Dissecting the multivariate extremal index and tail dependence
aa r X i v : . [ m a t h . S T ] F e b Dissecting the multivariate extremal index and tail dependence
Helena FerreiraUniversidade da Beira Interior, Centro de Matemática e Aplicações (CMA-UBI),Avenida Marquês d’Avila e Bolama, 6200-001 Covilhã, Portugal [email protected]
Marta FerreiraCenter of Mathematics of Minho UniversityCenter for Computational and Stochastic Mathematics of University of LisbonCenter of Statistics and Applications of University of Lisbon, Portugal [email protected]
Abstract
A central issue in the theory of extreme values focuses on suitable conditions such that the well-known results for the limiting distributions of the maximum of i.i.d. sequences can be applied tostationary ones. In this context, the extremal index appears as a key parameter to capture theeffect of temporal dependence on the limiting distribution of the maxima. The multivariate extremalindex corresponds to a generalization of this concept to a multivariate context and affects the taildependence structure within the marginal sequences and between them. As it is a function, theinference becomes more difficult, and it is therefore important to obtain characterizations, namelybounds based on the marginal dependence that are easier to estimate. In this work we present twodecompositions that emphasize different types of information contained in the multivariate extremalindex, an upper limit better than those found in the literature and we analyze its role in dependenceon the limiting model of the componentwise maxima of a stationary sequence. We will illustrate theresults with examples of recognized interest in applications. keywords: multivariate extreme values, multivariate extremal index, tail dependence, extremal co-efficients, madogram MS 2000 Subject Classification : 60G70
Let F be a multivariate distribution function (df), with continuous marginal dfs, in the domain ofattraction of a multivariate extreme values (MEV) df b H having unit Fréchet marginals. Thus F n ( nx , . . . , nx d ) → b H ( x , . . . , x d ) (1)and b H j ( x j ) = exp( − x − j ) , x j > .Consider { X n = ( X n , . . . , X nd ) } a stationary sequence such that F X n = F and let { M n =( M n , . . . , M nd ) } be a componentwise maxima sequence generated from X , . . . , X n and therefore M nj = W ni =1 X ij , j = 1 , . . . , d . If lim n →∞ P ( M n ≤ nx , . . . , M nd ≤ nx d ) = H ( x , . . . , x d ) , (2)for some MEV df H , we can relate H ( x , . . . , x d ) and b H ( x , . . . , x d ) through the so called mul-tivariate extremal index of { X n } . This is possible, even if the marginals b H j are not unit Fréchetdistributed, as considered for simplicity and without loss of generality. Indeed, to have (1) or mutatismutandis (2), it is sufficient that, as n → ∞ , the sequence of copulas C nF , with C F ( u , . . . , u d ) = F ( F − ( u ) , . . . , F − ( u d )) , converges to C b H , as well as, F nj ( nx j ) → b H j ( x j ) , j = 1 , . . . , d , which canbe reduced to the case of convergence to the Fréchet without affecting the convergence of C nF .We recall the definition of multivariate extremal index of { X n } and its role in the relationbetween H and b H (Nandagopalan [14] 1994). The sequence { X n } has multivariate extremal in-dex θ ( τττ ) ∈ (0 , , τττ = ( τ , . . . , τ d ) ∈ R d + , when for each τττ there is a sequence of real levels { u ( τττ ) n = ( u ( τ ) n , . . . , u ( τ d ) nd ) } satisfying nP ( X j > u ( τ j ) nj ) → τ j , j ∈ D = { , . . . , d } ,P ( c M n ≤ u ( τττ ) n ) → b γ ( τττ ) and P ( M n ≤ u ( τττ ) n ) → γ ( τττ ) = ( b γ ( τττ )) θ ( τττ ) , where c M n = ( c M n , . . . , c M nd ) , c M nj = W ni =1 b X ij , j = 1 , . . . , d , and { b X n } is a sequence of independentvectors such that F b X n = F X n . bserve that b γ ( τττ ) = exp (cid:16) − lim n →∞ nP ( X u n ) (cid:17) = exp ( − Γ( τττ )) , with Γ( τττ ) = lim n →∞ nP d [ j =1 { X ij > u ( τ j ) nj } ! = X ∅6 = J ⊂ D ( − | J | +1 lim n →∞ nP \ j ∈ J { X ij > u ( τ j ) nj } ! = X ∅6 = J ⊂ D ( − | J | +1 Γ ∗ J ( τττ J ) , where Γ ∗ J ( τττ J ) ≡ Γ ∗ ( τ j , j ∈ J ) = lim n →∞ nP \ j ∈ J { X ij > u ( τ j ) nj } ! and, in particular, Γ ∗{ j } ( τ j ) = τ j , j ∈ D . So, to say that Γ( τττ ) exists is equivalent to say that b γ ( τττ ) exists and we have γ ( τττ ) = exp ( − θ ( τττ )Γ( τττ )) = exp − θ ( τττ ) X ∅6 = J ⊂ D ( − | J | +1 Γ ∗ J ( τττ J ) , If { X n } has multivariate extremal index θ ( τττ ) then any sequence of subvectors { ( X n ) A } withindexes in A ⊂ { , . . . , d } has multivariate extremal index θ A ( τττ A ) , with θ A ( τττ A ) = lim τ i → + i A θ ( τ , . . . , τ d ) , τττ A ∈ R | A | + . In particular, for each j = 1 , . . . , d , { X nj } n ≥ has extremal index θ j .If θ ( τττ ) , τττ ∈ R d + , exists for { X n } we have H ( x , . . . , x d ) = b H ( x , . . . , x d ) θ ( − log b H ( x ) ,..., − log b H d ( x d )) (3)and H j ( x j ) = b H j ( x j ) θ j , j ∈ D .From inequalities d Y j =1 b H j ( x j ) θ j ≤ b H ( x , . . . , x d ) θ ( τ ( x ) ,...,τ d ( x d )) ≤ min j =1 ,...,d b H j ( x j ) θ j , e obtain W dj =1 θ j τ j Γ( τττ ) ≤ θ ( τττ ) ≤ P dj =1 θ j τ j Γ( τττ ) . (4)Besides the relation between H and b H , θ ( τττ ) also informs about the existence of clustering ofevents “at least some exceedance of u ( τ j ) nj by X nj , for some j ", since θ ( τττ ) = lim n →∞ E r n X i =1 { X i u ( τττ ) n } | r n X i =1 { X i u ( τττ ) n } > ! , (5)for sequences r n = [ n/k n ] and k n = o ( n ) provided that { X n } satisfies condition strong-mixing.The multivariate extremal index thus preserves, with the natural adaptations, the characteristicsthat