Extremal attractors of Liouville copulas
EExtremal attractors of Liouville copulas
Léo R. Belzile ∗ and Johanna G. Nešlehová † This is a copyedited, author-produced version of an article accepted for publication following peer-review in the
Journal of Multivariate Analysis , an Elsevier publication, c (cid:13)
Journal of MultivariateAnalysis (2017), 160C, pp. 68–92. doi:10.1016/j.jmva.2017.05.008
Abstract
Liouville copulas introduced in [31] are asymmetric generalizations of the ubiquitous Archimedean copula class.They are the dependence structures of scale mixtures of Dirichlet distributions, also called Liouville distributions. Inthis paper, the limiting extreme-value attractors of Liouville copulas and of their survival counterparts are derived.The limiting max-stable models, termed here the scaled extremal Dirichlet, are new and encompass several existingclasses of multivariate max-stable distributions, including the logistic, negative logistic and extremal Dirichlet. Asshown herein, the stable tail dependence function and angular density of the scaled extremal Dirichlet model have atractable form, which in turn leads to a simple de Haan representation. The latter is used to design e ffi cient algorithmsfor unconditional simulation based on the work of [10] and to derive tractable formulas for maximum-likelihoodinference. The scaled extremal Dirichlet model is illustrated on river flow data of the river Isar in southern Germany.
1. Introduction
Copula models play an important role in the analysis of multivariate data and find applications in many areas, includingbiostatistics, environmental sciences, finance, insurance, and risk management. The popularity of copulas is rootedin the decomposition of Sklar [39], which is at the heart of flexible statistical models and various measures, conceptsand orderings of dependence between random variables. According to Sklar’s result, the distribution function of anyrandom vector X = ( X , . . . , X d ) with continuous univariate margins F , . . . , F d satisfies, for any x , . . . , x d ∈ R ,Pr( X ≤ x , . . . , X d ≤ x d ) = C { F ( x ) , . . . , F d ( x d ) } , for a unique copula C , i.e., a distribution function on [0 , d whose univariate margins are standard uniform. Alterna-tively, Sklar’s decomposition also holds for survival functions, i.e., for any x , . . . , x d ∈ R ,Pr( X > x , . . . , X d > x d ) = ˆ C { ¯ F ( x ) , . . . , ¯ F d ( x d ) } , where ¯ F , . . . , ¯ F d are the marginal survival functions and ˆ C is the survival copula of X , related to the copula of X asfollows. If U is a random vector distributed as the copula C of X , ˆ C is the distribution function of 1 − U .In risk management applications, the extremal behavior of copulas is of particular interest, as it describes the de-pendence between extreme events and consequently the value of risk measures at high levels. Our purpose is to studythe extremal behavior of Liouville copulas. The latter are defined as the survival copulas of Liouville distributions[14, 17, 38], i.e., distributions of random vectors of the form R D α , where R is a strictly positive random variableindependent of the Dirichlet random vector D α = ( D , . . . , D d ) with parameter vector α = ( α , . . . , α d ). Liouvillecopulas were proposed by McNeil and Nešlehová [31] in order to extend the widely used class of Archimedeancopulas and create dependence structures that are not necessarily exchangeable. The latter property means that for ∗ École Polytechnique Fédérale de Lausanne, EPFL-SB-MATH-STAT, Station 8, CH-1015 Lausanne, Switzerland. [email protected] † Department of Mathematics and Statistics, McGill University, 805 rue Sherbrooke Ouest, Montréal, Québec, H3A 0B9, Canada. [email protected] a r X i v : . [ m a t h . S T ] J u l L. R. B elzile and
J. G. N e ˇ slehov ´ a any u , . . . , u d ∈ [0 ,
1] and any permutation π of the integers 1 , . . . , d , C ( u , . . . , u d ) = C ( u π (1) , . . . , u π ( d ) ). When α = d ≡ (1 , . . . , D α = D d is uniformly distributed on the unit simplex S d = { x ∈ [0 , d : x + · · · + x d = } . (1)In this special case, one recovers Archimedean copulas. Indeed, according to [30], the latter are the survival copulasof random vectors R D d , where R is a strictly positive random variable independent of D d . When α (cid:44) d , the survivalcopula of R D α is not Archimedean anymore. It is also no longer exchangeable, unless α = · · · = α d .In this article, we determine the extremal attractor of a Liouville copula and of its survival counterpart. As a by-product, we also obtain the lower and upper tail dependence coe ffi cients of Liouville copulas that quantify the strengthof dependence at extreme levels [25]. These results are complementary to [21], where the upper tail order functionsof a Liouville copula and its density are derived when α = · · · = α d , and to [19], where the extremal attractor of R D α is derived when R is light-tailed. The extremal attractors of Liouville copulas are interesting in their own right.Because non-exchangeability of Liouville copulas carries over to their extremal limits, the latter can be used to modelthe dependence between extreme risks in the presence of causality relationships [15]. The limiting extreme-valuemodels can be embedded in a single family, termed here the scaled extremal Dirichlet, whose members are new, non-exchangeable generalizations of the logistic, negative logistic, and Coles–Tawn extremal Dirichlet models given in[7]. We examine the scaled extremal Dirichlet model in detail and derive its de Haan spectral representation. The latteris simple and leads to feasible stochastic simulation algorithms and tractable formulas for likelihood-based inference.The article is organized as follows. The extremal behavior of the univariate margins of Liouville distributions isfirst studied in Section 2. The extremal attractors of Liouville copulas and their survival counterparts are then derivedin Section 3. When α is integer-valued, the results of [27, 31] lead to closed-form expressions for the limiting stabletail dependence functions, as shown in Section 4. Section 5 is devoted to a detailed study of the scaled extremalDirichlet model. In Section 6, the de Haan representation is derived and used for stochastic simulation. Estimationis investigated in Section 7, where expressions for the censored likelihood and the gradient score are also given. Anillustrative data analysis of river flow of the river Isar is presented in Section 8, and the paper is concluded by adiscussion in Section 9. Lengthy proofs are relegated to the Appendices.In what follows, vectors in R d are denoted by boldface letters, x = ( x , . . . , x d ); d and d refer to the vectors(0 , . . .
