain
α/η
ε (1,...,1)= βη/α+...+βη/α ,
XI1,...,XIp 1 d
(cid:16) (cid:17)
α/η α/η
ε (1)= βη/α , ε (1,1)= βη/α
XIj  i  XIj,XIj′  i 
iX∈Ij i∈IXj∪Ij′
   
8
and
α/η α/η α/η
χ = βη/α + βη/α − βη/α ,
XIj,XIj′  i   i   i 
iX∈Ij iX∈Ij′ i∈IXj∪Ij′
     
for all 1≤j <j′ ≤d.
Thepreviousexamples consist in asymmetriclogistic models. Inthefollowing weconsiderβ =
ji
β , i=1,...,d, and r>1, i.e., weighted geometric means.
j
Example 3.3. Consider r=2, C = and C = . We have
1 2
V Q
FXI(t1,...,td)= j=21exp(−βj(cid:18)−lnCj(cid:18)e−(cid:16)σ1t−11/η(cid:17)η/α,...,e−(cid:16)σdt−d1/η(cid:17)η/α(cid:19)(cid:19)α/η)
Y
d α/η d α/η
η/α η/α
= exp −β σ t−1/η −(1−β ) σ t−1/η
 1 i i 1 i i 
! !
 i_=1(cid:16) (cid:17) Xi=1(cid:16) (cid:17) 
 
d d α/η
η/α
= exp −β σ t−1/η −(1−β ) σ t−1/η .
 1 i i 1 i i ! 
 i_=1(cid:16) (cid:17) Xi=1(cid:16) (cid:17) 
 
Thus we obtain
ε (1,...,1)=β +(1−β )dα/η =β 1−dα/η +dα/η,
XI1,...,XIp 1 1 1
(cid:16) (cid:17)
ε (1)=β +(1−β )|I |α/η
XIj 1 1 j
and
χXIj,XIj′ = β1+(1−β1)|Ij|α/η+β1+(1−β1)|Ij′|α/η−β1−(1−β1)|Ij∪Ij′|α/η
= β1 1−|Ij|α/η−|Ij′|α/η+(|Ij|+|Ij′|)α/η +|Ij|α/η+|Ij′|α/η−(|Ij|+|Ij′|)α/η,
(cid:16) (cid:17)
for all 1≤j <j′ ≤d.
4 A note on asymptotic tail independence
In MEV models satisfying (i) and (ii), we only have tail dependence or tail independence between
two marginals Xj and Xj′ in the sense of
χXj,Xj′ =tl→im∞P(Fj(Xj)>1−1/t,Fj′(Xj′)>1−1/t),
9
being, respectively,positive and null. Just observe that
ln(1−1/t) −η ln(1−1/t) −η
P(Fj(Xj)>1−1/t,Fj′(Xj′)>1−1/t)=2t−1−1+P Xj <(cid:18)− σj (cid:19) ,Xj′ <(cid:18)− σj′ (cid:19) !
∼2t−1−1+P Xj < tσ−j1 −η,Xj′ < tσ−j′1 −η =2t−1−1+exp −ℓ(Xj,Xj′)((tσj)η,(tσj′)η)
(cid:18) (cid:16) (cid:17) (cid:16) (cid:17) (cid:19) n o
∼2t−1−t−1ℓ(Xj,Xj′)(cid:16)σjη,σjη′(cid:17)+t−2 ℓ(Xj,Xj′)2(cid:16)σjη,σjη′(cid:17)!2 ∼ tt−−12( 2ℓ(−Xjℓ,(XXjj′,)X(cid:16)jσ′jη),)σjη′(cid:17)!2 ,,iiff ℓℓ(Xj,Xj′) <=22,
2 (Xj,Xj′)

the first branch corresponding to tail dependence (χ = 2−ℓ ) and the second to in-
Xj,Xj′ (Xj,Xj′)
dependence (χ = 0). However, non-negligible dependence may occur even when we have
Xj,Xj′
independence in the limit. A classical example in this context is the multivariate Gaussian model,
whosebivariatemarginalsareasymptoticindependentwhateverthecorrelationparametersρjj′ <1.
This phenomenon was also noticed in real data applications (see, e.g., Tawn ([16] 1990), Guillou et
al. [9] 2014 and references therein). Ledford and Tawn ([10] 1996) addresses the modeling of the
decay rate of the dependenceunderasymptotic independence. More precisely, they consider
−1/κ
P(Fj(Xj)>1−1/t,Fj′(Xj′)>1−1/t)=t Xj,Xj′L(t), (8)
whereLisaslowlyvaryingfunction(i.e.,L(s),s>0,isarealfunctionsuchthatL(tx)/L(t)→1,as
t→∞, ∀x>0) and κ ∈(0,1] is denoted coefficient of asymptotic tail independence. Observe
Xj,Xj′
that MEV sub-vectors (Xj,Xj′) satisfy (8) with κXj,Xj′ = 1 and L(t) = 2−ℓ(Xj,Xj′) under tail
dependenceand κ =1/2 and L(t)=2 underindependence.
Xj,Xj′
In ourcontext of MEV models, we also have
χXIj,XIj′ =tl→im∞P(M(Ij)>1−1/t,M(Ij′)>1−1/t)>0,
unlessthemarginalsareindependent. IfwemovetoabroaderframeworkthantheMEVmodels,by
a similar reasoning as in Ledford and Tawn ([10] 1996), 2012), we assume
P(M(Ij)>1−1/t,M(Ij′)>1−1/t)=t−1/κXIj,XIj′ LXIj,XIj′(t), (9)
wherefunctionL isslowlyvaryingandκ ∈(0,1]correspondstotheblockcoefficient
XIj,XIj′ XIj,XIj′
of asymptotic tail independence introduced in Ferreira and Ferreira ([4]). Under the validity of
condition
P(mj∈iSn{Fj(Xj)}>1−1/t,jm′∈inT{Fj′(Xj′)}>1−1/t)=t−1/κXS,XTLXS,XT(t), (10)
10