made famous the univariate extremal index. Additionally to these similar characteristics to theunivariate extremal index, it plays an unavoidable role in the tail dependence characterization of H .If the tail dependence coefficients applied to F remain unchanged when applied to b H (Li [12] 2009),we can not guarantee the same for H , as will be seen in Section 3. The presence of serial dependencewithin each marginal sequence and between marginal sequences, makes it impossible to approximatethe dependence coefficients in the tail of M n to those of F .The dependence modeling between the marginals of F has received considerably more attentionin literature than the dependence between the marginals of F M n , which differs from F c M n = F n forbeing affected by θ ( τττ ) . The need to characterize this dependence appears, for instance, when we havea random field { X i ,n , i ∈ Z , n ≥ } and we consider random vectors ( X i ,n , . . . , X i s ,n ) correspondingto locations ( i , . . . , i s ) at time instant n . The sequence { ( X i ,n , . . . , X i s ,n ) } n ≥ presents in generala multivariate extremal index θ i ,...,i s ( τττ ) encompassing information about dependence in the spaceof locations i , . . . , i s and when the time n varies (Ferreira et al. [4] 2016). Relation (3) applied toMEV distributions b H and functions θ ( x , . . . , x d ) compatible with the properties of a multivariateextremal index, provide a means of constructing MEV distributions (Martins and Ferreira [13] 2005).Notwithstanding all these challenges posed by and for the multivariate extremal index, the liter-ature proves that it remained on the theoretical shelves of the study of extreme values.The main difficulty of applying the multivariate extremal index lies in the fact that it is a func-tion, unlike what happens with the marginal univariate extremal indexes, for which we have severalestimation methods in the literature (see, e.g., Gomes et al. [7] 2008, Northrop [16] 2015, Ferreiraand Ferreira [6] 2016 and references therein).Since it remains present the need to estimate the propensity for clustering in a context of mul-tivariate sequences, we propose in this work: (a) decompose it, highlighting different types of in-formation contained in it; (b) bound it in order to obtain a better upper limit than those availablein the literature; (c) enhance its role in the dependence of the tail of H ; (d) apply it to models of ecognized interest in applications. Based on (5) the multivariate extremal index can be seen as the number of the limiting meandimension of clustering of events counted by the point process N n = n X i =1 { X i u ( τττ ) n } . We are going to consider two point processes of more restricted events, corresponding to jointexceedances for various marginals of X i and enhance the contribution of the extremal indexes ofthese events in the value of θ ( τττ ) .Let, for each ∅ 6 = J ⊂ D = { , . . . , d } , N ∗ n,J = n X i =1 { T j ∈ J { X ij >u nj }} , n ≥ , and N ∗∗ n,J = n X i =1 { V j ∈ J X ij > W j ∈ J u nj } , n ≥ , where notations ∧ and ∨ stand for minimum and maximum, respectively.We denote the respective limiting mean number of occurrences by Γ ∗ J ( τττ J ) = lim n →∞ nP \ j ∈ J { X ij > u nj } ! and Γ ∗∗ J ( τττ J ) = lim n →∞ nP \ j ∈ J { X ij > _ j ∈ J u nj } ! . Observe that Γ ∗∗ J ( τττ J ) = lim n →∞ nP \ j ∈ J ( X ij > n V j ∈ J τ j )! . Thus Γ ∗∗ J ( τττ J ) = τ ∗∗ J ^ j ∈ J τ j ! , ith τ ∗∗ J an increasing function in V j ∈ J τ j and homogeneous of order . Therefore, we have τ ∗∗ J V j ∈ J τ j s ! = τ ∗∗ J (cid:16)V j ∈ J τ j (cid:17) s , (6)for all s = 0 , a relation that will be fundamental for the independence of θ ∗∗ from τ .In case J = D , we will omit the index J in notation.For each of these processes, we can define an index of clustering of occurrences, which we willalso call extremal indexes, θ ∗ J ( τττ J ) and θ ∗∗ J ( τττ J ) , being the latter a constant independent of τττ J , as wewill see.Let us assume that sequence { X n } n ≥ satisfies the strong-mixing condition (Leadbetter et al. [9]1983) and, as consequence, we have, as n → ∞ , P (cid:0) N n,J = 0 (cid:1) − P k n (cid:0) N [ n/k n ] ,J = 0 (cid:1) → ,P (cid:0) N ∗ n,J = 0 (cid:1) − P k n (cid:0) N ∗ [ n/k n ] ,J = 0 (cid:1) → and P (cid:0) N ∗∗ n,J = 0 (cid:1) − P k n (cid:0) N ∗∗ [ n/k n ] ,J = 0 (cid:1) → , for any integers sequence { k n } , such that, k n → ∞ , k n α n ( l n ) → and k n l n /n → , as n → ∞ ,where α n ( · ) and l n are the sequences of the strong-mixing condition. Thus P (cid:0) N n,J = 0 (cid:1) → exp ( − θ J ( τττ J )Γ J ( τττ J )) ,P (cid:0) N ∗ n,J = 0 (cid:1) → exp ( − θ ∗ J ( τττ J )Γ ∗ J ( τττ J )) and P (cid:0) N ∗∗ n,J = 0 (cid:1) → exp − θ ∗∗ J ( τττ J ) τ ∗∗ J ^ j ∈ J τ j !! , (7)with θ J ( τττ J ) = lim n →∞ k n P (cid:0) N [ n/k n ] ,J > (cid:1) / Γ J ( τττ J ) ,θ ∗ J ( τττ J ) = lim n →∞ k n P (cid:0) N ∗ [ n/k n ] ,J > (cid:1) / Γ ∗ J ( τττ J ) , ∗∗ J ( τττ J ) = lim n →∞ k n P (cid:0) N ∗∗ [ n/k n ] ,J > (cid:1) /τ ∗∗ J ^ j ∈ J τ j ! and θ ∗∗ J ( τττ J ) τ ∗∗ J ^ j ∈ J τ j ! ≤ θ ∗ J ( τττ J )Γ ∗ J ( τττ J ) ≤ _ j ∈ J θ j τ j ≤ θ J ( τττ J )Γ J ( τττ J ) . In the following we present relations between θ ∗∗ J ( τττ J ) , θ ∗ J ( τττ J ) and θ J ( τττ J ) , which will allow us adetailed interpretation of the information contained in θ ( τττ ) and an upper bound better than the onein (4). But first, we start by proving that θ ∗∗ J ( τττ J ) = θ ∗∗ J , i.e., these extremal indexes are independentof τττ , which is already known for J = { j } (Leadbetter et al. [9] 1983), j = 1 , . . . , d , since θ ∗∗{ j } = θ j .Indeed the proof runs along the same lines. Proposition 2.1.