0) and (1 , . . . ,
1) in R d , respectively. Binary operations such as x + y or a · x , x a are understood as component-wise operations. (cid:107) · (cid:107) stands for the (cid:96) -norm, viz. (cid:107) x (cid:107) = | x | + · · · + | x d | , ⊥⊥ for statistical independence. For any x , y ∈ R ,let x ∧ y = min( x , y ) and x ∨ y = max( x , y ). The Dirac delta function I i j is 1 if i = j and zero otherwise. Finally, R d + isthe positive orthant [0 , ∞ ) d and for any x ∈ R , x + denotes the positive part of x , max(0 , x ).
2. Marginal extremal behavior
A Liouville random vector X = R D α is a scale mixture of a Dirichlet random vector D α = ( D , . . . , D d ) withparameters α = ( α , . . . , α d ) > d . In what follows, R is referred to as the radial variable of X and ¯ α denotes the sumof the Dirichlet parameters, viz. ¯ α = (cid:107) α (cid:107) = α + · · · + α d . Recall that D α has the same distribution as Z / (cid:107) Z (cid:107) , where Z i ∼ Ga ( α i , i = , . . . , d are independent Gamma variables with scaling parameter 1. The margins of X are thusscale mixtures of Beta distributions, i.e., for i = , . . . , d , X i = RD i with D i ∼ Beta ( α i , ¯ α − α i ).As a first step towards the extremal behavior of Liouville copulas, this section is devoted to the extreme-valueproperties of the univariate margins of the vectors X and 1 / X , where X is a Liouville random vector with parameters α and a strictly positive radial part R , i.e., such that Pr( R ≤ =
0. To this end, recall that a univariate random variable X with distribution function F is in the maximum domain of attraction of a non-degenerate distribution F , denoted F ∈ M ( F ) or X ∈ M ( F ), if and only if there exist sequences of reals ( a n ) and ( b n ) with a n >
0, such that, for any x ∈ R , lim n →∞ F n ( a n x + b n ) = F ( x ) . By the Fisher–Tippett Theorem, F must be, up to location and scale, either the Fréchet ( Φ ρ ), the Gumbel ( Λ ) or theWeibull distribution ( Ψ ρ ) with parameter ρ >
0. Further recall that a measurable function f : R + → R + is calledregularly varying with index ρ ∈ ( −∞ , ∞ ), denoted f ∈ R ρ , if for any x > f ( tx ) / f ( t ) → x ρ as t → ∞ . If ρ = f iscalled slowly varying. For more details and conditions for F ∈ M ( F ), see, e.g., [12, 35]. xtremal attractors of Liouville copulas X are scale mixtures of Beta distributions, their extremal behavior, detailed inProposition 1, follows directly from Theorems 4.1, 4.4. and 4.5 in [20]. Proposition 1
Let X = R D α be a Liouville random vector with parameters α = ( α , . . . , α d ) and a strictly positiveradial variable R, i.e., Pr( R ≤ = . Then the following statements hold for any ρ > : (a) R ∈ M ( Φ ρ ) if and only if X i ∈ M ( Φ ρ ) for all i = , . . . , d. (b) R ∈ M ( Λ ) if and only if X i ∈ M ( Λ ) for all i = , . . . , d. (c) R ∈ M ( Ψ ρ ) if and only if X i ∈ M ( Ψ ρ + ¯ α − α i ) for all i = , . . . , d. Proposition 1 implies that the univariate margins of X are all in the domain of attraction of the same distribution if thelatter is Gumbel or Fréchet. This is not the case when R is in the Weibull domain of attraction. Note also that there arecases not covered by Proposition 1, in which the univariate margins X i are in the Weibull domain while R is not in thedomain of attraction of any extreme-value distribution. For example, when d = α = (1 ,
1) and R = X are standard uniform and hence in the maximum domain of attraction of Ψ ; see Example 3.3.15 in[12]. At the same time, R is clearly neither in the Weibull, nor the Gumbel, nor the Fréchet domain of attraction.In subsequent sections, we shall also need the extremal behavior of the univariate margins of 1 / X . The propositionbelow shows that the latter is determined by the properties of 1 / R . In contrast to Proposition 1, however, the univariatemargins of 1 / X are always in the Fréchet domain. The proof may be found in A. Proposition 2
Let X = R D α be a Liouville random vector with parameters α = ( α , . . . , α d ) and a strictly positiveradial variable R with Pr( R ≤ = . The following statements hold for any i = , . . . , d.(a) If / R ∈ M ( Φ ρ ) for ρ ∈ (0 , α i ] , then / X i ∈ M ( Φ ρ ) .(b) If E(1 / R α i + ε ) < ∞ for some ε > , then / X i ∈ M ( Φ α i ) .