For stationary sequences { X n } satisfying the strong-mixing condition, if thereexists the limit (7) for some τττ , then it exists for any τττ > and we have P (cid:0) N ∗∗ n,A = 0 (cid:1) → exp − θ ∗∗ A τ ∗∗ A ^ j ∈ A τ j !! , with θ ∗∗ A ∈ [0 , constant.Proof. From the strong-mixing condition, we have lim inf n →∞ P (cid:0) N ∗∗ n,A = 0 (cid:1) = lim inf n →∞ P k n (cid:0) N ∗∗ [ n/k n ] ,A = 0 (cid:1) = lim inf n →∞ − k n P (cid:0) N ∗∗ [ n/k n ] ,A > (cid:1) k n ! k n ≥ lim inf n →∞ − nP (cid:16)V j ∈ A X j > W j ∈ A u nj (cid:17) k n k n = − τ ∗∗ A (cid:16)V j ∈ A (cid:17) k n k n . Thus, if there exists Ψ( τ ∗∗ A ) = lim sup n →∞ P (cid:0) N ∗∗ n,A = 0 (cid:1) , we have Ψ (cid:16) τ ∗∗ A (cid:16)V j ∈ A (cid:17)(cid:17) ≥ exp (cid:16) − τ ∗∗ A (cid:16)V j ∈ A (cid:17)(cid:17) ,and so Ψ( τ ∗∗ A ) is a strictly positive function.We also have that function Ψ( τ ∗∗ A ) would have to satisfy Ψ( τ ∗∗ A /k ) = Ψ /k ( τ ∗∗ A ) , for all τ ∗∗ A > and k = 1 , , . . . , since, representing P ni =1 { V j ∈ A X ij >m/ V j ∈ A τ j } by N ∗∗ n (cid:18) u ( τ ∗∗ A ( V j ∈ A τ j ) ) m (cid:19) andapplying (7), it holds (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) N ∗∗ [ n/k n ] ,A (cid:18) u ( τ ∗∗ A ( V j ∈ A τ j ) ) n (cid:19) = 0 (cid:19) − P (cid:18) N ∗∗ [ n/k n ] ,A (cid:18) u ( τ ∗∗ A ( V j ∈ A τ j ) /k n ) [ n/k n ] (cid:19) = 0 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:20) nk n (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ^ j ∈ A X j > n V j ∈ A τ j ! − P ^ j ∈ A X j > [ n/k n ] V j ∈ A τ j /k n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:20) nk n (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V j ∈ A τ j n (1 + o (1)) − V j ∈ A τ j /k n [ n/k n ] (1 + o (1)) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o (1) nd thus we would have Ψ (cid:18) τ ∗∗ A k n (cid:19) = lim sup n →∞ P (cid:16) N ∗∗ [ n/k n ] ,A (cid:16) u ( τ ∗∗ A /k n )[ n/k n ] ,A (cid:17) = 0 (cid:17) = lim sup n →∞ P (cid:16) N ∗∗ n,A (cid:16) u ( τ ∗∗ A ) n,A (cid:17) = 0 (cid:17) = Ψ ( τ ∗∗ A ) /k n . On the other hand,
Ψ ( τ ∗∗ A ) would have to be a non increasing function because if τ ∗∗ ,A ^ j ∈ A τ ,j ! = lim n →∞ nP ^ j ∈ A X j > n V j ∈ A τ ,j ! > τ ∗∗ A ^ j ∈ A τ j ! = lim n →∞ nP ^ j ∈ A X j > n V j ∈ A τ j ! and τ ∗∗ A (cid:16)V j ∈ A τ j (cid:17) is increasing in V j ∈ A τ j , then from some order, ( ^ j ∈ A X j > n V j ∈ A τ j ) ⊂ ( ^ j ∈ A X j > n V j ∈ A τ ,j ) and thus (cid:26) N ∗∗ n,A (cid:18) u ( τ ∗∗ ,A ) n (cid:19) = 0 (cid:27) ⊂ (cid:26) N ∗∗ n,A (cid:18) u ( τ ∗∗ A ) n (cid:19) = 0 (cid:27) and Ψ (cid:0) τ ∗∗ ,A (cid:1) ≤ Ψ ( τ ∗∗ A ) . If Ψ ( τ ∗∗ A ) is a strictly positive function, non increasing and such that Ψ ( τ ∗∗ A /k ) = Ψ ( τ ∗∗ A ) /k , then Ψ ( τ ∗∗ A ) = exp ( − θ ∗∗ A τ ∗∗ A ) , with θ ∗∗ A a non negative constant. Since Ψ ( τ ∗∗ A ) > exp ( − τ ∗∗ A ) , it also comes θ ∗∗ A ≤ . For the lower limit, we can make the same reasoningto obtain the result.Let us start by emphasizing that, to θ ( τττ )Γ( τττ ) , we have the contribution of the clustering of thejoint exceedances of all levels by the respective marginals, including the particular case of the cluster-ing of exceedances of the largest level by the lower marginal, as well as, the clustering of exceedancesof one or more levels by the respective marginals without joint exceedances of all levels. Proposition 2.2.
Let { X n } be a stationary sequence satisfying the strong-mixing condition and { u ( τττ ) n = ( u ( τ ) n , . . . , u ( τ d ) n ) } a sequence of normalized real levels for which there exists Γ( τττ ) . Then θ ( τττ )Γ( τττ ) = θ ∗∗ τ ∗∗ d ^ j =1 τ j ! + θ ∗ ( τττ )Γ ∗ ( τττ ) β (1) ( τττ ) + X ∅6 = J ⊂ D ( − | J | +1 Θ J ( τττ J ) , where β (1) ( τττ ) = lim n →∞ P ( N ∗∗ r n = 0 | N ∗ r n > and Θ J ( τττ J ) = lim n →∞ k n P \ j ∈ J { N r n , { j } > }| N ∗ r n = 0 ! . roof. We have k n P ( N r n >
0) = k n P ( N ∗∗ r n >
0) + k n P ( N ∗ r n > , N ∗∗ r n = 0) + k n P ( N r n > , N ∗ r n = 0)= k n P ( N ∗∗ r n >
0) + k n P ( N ∗ r n > P ( N ∗∗ r n = 0 | N ∗ r n > k n P (cid:16)S dj =1 { N r n , { j } > } , N ∗ r n = 0 (cid:17) = k n P ( N ∗∗ r n >
0) + k n P ( N ∗ r n > P ( N ∗∗ r n = 0 | N ∗ r n > X ∅6 = J ⊂ D ( − | J | +1 k n P \ j ∈ J { N r n , { j } > } , N ∗ r n = 0 ! . In what concerns the last term, observe that d X j =1 k n P (cid:0) N r n , { j } > , N ∗ r n = 0 (cid:1) = d X j =1 k n P (cid:0) N r n , { j } > (cid:1) − d X j =1 k n P (cid:0) N r n , { j } > , N ∗ r n > (cid:1) and since lim n →∞ P ( N ∗ r n = 0) = 1 , we have the result.Observe that β (1) ( τττ ) reduces θ ∗ ( τττ ) from the joint exceedances of W dj =1 n/τ j accounted for θ ∗∗ .We can say that in the last term of representation of θ ( τττ )Γ( τττ ) we are accounting the clusteringpropensity concerning one or more marginals, without joint exceedances of all the marginals.We illustrate the previous result with a bivariate sequence with unit Fréchet marginals and suchthat the joint tail is regularly varying at ∞ with index η ∈ (0 , measuring a penultimate taildependence, as the (sub)model presented in Ledford and Tawn ([10, 11] 1996,1997). Example 2.1.