3. Extremal behavior of Liouville copulas
In this section, we will identify the extremal behavior of a Liouville random vector X = R D α and of the randomvector 1 / X , assuming that Pr( R ≤ =
0. As a by-product, we will obtain the extremal attractors of Liouville copulasand their survival counterparts. To this end, recall that a random vector Y with joint distribution function H is in themaximum domain of attraction of a non-degenerate distribution function H , in notation H ∈ M ( H ) or Y ∈ M ( H ),i ff there exist sequences of vectors ( a n ) in (0 , ∞ ) d and ( b n ) in R d such that for all x ∈ R d ,lim n →∞ H n ( a n x + b n ) = H ( x ) . When the univariate margins F , . . . , F d of H are continuous, H ∈ M ( H ) holds if and only if F i ∈ M ( F i ) for all i = , . . . , d , where F , . . . , F d are the univariate margins of H , and further if the unique copula C of H is in thedomain of attraction of the unique copula C of H , denoted C ∈ M ( C ), i.e., i ff for all u ∈ [0 , d ,lim n →∞ C n ( u / n ) = C ( u ) . In particular, the univariate margins of the max-stable distribution H must each follow a generalized extreme-valuedistribution, and C must be an extreme-value copula. This means that for all u ∈ [0 , d , C ( u ) = exp[ − (cid:96) {− log( u ) , . . . , − log( u d ) } ] , (2)where (cid:96) : R d + → [0 , ∞ ) is a stable tail dependence function, linked to the so-called exponent measure ν viz. ν { [ d , x ) c } = (cid:96) (1 / x ), see, e.g., [35]. The latter can be characterized through an angular (or spectral) probabilitymeasure σ d on S d given in Eq. (1) which satisfies (cid:82) S d w i d σ d ( w ) = / d for all i = , . . . , d . For all x ∈ R d + , one has (cid:96) ( x ) = d (cid:90) S d max( w x , . . . , w d x d ) d σ d ( w ) . (3) L. R. B elzile and J. G. N e ˇ slehov ´ a Because (cid:96) is homogeneous of order 1, i.e., for any c > x ∈ R d + , (cid:96) ( c x ) = c (cid:96) ( x ), C can also be expressed via thePickands dependence function A : S d → [0 , ∞ ) related to (cid:96) through (cid:96) ( x ) = (cid:107) x (cid:107) A( x / (cid:107) x (cid:107) ). Then at any u ∈ [0 , d , C ( u ) = exp (cid:34) log( u · · · u d )A (cid:40) log( u )log( u · · · u d ) , . . . , log( u d )log( u · · · u d ) (cid:41)(cid:35) . When d =
2, it is more common to define the Pickands dependence function A : [0 , → [0 ,
1] through (cid:96) ( x , x ) = ( x + x )A { x / ( x + x ) } so that, for all u , u ∈ [0 , C ( u , u ) = exp (cid:34) log( u u )A (cid:40) log( u )log( u u ) (cid:41)(cid:35) . (4)Now consider a Liouville vector X = R D α with a strictly positive radial variable. Theorem 1 specifies when X ∈ M ( H ) and identifies H . While part (a) follows from regular variation of X , parts (b) and (c) are special casesof the results discussed in Section 2.2 in [19]. Details of the proof may be found in B. Theorem 1
Let X = R D α , D α = ( D , . . . , D d ) , α > d , and Pr( R ≤ = . Then the following statements hold.(a) If R ∈ M ( Φ ρ ) for some ρ > , then X ∈ M ( H ) , where H is a multivariate extreme-value distribution withunivariate margins F i = Φ ρ , i = , . . . , d, and a stable tail dependence function given, for all x ∈ R d + , by (cid:96) ( x ) = Γ ( ¯ α + ρ ) Γ ( ¯ α ) E max Γ ( α ) x D ρ Γ ( α + ρ ) , . . . , Γ ( α d ) x d D ρ d Γ ( α d + ρ ) . (b) If R ∈ M ( Λ ) , then X ∈ M ( H ) , where for all x ∈ R d , H ( x ) = (cid:81) di = Λ ( x i ) .(c) If R ∈ M ( Ψ ρ ) for some ρ > , then X ∈ M ( H ) , where for all x ∈ R d , H ( x ) = (cid:81) di = Ψ ρ + ¯ α − α i ( x i ) . The next result, also proved in B, specifies the conditions under which 1 / X ∈ M ( H ) and gives the form of the limitingextreme-value distribution H . Theorem 2
Let X = R D α , D α = ( D , . . . , D d ) , α > d , and assume that Pr( R ≤ = . Let α M = max( α , . . . , α d ) .The following cases can be distinguished:(a) If / R ∈ M ( Φ ρ ) for ρ ∈ (0 , α M ] , set I = { i : α i ≤ ρ } , I = { i : α i > ρ } and ¯ α = (cid:80) i ∈ I α i . Then / X ∈ M ( H ) , where the univariate margins of H are F i = Φ ρ ∧ α i , i = , . . . , d, and the stable tail dependencefunction is given, for all x ∈ R d + , by (cid:96) ( x ) = (cid:88) i ∈ I x i + Γ ( ¯ α − ρ ) Γ ( ¯ α ) E max i ∈ I Γ ( α i ) x i D − ρ i Γ ( α i − ρ ) = (cid:88) i ∈ I x i + Γ ( ¯ α − ρ ) Γ ( ¯ α ) E max i ∈ I Γ ( α i ) x i (cid:101) D − ρ i Γ ( α i − ρ ) , where ( (cid:101) D i , i ∈ I ) is a Dirichlet random vector with parameters ( α i , i ∈ I ) if | I | > and (cid:101) D i ≡ if I = { i } .(b) If E (cid:16) / R β (cid:17) < ∞ for β > α M , then / X ∈ M ( H ) , where for all x ∈ R d , H ( x ) = (cid:81) di = Φ α i ( x i ) . Remark 1
Note that in the case of asymptotic independence between the components of X (Theorem 1 (b–c)) or1 / X (Theorem 2 (b)), dependence between component-wise maxima of finitely many vectors may still be present.Refinements of asymptotic independence are then needed, but these considerations surpass the scope of this paper.One option would be to consider triangular arrays as in [23]; extremes of arrays of Liouville vectors can be obtainedas a special case of extremes of arrays of weighted Dirichlet distributions developed in [18]. Another avenue worthexploring might be the limits of scaled sample clouds, as in [2] and [32].The stable tail dependence functions appearing in Theorems 1 and 2 will be investigated in greater detail in thesubsequent sections. Before proceeding, we introduce the following terminology, emphasizing that they can in fact beembedded in one and the same parametric class. xtremal attractors of Liouville copulas Definition 1