Suppose that d = 2 and { ( X n , X n ) } n ≥ is a strong-mixing stationary sequence,with unit Fréchet marginals and such that P ( X n > x, X n > x ) ∼ x − /η L ( x ) , (8) as x → ∞ , where < η < and L is a slowly varying function, i.e., L ( tx ) /L ( x ) → , ∀ t > . Then θ ∗∗ = k n P ( N ∗∗ r n > ≤ nP (cid:18) X n > nτ ∧ τ , X n > nτ ∧ τ (cid:19) ∼ n (cid:18) nτ ∧ τ (cid:19) − /η L (cid:18) nτ ∧ τ (cid:19) → ,θ ∗ ( τ , τ ) ≤ nP (cid:18) X n > nτ , X n > nτ (cid:19) ≤ nP (cid:18) X n > nτ ∧ τ , X n > nτ ∧ τ (cid:19) → . Therefore, regardless of additional conditions on the serial dependence, the validity of (8) implies θ ( τττ )Γ( τττ ) = X ∅6 = J ⊂{ , } ( − | J | +1 lim n →∞ k n P \ j ∈ J { N r n , { j } > } , N ∗ r n = 0 ! nd Γ( τττ ) = τ + τ . Since k n P ( N ∗ r n > → we can thus write in this model θ ( τττ ) = 1 τ + τ lim n →∞ k n (cid:0) P (cid:0) N r n , { } > (cid:1) + P (cid:0) N r n , { } > (cid:1) − P (cid:0) N r n , { } > , N r n , { } > (cid:1)(cid:1) . (9) We now consider several particular situations.(a) In the case of independent vectors ( X n , X n ) , n ≥ , we have θ ( τττ ) = 1 τ + τ θ τ + θ τ − lim n →∞ k n P [ ≤ i nτ , X i ≤ nτ , X i ′ ≤ nτ , X i ′ > nτ , (cid:27)[ (cid:26) X i ≤ nτ , X i > nτ , X i ′ > nτ , X i ′ ≤ nτ (cid:27)(cid:27)(cid:19)(cid:19) = τ + τ τ + τ = 1 . It will then come P ( M n ≤ n/τ , M n ≤ n/τ ) → exp( − Γ( τττ )) = exp( − τ ) exp( − τ ) , that is, M n and M n are also asymptotically independent.(b) Suppose that { ( X n , X n ) } n ≥ , satisfies condition D ( m ) { , } defined by lim n →∞ n [ n/k n ] X j = m +1 P ( X > n/τ , X j > n/τ ) = 0 , which extends D ′ { , } of Davis ([2] 1982), satisfied by i.i.d. sequences. Then θ ( τττ ) = 1 τ + τ θ τ + θ τ − lim n →∞ n m X i =2 P ( X > n/τ , X i > n/τ ) ! , where the last part reflects the cross dependence.(c) If we assume an analogous hypothesis of (8) for ( X , X i ) with different η i , we will alsoobtain asymptotic independence between M n and M n , since the last term has null limit. We have P ( M n ≤ n/τ , M n ≤ n/τ ) → exp( − Γ( τττ ) θ ( τττ )) = exp( − θ τ ) exp( − θ τ ) .(d) If θ ( τττ ) = θ , ∀ τττ ∈ R , then θ = θ = θ and, from (9), θ = θ − lim n →∞ k n P (cid:0) N r n , { } > , N r n , { } > (cid:1) , which implies that this limit is null and thus P ( M n ≤ n/τ , M n ≤ n/τ ) → exp( − θ ( τ + τ )) =exp( − θτ ) exp( − θτ ) . We present below a relation between θ ( τττ ) and the extremal indexes θ ∗∗{ j,...,d } and θ ∗{ j,...,d } (cid:0) τττ { j,...,d } (cid:1) , j = 1 , . . . , d , which discriminates different informations contained in function θ ( τττ ) and provides anupper bound for θ ( τττ ) better than the one in (4). In Example 2.2 we show that the proposed upperbound for the M4 processes, can be better than the one presented in Ehlert and Schlather ([3] 2008). he new upper bound has also the advantage of depending only on constant extremal indexes whichcan be estimated by known methods of literature. Proposition 2.3.
Let { X n } be a stationary sequence satisfying the strong-mixing condition and { u ( τττ ) n = ( u ( τ ) n , . . . , u ( τ d ) n ) } a sequence of normalized real levels for which there exists Γ( τττ ) . Then(a) θ ( τττ )Γ( τττ ) = lim n →∞ k n P ( N r n >
0) = d X j =1 θ j τ j − d − X j =1 θ ∗∗{ j,...,d } τ ∗∗{ j,...,d } d ^ i = j τ i ! − d − X j =1 θ ∗{ j,...,d } (cid:0) τττ { j,...,d } (cid:1) Γ ∗{ j,...,d } (cid:0) τττ { j,...,d } (cid:1) β (1) j (cid:0) τττ { j,...,d } (cid:1) − d − X j =1 X J ⊂{ j +1 ....,d } ( − | J | +1 β (2) { j }∪ J (cid:0) τττ { j }∪ J (cid:1) , where β (1) j (cid:0) τττ { j,...,d } (cid:1) = lim n →∞ P (cid:0) N ∗∗ r n , { j,...,d } = 0 | N ∗ r n , { j,...,d } > (cid:1) and β (2) { j }∪ J (cid:0) τττ { j }∪ J (cid:1) =lim n →∞ k n P (cid:16)T i ∈{ j }∪ J { N r n , { i } > }| N ∗ r n , { j,...,d } = 0 (cid:17) , provided that the limiting constants ex-ist.(b) θ ( τττ ) ≤ τττ ) (cid:16)P dj =1 θ j τ j − P d − j =1 θ ∗∗{ j,...,d } τ ∗∗{ j,...,d } (cid:16)V di = j τ i (cid:17)(cid:17) . Proof.
We have k n P ( N r n >
0) = k n P d [ j =1 { N r n , { j } > } ! = d − X j =1 k n P N r n , { j } > , d \ i = j +1 { N r n , { i } = 0 } ! + k n P (cid:0) N r n , { d } > (cid:1) = d X j =1 k n P (cid:0) N r n , { j } > (cid:1) − d − X j =1 k n P N r n , { j } > , d [ i = j +1 { N r n , { i } > } ! . egarding the second term, we can also say that d − X j =1 k n P N r n , { j } > , d [ i = j +1 { N r n , { i } > } ! = d − X j =1 k n P N r n , { j } > , d [ i = j +1 { N r n , { i } > } , N ∗ r n , { j,...,d } > ! + d − X j =1 k n P N r n , { j } > , d [ i = j +1 { N r n , { i } > } , N ∗ r n , { j,...,d } = 0 ! = d − X j =1 k n P (cid:0) N ∗ r n , { j,...,d } > , N ∗∗ r n , { j,...,d } > (cid:1) + d − X j =1 k n P (cid:0) N ∗ r n , { j,...,d } > , N ∗∗ r n , { j,...,d } = 0 (cid:1) + d − X j =1 k n P N r n , { j } > , d [ i = j +1 { N r n , { i } > } , N ∗ r n , { j,...,d } = 0 ! = d − X j =1 k n P (cid:0) N ∗∗ r n , { j,...,d } > (cid:1) + d − X j =1 k n P (cid:0) N ∗ r n , { j,...,d } > , N ∗∗ r n , { j,...,d } = 0 (cid:1) + d − X j =1 k n P N r n , { j } > , d [ i = j +1 { N r n , { i } > } , N ∗ r n , { j,...,d } = 0 ! . Therefore, θ ( τττ )Γ( τττ ) = lim n →∞ k n P ( N r n >
0) = d X j =1 θ j τ j − d − X j =1 θ ∗∗{ j,...,d } τ ∗∗{ j,...,d } d ^ i = j τ i ! − d − X j =1 θ ∗{ j,...,d } (cid:0) τττ { j,...,d } (cid:1) Γ ∗{ j,...,d } (cid:0) τττ { j,...,d } (cid:1) lim n →∞ P (cid:0) N ∗∗ r n , { j,...,d } = 0 | N ∗ r n , { j,...,d } > (cid:1) − d − X j =1 lim n →∞ k n P d [ i = j +1 { N r n , { j } > , N r n , { i } > }| N ∗ r n , { j,...,d } = 0 ! , since P (cid:0) N ∗ r n , { j,...,d } = 0 (cid:1) → , as n → ∞ .The above result means that, for each j ∈ { , . . . , d } , the values θ ∗{ j,...,d } (cid:0) τττ { j,...,d } (cid:1) only contributeto θ ( τττ ) if it is not asymptotically almost surely the local occurrence of some joint exceedances of the argest level u ( τ j ) ni , i ∈ { , . . . , d } , among the joint exceedances of these levels. Otherwise, the jointexceedances clustering is considered only through the clustering of the joint exceedances of the largestlevel u ni , i ∈ { j, . . . , d } , and measured by θ ∗∗{ j,...,d } , disappearing the third term. Therefore, the sec-ond and third terms together account for the clustering of two situations of joint exceedances. Thefourth term measures the clustering of exceedances of u nj and of one or more u ni , i ∈ { j + 1 , . . . , d } ,in the absence of joint exceedances of levels u ni , i ∈ { j, . . . , d } , not accounted within the second andthird terms. All these clustering situations were accounted by excess in the first term.The function θ ( τττ ) is homogeneous of order zero and thus θ ( τ, . . . , τ ) = θ (1 , . . . , , ∀ τ ∈ R . Theconstant θ (1 , . . . , has been used as a dependence coefficient of the marginals of H (see, e.g., Martinsand Ferreira [13] 2005, Ehlert and Schlather [3] 2008, Ferreira and Ferreira [5] 2015, and referencestherein).We are going to analyze the consequences of the decompositions presented for θ ( τττ ) in the calcu-lation of θ ( ) .If τ = . . . = τ d = τ , then N ∗∗ n = N ∗ n , β (1) J ( τττ ) = 0 , Γ ∗ ( τττ ) = τ ∗∗ ( τττ ) and Γ( τττ ) = P ∅6 = J ⊂ D ( − | J | +1 τ ∗∗ J ( τττ J ) .The first decomposition θ ( )Γ( ) = θ ∗∗ τ ∗∗ ( ) + lim n →∞ k n P d [ j =1 { N r n , { j } > } , N ∗ r n = 0 ! , separates once again the contribution of the clustering of exceedances across all marginals from thecontribution of the clustering of exceedances of one or more marginals without exceedances of allmarginals.In the next section, we will give an important utility to the boundary of θ ( τττ )Γ ∗ ( τττ ) . It will serveto delimitate the difference between the tail dependence coefficients of H and b H .The second decomposition allow us to obtain an upper bound for θ ( ) , which can be better thanthe one presented in (4). From the previous result, we have θ ( )Γ( ) ≤ d X j =1 θ j − d − X j =1 θ ∗∗{ j,...,d } τ ∗∗{ j,...,d } ( ) . (10)From the proof of Proposition 2.3 we found that, instead of following the order , . . . , d to decomposeinitially the event { S dj =1 N r n , { j } > } in a reunion of disjoint events { N r n , { j } > , T di = j +1 { N r n , { i } > }} , j = 1 , . . . , d − and { N r n , { d } > }} , we can consider any other permutation ( i , . . . , i d ) from (1 , . . . , d ) and repeat the process. Therefore the previous upper limit can be improved in the followingsense: θ ( )Γ( ) ≤ d X j =1 θ j − _ ( i ,...,i d ) ∈P d i d − X j = i θ ∗∗{ j,...,i d } τ ∗∗{ j,...,i d } ( ) , here P d denotes the set of all permutations of (1 , . . . , d ) . Example 2.2.