For any α > and ρ ∈ ( − α, ∞ ) , let c ( α, ρ ) = Γ ( α + ρ ) / Γ ( α ) denote the rising factorial. For d ≥ and α , . . . , α d > and let ( D , . . . , D d ) denote a Dirichlet random vector with parameters α = ( α , . . . , α d ) and set ¯ α = α + · · · + α d . For any − min( α , . . . , α d ) < ρ < ∞ , the scaled extremal Dirichlet stable tail dependence functionwith parameters ρ and α is given, for all x ∈ R d + , by (cid:96) D ( x ; ρ, α ) = c ( ¯ α, ρ )E max x D ρ c ( α , ρ ) , . . . , x d D ρ d c ( α d , ρ ) , (5) when ρ (cid:44) and by max( x , . . . , x d ) when ρ = . For any ρ > , the positive scaled extremal Dirichlet stable taildependence function (cid:96) pD with parameters ρ and α is given, for all x ∈ R d + , by (cid:96) pD ( x ; ρ, α ) = (cid:96) D ( x ; ρ, α ) , while for any < ρ < min( α , . . . , α d ) , the negative scaled extremal Dirichlet stable tail dependence function (cid:96) nD is given, for all x ∈ R d + , by (cid:96) nD ( x ; ρ, α ) = (cid:96) D ( x ; − ρ, α ) . Remark 2
As will be seen in Section 5, distinguishing between the positive and negative scaled extremal Dirichletmodels makes the discussion of their properties slightly easier because the sign of ρ impacts the shape of the corre-sponding angular measure. When ρ → (cid:96) D ( x ; ρ, α ) becomes max( x , . . . , x d ), the stable tail dependence functioncorresponding to comonotonicity, while when ρ → ∞ , (cid:96) D ( x ; ρ, α ) becomes x + · · · + x d , the stable tail dependencefunction corresponding to independence. Note also that ρ ∈ ( −∞ , ∞ ) can be allowed, with the convention that allvariables whose indices i are such that ρ ≤ − α i are independent, i.e., (cid:96) nD is then of the form given in Theorem 2 (a).From Theorems 1 and 2, we can now easily deduce the extremal behavior of Liouville copulas and their survivalcounterparts. To this end, recall that a Liouville copula C is defined as the survival copula of a Liouville randomvector X = R D α with Pr( R ≤ =
0. The following corollary follows directly from Theorem 2 upon noting that C isalso the unique copula of 1 / X . Corollary 1
Let C be the unique survival copula of a Liouville random vector X = R D α with Pr( R ≤ = . Let α M = max( α , . . . , α d ) and set I = { i : α i ≤ ρ } , I = { i : α i > ρ } . Then the following statements hold.(a) If / R ∈ M ( Φ ρ ) for ρ ∈ (0 , α M ] and | I | > , then C ∈ M ( C ) , where C is an extreme-value copula of the form (2) whose stable tail dependence function is given, for all x ∈ R d + , by (cid:96) ( x ) = (cid:88) i ∈ I x i + (cid:96) nD ( x { } ; ρ, α { } ) , where and x { } = ( x i , i ∈ I ) , α { } = ( α i , i ∈ I ) . If / R ∈ M ( Φ ρ ) for ρ ∈ (0 , α M ] and | I | ≤ , then C ∈ M ( Π ) ,where Π is the independence copula given, for all u ∈ [0 , d , by Π ( u ) = u · · · u d .(b) If E (cid:16) / R β (cid:17) < ∞ for β > α M , then C ∈ M ( Π ) . Remark 3
Observe that Corollary 1 (a) in particular implies that when d = α < α and 1 / R ∈ M ( Φ ρ ) for α ≤ ρ < α , C ∈ M ( Π ). Also note that when α = · · · = α d ≡ α and 1 / R ∈ M ( Φ ρ ) for ρ ∈ (0 , α ), the result inCorollary 1 (a) can be derived from formula (5) in Proposition 3 in [21] by relating the tail order function to the stabletail dependence function when the tail order equals 1.The survival counterpart ˆ C of a Liouville copula C is given as the distribution function of 1 − U , where U is arandom vector distributed as C . As C is the unique survival copula of X , ˆ C is the unique copula of X . The followingresult thus follows directly from Theorem 1. Corollary 2
Let ˆ C be the unique copula of a Liouville random vector X = R D α with Pr( R ≤ = . Then thefollowing statements hold.(a) If R ∈ M ( Φ ρ ) for ρ > , then ˆ C ∈ M ( C ) , where C is an extreme-value copula of the form (2) with the positivescaled extremal Dirichlet stable tail dependence function given, for all x ∈ R d + , by (cid:96) pD ( x ; ρ, α ) .(b) If R ∈ M ( Λ ) or R ∈ M ( Ψ ρ ) with ρ > , then ˆ C ∈ M ( Π ) , where Π is the independence copula. L. R. B elzile and
J. G. N e ˇ slehov ´ a
4. The case of integer-valued Dirichlet parameters
When α is integer-valued, Liouville distributions are particularly tractable because their survival function is explicit.In this section, we will use this fact to derive closed-form expressions for the positive and negative scaled extremalDirichlet stable tail dependence functions. To this end, first recall the notion of the Williamson transform. The latteris related to Weyl’s fractional integral transform and was used to characterize d -monotone functions in [43]; it wasadapted to non-negative random variables in [30]. Definition 2
Let X be a non-negative random variable with distribution function F, and let k ≥ be an arbitraryinteger. The Williamson k-transform of X is given, for all x > , by W k F ( x ) = (cid:90) ∞ x (cid:18) − xr (cid:19) k − dF ( r ) = E (cid:18) − xX (cid:19) k − + . For any k ≥
1, the distribution of a positive random variable X is uniquely determined by its Williamson k -transform,the formula for the inverse transform being explicit [30, 43]. If ψ = W k F , then, for all x > F ( x ) = W − k ψ ( x ) = − k − (cid:88) j = ( − j x j ψ ( j ) ( x ) j ! − ( − k − x k − ψ ( k − + ( x )( k − , where for j = , . . . , k − ψ ( j ) is the j th derivative of ψ and ψ ( k − + is the right-hand derivative of ψ ( k − . These deriva-tives exist because a Williamson k -transform ψ is necessarily k -monotone [43]. This means that ψ is di ff erentiable upto order k − , ∞ ) with derivatives satisfying ( − j ψ ( j ) ≥ j = , . . . , k − − k − ψ ( k − isnon-increasing and convex on (0 , ∞ ). Moreover, ψ ( x ) → x → ∞ and if F (0) = ψ ( x ) → x → C be a Liouville copula corresponding to a Liouville random vector X = R D α with integer-valuedparameters α = ( α , . . . , α d ) and a strictly positive radial part R , i.e., Pr( R ≤ =