Consider the M4 process, X n = 0 . Z n ∨ . Z n − X n = 0 . Z n − ∨ . Z n − ∨ . Z n − , with { Z n ≡ Z ,n } , where { Z l,n } , l ≥ , n ≥ , is an array of independent unit Fréchet randomvariables. We have θ = 0 . , θ = 0 . and θ ( )Γ( ) = 0 . . Since { X n } n ≥ is -independent,representing { X i > n/τ, X i > n/τ } by A i,n and τ ∧ τ = τ , we have that θ ∗∗{ , } τ ∗∗{ , } ( τ ) = lim n →∞ nP (cid:0) A ,n ∩ A ,n ∩ A ,n ∩ A ,n (cid:1) = lim n →∞ nP (cid:0) { . Z > n/τ } ∩ A ,n ∩ A ,n ∩ A ,n (cid:1) = lim n →∞ nP (cid:0) { . Z > n/τ } ∩ A ,n (cid:1) = lim n →∞ nP ( { . Z > n/τ, . Z ≤ n/τ } ∪ { . Z > n/τ, . Z > n/τ } )= 0 . τ = 0 . τ ∧ τ ) . Therefore, Proposition 2.3 indicates that θ ( )Γ( ) ≤ . . − . . . The upper limit in thistype of processes has no great interest since we have the theoretical expression for θ ( τττ ) . However,this example serves to show that our upper bound can be better than the one presented in Ehlert andSchlather ([3] 2008) for M4 processes. Indeed, by applying their Corollary 3, we obtain θ ( )Γ( ) ≤ Γ( ) − _ j =1 (1 − θ j ) ! ∧ d X j =1 θ j = ((0 . . . . − (0 . ∨ . ∧ .
2= 1 . ∧ . . . In the cases where the number of signatures of an M4 process exceeds the number of marginals,examples are easily constructed in which the Ehlert and Schlather upper limit is reduced to P dj =1 θ j ,being in these cases the lower limit of (10) below this. Our upper bound still has the advantage ofbeing applied to processes outside the max-stable class. For each pair ( j, j ′ ) , j < j ′ belonging to D , consider the bivariate (upper) tail dependence coefficient χ Fjj ′ ∈ [0 , for random pair ( X nj , X nj ′ ) with df F jj ′ , discussed in Sibuya ([17] 1960) and Joe ([8] χ Fjj ′ = lim u ↑ + P ( F j ( X ij ) > u | F j ′ ( X ij ′ ) > u ) and coefficient χ Fjj ′ ∈ [ − , of Coles et al. ([1] 1999), defined by χ Fjj ′ = lim u ↑ + P ( F j ′ ( X ij ′ ) > u )log P ( F j ( X ij ) > u, F j ′ ( X ij ′ ) > u ) − . We can say that χ Fjj ′ corresponds to the probability of one variable being high given that theother is high too. The case χ Fjj ′ > means asymptotic dependence between X nj and X nj ′ and when-ever χ Fjj ′ = 0 the variables are said to be asymptotically independent. Assuming χ Fjj ′ > withinasymptotically independent data may carry to an over-estimation of probabilities of extreme jointevents (see, e.g., Ledford and Tawn [10, 11] 1996, 1997). Asymptotically independent models, i.e.,having χ Fjj ′ = 0 , may exhibit a residual tail dependence rendering different degrees of dependence atfinite levels. Coefficient χ Fjj ′ is a suitable tail measure within this class. Thus the pair ( χ Fjj ′ , χ Fjj ′ ) is a useful tool in characterizing the extremal dependence: under asymptotic dependence we have χ Fjj ′ = 1 and < χ Fjj ′ ≤ quantifies the strength of dependence between the variables ( X nj , X nj ′ ) and, within the class of asymptotic independence, we have χ Fjj ′ = 0 and − ≤ χ Fjj ′ < measures thestrength of dependence of the random pair.Observe that, both measures can be calculated from the copula C Fjj ′ ( u, u ) = F jj ′ ( F − j ( u ) , F − j ′ ( u )) ,with χ Fjj ′ = 2 − lim u ↑ + log C Fjj ′ ( u, u )log u and χ Fjj ′ = lim u ↑ + − u )log (cid:16) − u + C Fjj ′ ( u, u ) (cid:17) − . If F belongs to the domain of attraction of b H , then χ Fjj ′ = χ b Hjj ′ and χ Fjj ′ = χ b Hjj ′ . This resultsfrom the uniform convergence of C nF to C b H and from C Fnjj ′ ( u, u ) = (cid:16) C Fjj ′ ( u /n , u /n ) (cid:17) n . We willthen have lim u ↑ + lim n →∞ (cid:16) C Fjj ′ ( u /n , u /n ) (cid:17) n C Fjj ′ ( u, u ) = lim n →∞ lim u ↑ + (cid:16) C Fjj ′ ( u /n , u /n ) (cid:17) n C Fjj ′ ( u, u ) = 1 , which guarantees the constancy of χ F n jj ′ and χ F n jj ′ , as n → ∞ .The presence of dependence among the variables of { X n } expressed by a function θ ( τττ ) with values ess than one, may affect the limiting behavior of χ F n jj ′ but not the limiting behavior of χ F n jj ′ , where F n denotes de df of M n . Proposition 3.1.
For stationary sequences { X n } , with multivariate extremal index θ ( τττ ) , τττ ∈ R d + ,for any choice j < j ′ in D , we have, χ Hjj ′ = χ b Hjj ′ .Proof. Based on the spectral representation of MEV copulas and relation C Hjj ′ ( u j , u j ′ ) = (cid:18) C c Hjj ′ (cid:16) u /θ j j , u /θ j ′ j ′ (cid:17)(cid:19) θ (cid:18) − log ujθj , − log uj ′ θj ′ (cid:19) , (11)we have χ b Hjj ′ = lim u ↑ + − u )log (cid:18) − u − C c Hjj ′ ( u, u ) (cid:19) −
1= lim u ↑ + − u )log (cid:16) − u − exp (cid:16) − R ( w ( − log u ) ∨ (1 − w )( − log u )) d c W ( w ) (cid:17)(cid:17) −
1= lim u ↑ + − u )log − u − u − log C c Hjj ′ ( e − ,e − ) ! − where c W is the spectral measure of b H . On the other hand χ Hjj ′ = lim u ↑ + − u )log − u − u θ jj ′ (cid:18) θj , θj ′ (cid:19) − log C c Hjj ′ (cid:16) exp ( − θ − j ) , exp (cid:16) − θ − j ′ (cid:17)(cid:17)! − Therefore, (1 − χ Hjj ′ ) = (1 − χ b Hjj ′ ) A with A = lim u ↑ + log (cid:16) − u − u Γ(1 , (cid:17) log − u − u θ jj ′ (cid:18) θj , θj ′ (cid:19) Γ (cid:18) θj , θj ′ (cid:19) ! = lim u ↑ + log (1 − u − u a )log (1 − u − u b ) = lim u ↑ + − au a − − bu b − lim u ↑ + − u + u b − u + u a = 1 , with a = Γ(1 , and b = θ jj ′ (cid:16) θ j , θ j ′ (cid:17) Γ (cid:16) θ j , θ j ′ (cid:17) . Proposition 3.2.
For stationary sequences { X n } , with multivariate extremal index θ ( τττ ) , τττ ∈ R d + ,we have, for any choice j < j ′ in D , a) χ Hjj ′ = 2 − θ jj ′ (cid:16) θ j , θ j ′ (cid:17) Γ jj ′ (cid:16) θ j , θ j ′ (cid:17) ;(b) χ Hjj ′ − χ b Hjj ′ = Γ jj ′ (1 , − θ jj ′ (cid:16) θ j , θ j ′ (cid:17) Γ jj ′ (cid:16) θ j , θ j ′ (cid:17) .Proof. Using the spectral representation of MEV copulas and relation (11), we have χ Hjj ′ = 2 − θ jj ′ (cid:18) θ j , θ j ′ (cid:19) lim u ↑ + R (cid:16) − log uwθ j ∨ − log u (1 − w ) θ j ′ (cid:17) d c W ( w ) − log u = 2 − θ jj ′ (cid:18) θ j , θ j ′ (cid:19) Z (cid:18) wθ j ∨ − wθ j ′ (cid:19) d c W ( w )= 2 − (cid:18) − θ jj ′ (cid:18) θ j , θ j ′ (cid:19) log C c Hjj ′ (exp( − /θ j ) , exp( − /θ j ′ )) (cid:19) = 2 − θ jj ′ (cid:18) θ j , θ j ′ (cid:19) Γ jj ′ (cid:18) θ j , θ j ′ (cid:19) , where c W is the spectral measure of b H .The previous proposition can be rewritten in terms of the extremal coefficients ε Hjj ′ and ε b Hjj ′ , suchthat, C c Hjj ′ ( u, u ) = u ε c Hjj ′ and C Hjj ′ ( u, u ) = u ε Hjj ′ , since these satisfy the relations χ Hjj ′ = 2 − ε Hjj ′ and χ b Hjj ′ = 2 − ε b Hjj ′ . From (a) we conclude that ε Hjj ′ = θ jj ′ (cid:16) θ j , θ j ′ (cid:17) Γ jj ′ (cid:16) θ j , θ j ′ (cid:17) . Consequently, for themeasure of asymptotic independence called madogram (Naveau et al. [15] 2009), defined by ν Fjj ′ = 12 E | F j ( X nj ) − F j ′ ( X nj ′ ) | and satisfying ν Fjj ′ = 12 ε Fjj ′ − ε Fjj ′ + 1 , we have(a) ν Fjj ′ = ν b Hjj ′ =
12 Γ jj ′ (1 , − jj ′ (1 , ;(b) ν Hjj ′ = θ jj ′ (cid:18) θj , θj ′ (cid:19) Γ jj ′ (cid:18) θj , θj ′ (cid:19) − θ jj ′ (cid:18) θj , θj ′ (cid:19) Γ jj ′ (cid:18) θj , θj ′ (cid:19) +1 . Therefore, for large n , the madogram of ( M nj , M nj ′ ) can not be taken by the madogram of ( c M nj , c M nj ′ ) .From relation (b) in Proposition 2.3, we conclude that χ Hjj ′ ≥ θ ∗∗ jj ′ τ ∗∗ jj ′ (cid:18) θ j ∨ θ j ′ (cid:19) (12) nd we can establish the following consequence about the value of the difference between χ Hjj ′ and χ b Hjj ′ . Corollary 3.3.
For stationary sequences { X n } satisfying the strong-mixing condition, with multi-variate extremal index θ ( τττ ) , τττ ∈ R d + , we have, for any choice j < j ′ in D ,(a) θ ( τττ ) = θ , ∀ τττ ∈ R d + implies χ Hjj ′ = χ b Hjj ′ ;(b) (cid:12)(cid:12)(cid:12) χ Hjj ′ − χ b Hjj ′ (cid:12)(cid:12)(cid:12) ≥ max n θ ∗∗ jj ′ τ ∗∗ jj ′ (cid:16) θ j ∨ θ j ′ (cid:17) − jj ′ (1 , , − Γ jj ′ (1 , o .Proof. (a) If θ ( τττ ) is constant equal to θ , then θ j = θ j ′ = θ and, since Γ is homogeneous of order 1,from (b) of Proposition 3.2, we have χ Hjj ′ − χ b Hjj ′ = Γ jj ′ (1 , − Γ jj ′ (cid:0) θθ , θθ (cid:1) = 0 ;(b) The inequality follows from (b) of Proposition 3.2 and from (12).We emphasize that the quantity θ ∗∗ jj ′ τ ∗∗ jj ′ (cid:16) θ j ∨ θ j ′ (cid:17) that we find in (12) and in (b) of the previousproposition reflects a clustering propensity of X nj ∧ X nj ′ through the extremal index θ ∗∗ jj ′ and τ ∗∗ jj ′ (cid:18) θ j ∨ θ j ′ (cid:19) = lim n →∞ nP ( X nj > n ( θ j ∨ θ j ′ ) , X nj ′ > n ( θ j ∨ θ j ′ )) . From this discussion we conclude that:(i) The tail dependencies of (cid:16) c M n , c M n (cid:17) and of ( M n , M n ) , for large n , evaluated through coef-ficient χ , can be considered equal when the multivariate extremal index is constant, otherwisethey differ in at least max n θ ∗∗ jj ′ τ ∗∗ jj ′ (cid:16) θ j ∨ θ j ′ (cid:17) − jj ′ (1 , , − Γ jj ′ (1 , o , where the previ-ous quantities can be estimated from the existing methods in literature.(ii) If we estimate the dependence χ Fjj ′ on the tail of ( X nj , X nj ′ ) , we do not obtain the depen-dence on the tail of ( M n , M n ) , unless we correct the result with an estimate of Γ jj ′ (1 , − θ jj ′ (cid:16) θ j , θ j ′ (cid:17) Γ jj ′ (cid:16) θ j , θ j ′ (cid:17) .In cases where b H has totally dependent marginals ( χ b Hjj ′ = 1 ) or has independent marginals( χ b Hjj ′ = 0 ), the previous lower limit loses interest by triviality. We underline the expression of χ Hjj ′ in these two cases in the next result, which is derived from (a) of Proposition 3.2. Corollary 3.4.