0. Let ψ be the Williamson¯ α -transform of R and set I α = { , . . . , α − } × · · · × { , . . . , α d − } . By Theorem 2 in [31], one then has, for all x ∈ R d + ,Pr( X > x ) = ¯ H ( x ) = (cid:88) ( j ,..., j d ) ∈ I α ( − j + ··· + j d ψ ( j + ··· + j d ) ( x + · · · + x d ) j ! · · · j d ! d (cid:89) i = x j i i . (6)In particular, the margins of X have survival functions satisfying, for all x > i = , . . . , d ,Pr( X i > x ) = ¯ H i ( x ) = α i − (cid:88) j = ( − j x j ψ ( j ) ( x ) j ! = − W − α i ψ ( x ) . (7)By Sklar’s Theorem for survival functions, the Liouville copula C is given, for all u ∈ [0 , d , by C ( u ) = ¯ H { ¯ H − ( u ) , . . . , ¯ H − d ( u d ) } . Although this formula is not explicit, it is clear from Equations (6) and (7) that C depends on the distribution of X only through the Williamson ¯ α -transform ψ of R and the Dirichlet parameters α . For this reason, we shall denotethe Liouville copula in this section by C ψ, α and refer to ψ as its generator, reiterating that ψ must be an ¯ α -monotonefunction satisfying ψ (1) = ψ ( x ) → x → ∞ . When α = d , C ψ, is the Archimedean copula with generator ψ , given, for all u ∈ [0 , d by C ψ, ( u ) = ψ { ψ − ( u ) + · · · + ψ − ( u d ) } . Because the relationship between ψ and R isone-to-one [30, Proposition 3.1], we will refer to R as the radial distribution corresponding to ψ .Now suppose that 1 / R ∈ M ( Φ ρ ) with ρ ∈ (0 , − ψ (1 / · ) ∈R − ρ . It further follows from Corollary 1 (a) that C ψ, α ∈ M ( C ) where C is an extreme-value copula with the negativescaled extremal Dirichlet stable tail dependence function (cid:96) nD ( · ; ρ, α ). This is because ρ < ≤ min( α , . . . , α d ) so that I = ∅ in Corollary 1 (a). Eq. (6) and the results of [27] can now be used to derive the following explicit expressionfor (cid:96) nD , as detailed in C. xtremal attractors of Liouville copulas Proposition 3
Let C ψ, α be a Liouville copula with integer-valued parameters α = ( α , . . . , α d ) and generator ψ . If − ψ (1 / · ) ∈ R − ρ for some ρ ∈ (0 , , then C ψ, α ∈ M ( C ) , where C is an extreme-value copula with scaled negativeextremal Dirichlet stable tail dependence function (cid:96) nD as given in Definition 1. Furthermore, for all x ∈ R d + , (cid:96) nD ( x ; ρ, α ) = Γ (1 − ρ ) d (cid:88) j = (cid:40) x j c ( α j , − ρ ) (cid:41) /ρ ρ − ρ (cid:88) ( j ,..., j d ) ∈ I α ( j ,..., j d ) (cid:44) (0 ,..., Γ ( j + · · · + j d − ρ ) Γ (1 − ρ ) d (cid:89) i = Γ ( j i + (cid:110) x i c ( α i , − ρ ) (cid:111) /ρ (cid:80) dk = (cid:110) x k c ( α k , − ρ ) (cid:111) /ρ j i . When α = d , the index set I α reduces to the singleton { } , and the expression for (cid:96) nD given in Proposition 3 simplifies,for all x ∈ R d + , to the stable tail dependence function of the Gumbel–Hougaard copula, viz. (cid:96) nD ( x ; ρ, d ) = (cid:0) x /ρ + · · · + x /ρ d (cid:1) ρ . The Liouville copula C ψ, d , which is the Archimedean copula with generator ψ , is thus indeed in the domain ofattraction of the Gumbel–Hougaard copula with parameter 1 /ρ , as shown, e.g., in [6, 27]. Remark 4
When α = d and 1 − ψ (1 / · ) ∈ R − , it is shown in Proposition 2 of [27] that C ψ, is in the domain ofattraction of the independence copula. However, when α is integer-valued but such that max( α , . . . , α d ) >
1, regularvariation of 1 − ψ (1 / · ) does not su ffi ce to characterize those cases in Corollary 1 that are not covered by Proposition 3.This is because by Theorem 2 of [27], 1 / R ∈ M ( Φ ρ ) for ρ ≥
1, 1 / R ∈ M ( Λ ) and 1 / R ∈ M ( Ψ ρ ) for ρ > − ψ (1 / · ) ∈ R − . At the same time, by Corollary 1, C ψ, α ∈ M ( Π ) clearly does not hold in all these cases.Next, let ˆ C ψ, α be the survival copula of a Liouville copula C ψ, α , i.e., the distribution function of 1 − U , where U is arandom vector with distribution function C ψ, α . The results of [27] can again be used to restate the conditions underwhich ˆ C ψ, α ∈ M ( C ) in terms of ψ and to give an explicit expression for the stable tail dependence function of C . Proposition 4
Let ˆ C ψ, α be the survival copula of a Liouville copula C ψ, α with integer-valued parameters α and agenerator ψ . Then the following statements hold.(a) If ψ ∈ R − ρ for some ρ > , then ˆ C ψ, α ∈ M ( C ) , where C has a positive scaled extremal Dirichlet stable taildependence function (cid:96) pD as given in Definition 1. The latter can be expressed, for all x ∈ R d + , as (cid:96) pD ( x ; ρ, α ) = Γ (1 + ρ ) Γ ( ρ ) d (cid:88) k = (cid:88) ≤ i < ··· < i k ≤ d ( − k + k (cid:88) j = x i j c ( α i j , ρ ) − /ρ − ρ × (cid:88) ( j ,..., j k ) ∈ I ( α i ,...,α ik ) Γ ( j + · · · + j k + ρ ) j ! · · · j k ! k (cid:89) m = (cid:110) x im c ( α im , ρ ) (cid:111) − /ρ (cid:80) kj = (cid:26) x ij c ( α ij , ρ ) (cid:27) − /ρ j m . (b) If ψ ∈ M ( Λ ) or ψ ∈ M ( Ψ ρ ) for some ρ > , ¯ C ψ, α ∈ M ( Π ) , where Π is the independence copula. When α = d , the expression for (cid:96) pD in part (a) of Proposition 4 simplifies, for all x ∈ R d + , to (cid:96) pD ( x ; ρ, d ) = (cid:88) A ⊆{ ,..., d } , A (cid:44) ∅ ( − | A | + (cid:88) i ∈ A x − /ρ i − ρ , which is the stable tail dependence function of the Galambos copula [24]. When ψ ∈ R − ρ for some ρ >
0, ˆ C ψ, d is thusindeed in the domain of attraction of the Galambos copula, as shown, e.g., in [27].