For stationary sequences { X n } , with multivariate extremal index θ ( τττ ) , τττ ∈ R d + , wehave, for any choice j < j ′ in D ,(a) If H has independent marginals, then χ Hjj ′ = 2 − (cid:16) θ j + θ j ′ (cid:17) θ jj ′ (cid:16) θ j , θ j ′ (cid:17) ;(b) If H has totally dependent marginals, then χ Hjj ′ = 2 − (cid:16) θ j ∨ θ j ′ (cid:17) θ jj ′ (cid:16) θ j , θ j ′ (cid:17) . Now we construct some examples that illustrate the cases χ Hjj ′ > χ b Hjj ′ and χ Hjj ′ < χ b Hjj ′ . xample 3.1. We first consider the following bivariate M4 process with one moving pattern, X n = Z n − ∨ Z n ∨ Z n +1 X n = Z n − ∨ Z n ∨ Z n +1 , where Z n ≡ Z ,n , ∀ n ≥ . We have in this case C F ( u , u ) = (cid:16) u / ∧ u / (cid:17) (cid:16) u / ∧ u / (cid:17) (cid:16) u / ∧ u / (cid:17) and χ F = χ b H = 2 − (cid:18)
28 + 18 + 68 (cid:19) = 78 . Otherwise H ( x , x ) = exp (cid:18) − (cid:18) x − ∨ x − (cid:19)(cid:19) . Therefore, C H ( u , u ) = u ∧ u and χ H = 1 > χ b H . Example 3.2.
Now consider a modification in the above example through the introduction of onemore pattern, X n = Z ,n ∨ Z ,n +1 ∨ Z ,n X n = Z ,n ∨ Z ,n +1 ∨ Z ,n , We have the same C F and χ F = as in the previous example, but here H ( x , x ) = exp (cid:18) − (cid:18) x − ∨ x − (cid:19)(cid:19) exp (cid:18) − (cid:18) x − ∨ x − (cid:19)(cid:19) . and therefore, C H ( u , u ) = (cid:16) u / ∧ u / (cid:17) (cid:16) u / ∧ u / (cid:17) . Then χ H = 2 − (cid:0) + (cid:1) = < χ b H . References [1] Coles, S., Heffernan, J., Tawn, J. (1999). Dependence Measures for Extreme Value Analyses.Extremes 2(4), 339-365[2] Davis, R.A. (1982). Limit laws for the maximum and minimum of stationary sequences. Z.Wahrsch. verw. Gebiete. 61, 31-42[3] Ehlert, A., Schlather, M. (2008). Capturing the multivariate extremal index: Bounds andinterconnections. Extremes 11(4), 353-377[4] Ferreira, H., Pereira, L., Martins, A.P. (2016). Clustering of high values in random fields.Accepted for publication in Extremes
5] Ferreira, H., Ferreira, M. (2015). Extremes of scale mixtures of multivariate time series. Journalof Multivariate Analysis 137, 82-99[6] Ferreira, H., Ferreira, M. (2016). Estimating the extremal index through local dependence.Accepted for publication in Annales de l’Institut Henri Poincaré[7] Gomes, M.I., Hall, A., Miranda, M.C. (2008). Subsampling techniques and the Jackknifemethodology in the estimation of the extremal index. Computational Statistics & Data Anal-ysis 52(4), 2022-2041[8] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.[9] Leadbetter, M.R., Lindgren, G., Rootzén, H. (1983). Extremes and Related Properties ofRandom Sequences and Processes, Springer, Berlin.[10] Ledford, A.W., Tawn, J.A. (1996). Statistics for near independence in multivariate extremevalues. Biometrika 83, 169-187[11] Ledford, A.W., Tawn, J.A. (1997). Modelling dependence within joint tail regions. J. R. Statist.Soc. B 59, 475-499[12] Li, H. (2009). Orthant tail dependence of multivariate extreme value distributions. Journal ofMultivariate Analysis 100(1), 243-256[13] Martins, A.P., Ferreira, H. (2005). The multivariate extremal index and the dependence struc-ture of a multivariate extreme value distribution. TEST 14(2), 433-448[14] Nandagopalan, S. (1994). On the multivariate extremal index. J. of Research, National Inst. ofStandards and Technology 99, 543-550[15] Naveau, P., Guillou, A., Cooley, D. and Diebolt, J. (2009). Modelling pairwise dependence ofmaxima in space. Biometrika 96, 1-17[16] Northrop, P.J. (2015) An efficient semiparametric maxima estimator of the extremal index.Extremes 18(4), 585-603[17] Sibuya, M. (1960). Bivariate extreme statistics. Annals of the Institute of Statistical Mathe-matics 11, 195-2105] Ferreira, H., Ferreira, M. (2015). Extremes of scale mixtures of multivariate time series. Journalof Multivariate Analysis 137, 82-99[6] Ferreira, H., Ferreira, M. (2016). Estimating the extremal index through local dependence.Accepted for publication in Annales de l’Institut Henri Poincaré[7] Gomes, M.I., Hall, A., Miranda, M.C. (2008). Subsampling techniques and the Jackknifemethodology in the estimation of the extremal index. Computational Statistics & Data Anal-ysis 52(4), 2022-2041[8] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.[9] Leadbetter, M.R., Lindgren, G., Rootzén, H. (1983). Extremes and Related Properties ofRandom Sequences and Processes, Springer, Berlin.[10] Ledford, A.W., Tawn, J.A. (1996). Statistics for near independence in multivariate extremevalues. Biometrika 83, 169-187[11] Ledford, A.W., Tawn, J.A. (1997). Modelling dependence within joint tail regions. J. R. Statist.Soc. B 59, 475-499[12] Li, H. (2009). Orthant tail dependence of multivariate extreme value distributions. Journal ofMultivariate Analysis 100(1), 243-256[13] Martins, A.P., Ferreira, H. (2005). The multivariate extremal index and the dependence struc-ture of a multivariate extreme value distribution. TEST 14(2), 433-448[14] Nandagopalan, S. (1994). On the multivariate extremal index. J. of Research, National Inst. ofStandards and Technology 99, 543-550[15] Naveau, P., Guillou, A., Cooley, D. and Diebolt, J. (2009). Modelling pairwise dependence ofmaxima in space. Biometrika 96, 1-17[16] Northrop, P.J. (2015) An efficient semiparametric maxima estimator of the extremal index.Extremes 18(4), 585-603[17] Sibuya, M. (1960). Bivariate extreme statistics. Annals of the Institute of Statistical Mathe-matics 11, 195-210