5. Properties of the scaled extremal Dirichlet models
In this section, the scaled extremal Dirichlet model with stable tail dependence function given in Definition 1 isinvestigated in greater detail. In Section 5.1 we derive formulas for the so-called angular density and relate thepositive and negative scaled extremal Dirichlet models to classical classes of stable tail dependence functions. InSection 5.2 we focus on the bivariate case and derive explicit expressions for the stable tail dependence functions and,as a by-product, obtain formulas for the tail dependence coe ffi cients of Liouville copulas. L. R. B elzile and J. G. N e ˇ slehov ´ a . . . . w A ngu l a r d e n s it y scaled Dirichlet, ρ > . . . . . . . w A ngu l a r d e n s it y scaled Dirichlet, ρ < Figure 1: Angular density of the scaled extremal Dirichlet model. Left panel: ρ = / α = (2 , /
2) (black full), ρ = / α = (1 / , /
10) (red dashed), ρ = / α = (1 / , /
2) (blue dotted). Right panel: ρ = − / α = (2 , /
2) (black full), α = (2 / , /
5) (red dashed) and α = (1 / , /
2) (blue dotted).
The first property worth noting is that the positive and negative scaled extremal Dirichlet models are closed undermarginalization. Indeed, letting x i → ≤ i ≤ d , we can easily derive from Lemma 2 that forany α > d , ρ >
0, and any x ∈ R d + , (cid:96) pD ( x ; ρ, α ) → (cid:96) pD ( x − i ; ρ, α − i ) as x i →
0, where for any y ∈ R d , y − i denotes thevector ( y , . . . , y i − , y i + , . . . , y d ). Similarly, for any 1 ≤ i ≤ d , α > d , 0 < ρ < min( α , . . . , α d ), and any x ∈ R d + , (cid:96) nD ( x ; ρ, α ) → (cid:96) nD ( x − i ; ρ, α − i ) as x i → S d when ρ (cid:44)
0, the density of the angular measure σ d completely characterizes the stable tail dependence function and hencealso the associated extreme-value copula. This so-called angular density of the scaled extremal Dirichlet models isgiven below and derived in D. Proposition 5
Let d ≥ and set α , . . . , α d > and ¯ α = α + · · · + α d . For any ρ > − min( α , . . . , α d ) , let also c ( α , ρ ) = ( c ( α , ρ ) , . . . , c ( α d , ρ )) , where c ( α, ρ ) is as in Definition 1. Then for any − min( α , . . . , α d ) < ρ < ∞ , ρ (cid:44) ,the angular density of the scaled extremal Dirichlet model with parameters ρ > and α is given, for all w ∈ S d , byh D ( w ; ρ, α ) = Γ ( ¯ α + ρ ) d | ρ | d − (cid:81) di = Γ ( α i ) d (cid:88) j = (cid:110) c ( α j , ρ ) w j (cid:111) /ρ − ρ − ¯ α d (cid:89) i = { c ( α i , ρ ) } α i /ρ w α i /ρ − i . The angular density of the positive scaled extremal Dirichlet model with parameters ρ > and α is given, for all w ∈ S d , by h pD ( w ; ρ, α ) = h D ( w ; ρ, α ) , while the angular density of the negative scaled extremal Dirichlet model withparameters < ρ < min( α , . . . , α d ) and α is given, for all w ∈ R d + , by h nD ( w ; ρ, α ) = h D ( w ; − ρ, α ) . From Proposition 5, it is easily seen that when α = d , the angular density h pD reduces, for any ρ > w ∈ S d ,to the angular density of the symmetric negative logistic model; see, e.g., Section 4.2 in [7]. In general, the angulardensity h pD is not symmetric unless α = α d .The positive scaled Dirichlet model can thus be viewed as a new asymmetric generalization of the negative logisticmodel which does not place any mass on the vertices or facets of S d , unless at independence or comonotonicity, i.e.,when ρ → ∞ and ρ →
0, respectively. Furthermore, h pD can also be interpreted as a generalization of the Coles–Tawn extremal Dirichlet model. Indeed, h pD ( x ; 1 , α ) is precisely the angular density of the latter model given, e.g., inEquation (3.6) in [7]. Similarly, the negative scaled extremal Dirichlet model is a new asymmetric generalization ofGumbel’s logistic model [16]. Indeed, when α = d , h nD simplifies to the logistic angular density, given, e.g., on p.381 in [7]. xtremal attractors of Liouville copulas scaled Dirichlet, α = (1 , / , / , ρ = 1 / w = 1 w = 1 w = 1 scaled Dirichlet, α = (5 / , , , ρ = − / w = 1 w = 1 w = 1 scaled Dirichlet, α = (1 / , / , / , ρ = 1 / w = 1 w = 1 w = 1 scaled Dirichlet, α = (5 / , / , / , ρ = − / w = 1 w = 1 w = 1 Figure 2: Angular density of the scaled extremal Dirichlet model with α = (1 , / , / , ρ = / α = (1 / , / , / , ρ = / α = (5 / , , ρ = − / α = (5 / , / , / , ρ = − / h pD and h nD that obtain through various choices of α and ρ . Theasymmetry when α (cid:44) α d is clearly apparent. For the same value of ρ , the shapes of the angular density can be quitedi ff erent depending on α . In view of the aforementioned closure of both the positive and negative scaled extremalDirichlet models under marginalization, this means that these models are able to capture strong dependence in somepairs of variables (represented by a mode close to 1 / When d =
2, the stable tail dependence functions of the positive and negative scaled extremal Dirichlet models havea closed-form expression in terms of the incomplete beta function given, for any t ∈ (0 ,
1) and α , α >
0, byB( t ; α , α ) = (cid:90) t x α − (1 − x ) α − d x . When t =
1, this integral is the beta function, viz. B( α , α ) = Γ ( α ) Γ ( α ) / Γ ( α + α ). A direct calculation yields thecorresponding Pickands dependence function, for any t ∈ [0 , pD ( t ; ρ, α , α ) = (cid:96) pD (1 − t , t ; ρ, α , α ), i.e.,0 L. R. B elzile and J. G. N e ˇ slehov ´ a t A ( t ) scaled Dirichlet, ρ > t A ( t ) scaled Dirichlet, ρ < Figure 3: Pickands dependence function of the scaled extremal Dirichlet model. Left panel: ρ = / α = (2 , / ρ = / α = (1 / , /
10) (red dashed), ρ = / α = (1 / , /
2) (blue dotted). Right panel: ρ = − / α = (2 , /
2) (black full), α = (2 / , /
5) (red dashed) and α = (1 / , /
2) (blue dotted).A pD ( t ; ρ, α , α ) = (1 − t )B( α , α + ρ ) B (cid:34) { c ( α , ρ )(1 − t ) } /ρ { c ( α , ρ )(1 − t ) } /ρ + { c ( α , ρ ) t } /ρ ; α , α + ρ (cid:35) + t B( α , α + ρ ) B (cid:34) { c ( α , ρ ) t } /ρ { c ( α , ρ )(1 − t ) } /ρ + { c ( α , ρ ) t } /ρ ; α , α + ρ (cid:35) . When α = α =
1, A pD becomes the Pickands dependence function of the Galambos copula, viz. A pD ( t ; ρ, , = − (cid:8) t − /ρ + (1 − t ) − /ρ (cid:9) − ρ , as expected given that the positive scaled extremal Dirichlet model becomes the symmetricnegative logistic model in this case.Similarly, for any t ∈ [0 , nD ( t ; ρ, α , α ) = (cid:96) nD (1 − t , t ; ρ, α , α ) equalsA nD ( t ; ρ, α , α ) = (1 − t )B( α − ρ, α ) B (cid:34) { (1 − t ) c ( α , − ρ ) } /ρ { tc ( α , − ρ ) } /ρ + { (1 − t ) c ( α , − ρ ) } /ρ ; α − ρ, α (cid:35) + t B( α − ρ, α ) B (cid:34) { tc ( α , − ρ ) } /ρ { tc ( α , − ρ ) } /ρ + { (1 − t ) c ( α , − ρ ) } /ρ ; α − ρ, α (cid:35) . When α = α =
1, A nD simplifies to the stable tail dependence function of the Gumbel extreme-value copula, viz.A nD ( t ; ρ, , = (cid:8) t /ρ + (1 − t ) /ρ (cid:9) ρ . This again confirms that the negative scaled extremal Dirichlet model becomesthe symmetric logistic model when α = α =
1. The Pickands dependence functions A pD and A nD are illustrated inFigure 3, for the same choices of parameters and the corresponding angular density shown in Figure 1.The above formulas for A pD and A nD now easily lead to expressions for their upper tail dependence coe ffi cients.Recall that for an arbitrary bivariate copula C , the lower and upper tail dependence coe ffi cients of [25] are given by λ (cid:96) ( C ) = lim u → C ( u , u ) u , λ u ( C ) = − lim u → C ( u , u ) − u − = lim u → ˆ C ( u , u ) u , where ˆ C is the survival copula of C , provided these limits exist. When C is bivariate extreme-value with Pickandsdependence function A, it follows easily from (4) that λ (cid:96) ( C ) = λ u ( C ) = − / xtremal attractors of Liouville copulas C pD ρ, α is a bivariate extreme-value copula with positive scaled extremal Dirichlet Pickands de-pendence function A pD and parameters ρ > α , α >
0. Then λ u ( C pD ρ, α ) = − α , α + ρ ) B (cid:40) c ( α , ρ ) /ρ c ( α , ρ ) /ρ + c ( α , ρ ) /ρ ; α , α + ρ (cid:41) − α , α + ρ ) B (cid:40) c ( α , ρ ) /ρ c ( α , ρ ) /ρ + c ( α , ρ ) /ρ ; α , α + ρ (cid:41) . (8)Similarly, if C nD ρ, α is a bivariate extreme-value copula with negative scaled extremal Dirichlet Pickands dependencefunction A nD and parameters α , α > < ρ < min( α , α ), then λ u ( C nD ρ, α ) = − α − ρ, α ) B (cid:40) c ( α , − ρ ) /ρ c ( α , − ρ ) /ρ + c ( α , − ρ ) /ρ ; α − ρ, α (cid:41) + α − ρ, α ) B (cid:40) c ( α , − ρ ) /ρ c ( α , − ρ ) /ρ + c ( α , − ρ ) /ρ ; α − ρ, α (cid:41) . (9)In the symmetric case α = α ≡ α , Expressions (8) and (9) simplify to λ u ( C pD ρ, α ) = − α, α + ρ ) B (cid:32)
12 ; α, α + ρ (cid:33) , λ u ( C nD ρ, α ) = − α − ρ, α ) B (cid:32)
12 ; α − ρ, α (cid:33) . Formulas (8) and (9) lead directly to expressions for the tail dependence coe ffi cients of Liouville copulas. This isbecause if C ∈ M ( C ), where C is an extreme-value copula with Pickands tail dependence function A , λ u ( C ) = − (1 /
2) [29, Proposition 7.51]. Similarly, if ˆ C ∈ M ( C ∗ ), where C ∗ is an extreme-value copula with Pickandstail dependence function A ∗ , λ (cid:96) ( C ) = − ∗ (1 / Corollary 3
Suppose that C is the survival copula of a Liouville random vector R D α with parameters α > and aradial part R such that Pr( R ≤ = . Then the following statements hold.(a) If R ∈ M ( Φ ρ ) for some ρ > , λ (cid:96) ( C ) = λ u ( C pD ρ, α ) is given by Eq. (8).(b) If R ∈ M ( Λ ) or R ∈ M ( Ψ ρ ) for some ρ > , λ (cid:96) ( C ) = .(c) If / R ∈ M ( Φ ρ ) for some < ρ < α ∧ α , λ u ( C ) = λ u ( C nD ρ, α ) is given by Eq. (9).(d) If / R ∈ M ( Φ ρ ) for ρ > α ∧ α or if E(1 / R β ) < ∞ for β > α ∨ α , λ u ( C ) = . The role of the parameters α and ρ is best explained if we consider the reparametrization ∆ α = | α − α | and Σ α = α + α . As is the case for the Dirichlet distribution, the level of dependence is higher for large values of Σ α .Furthermore, λ u is monotonically decreasing in ρ . Higher levels of extremal asymmetry, as measured by departuresfrom the diagonal on the copula scale, are governed by both Σ α and ∆ α . The larger Σ α , the lower the asymmetry.Likewise, the larger ∆ α , the larger the asymmetry. Contrary to the case of extremal dependence, the behavior in ρ isnot monotone. For the negative scaled extremal Dirichlet model, asymmetry is maximal when ρ ≈ α ∧ α . When Σ α is small, smaller values of ρ induce larger asymmetry, but this is not the case for larger values of Σ α where theasymmetry profile is convex with a global maximum attained for larger values of ρ .
6. de Haan representation and simulation algorithms
Random samples from the scaled extremal Dirichlet model can be drawn e ffi ciently using the algorithms recentlydeveloped in [10]. We first derive the so-called de Haan representation in Section 6.1 and adapt the algorithms from[10] to the present setting in Section 6.2.2 L. R. B elzile and J. G. N e ˇ slehov ´ a First, introduce the following family of univariate distributions, which we term the scaled Gamma family and denoteby sGa ( a , b , c ). It has three parameters a , c > b (cid:44) x >
0, by f ( x ; a , b , c ) = | b | Γ ( c ) a − bc x bc − exp (cid:40) − (cid:18) xa (cid:19) b (cid:41) . (10)Observe that when Z ∼ Ga ( c ,
1) is a Gamma variable with shape parameter c > Y d = aZ / b isscaled Gamma sGa ( a , b , c ). Consequently, E ( Y ) = a Γ ( c + / b ) / Γ ( c ) < ∞ provided that b < − / c . The scaled Gammafamily includes several well-known distributions as special cases, notably the Gamma when b =
1, the Weibull when c = b >
0, the inverse Gamma when b = −
1, and the Fréchet when c = b <
0. When b >
0, the scaledGamma is the generalized Gamma distribution of [41], albeit in a di ff erent parametrization.Now consider the parameters α = ( α , . . . , α d ) with α > d and ρ > − min( α , . . . , α d ), ρ (cid:44)
0. Let V be a randomvector with independent scaled Gamma margins V i ∼ sGa { / c ( α i , ρ ) , /ρ, α i } , where for α > c ( α, ρ ) = Γ ( α + ρ ) / Γ ( α )as in Definition 1. If Z is a random vector with independent Gamma margins Z i ∼ Ga ( α i ,
1) then for all i = , . . . , d , V i d = Z ρ i / c ( α i , ρ ). Furthermore, recall that (cid:107) Z (cid:107) ∼ Ga ( ¯ α,
1) is independent of Z / (cid:107) Z (cid:107) , which has the same distributionas the Dirichlet vector D α = ( D , . . . , D d ). One thus has, for all x ∈ R d + ,E (cid:26) max ≤ i ≤ d ( x i V i ) (cid:27) = E (cid:34) max ≤ i ≤ d (cid:40) x i Z ρ i c ( α i , ρ ) (cid:41)(cid:35) = E ( (cid:107) Z (cid:107) ρ ) E (cid:34) max ≤ i ≤ d (cid:40) x i D ρ i c ( α i , ρ ) (cid:41)(cid:35) = (cid:96) D ( x ; ρ, α ) , (11)where (cid:96) D is as in Definition 1, given that E ( (cid:107) Z (cid:107) ρ ) = c ( ¯ α, ρ ).When ρ =
1, the positive scaled Dirichlet extremal model becomes the Coles–Tawn Dirichlet extremal model, V i ∼ Ga ( α i ,
1) and Eq. (11) reduces to the representation derived in [37]. When α = d , (cid:96) D becomes the stable taildependence function of the negative logistic model, V i is Weibull and Eq. (11) is the representation in Appendix A.2.4of [10]. Similarly, when ρ < α = d , the negative scaled Dirichlet extremal model becomes the logistic model, V i is Fréchet and Eq. (11) is the representation in Appendix A.2.4 of [10]. The requirement that ρ > − min( α , . . . , α d )ensures that the expectation of V i is finite for all i ∈ { , . . . , d } .Eq. (11) implies that the max-stable random vector Y with unit Fréchet margins and extreme-value copula withstable tail dependence function (cid:96) D ( · ; ρ, α ) admits the de Haan [9] spectral representation Y d = max k ∈ N ζ k V k , (12)where Z = { ζ k } ∞ k = is a Poisson point process on (0 , ∞ ) with intensity ζ − d ζ and V k is an i.i.d. sequence of ran-dom vectors independent of Z . Furthermore, the univariate margins of V k are independent and such that V k j ∼ sGa { / c ( α j , ρ ) , /ρ, α j } for j = , . . . , d with E ( V k ) = d for all k ∈ N . The de Haan representation (12) o ff ers, among other things, an easy route to unconditional simulation of max-stablerandom vectors that follow the scaled Dirichlet extremal model, as laid out in [10] in the more general context of max-stable processes. To see how this work applies in the present setting, fix an arbitrary j ∈ { , . . . , d } and recall that the j th extremal function φ + j is given, almost surely, as ζ k V k such that Y j = ζ k V k j . From Eq. (12) and Proposition 1 in[10] it then directly follows that φ + j / Y j = ( W j / W j j , . . . , W j d / W j j ), where W j = ( W j , . . . , W j d ) is a randomvector with density given, for all x ∈ R d + , by | /ρ | Γ ( α j ) c ( α j , ρ ) α j /ρ x α j /ρ j exp (cid:104) −{ c ( α j , ρ ) x j } /ρ (cid:105) × d (cid:89) j = , j (cid:44) j | /ρ | Γ ( α j ) c ( α j , ρ ) α j /ρ x α j /ρ − j exp (cid:104) −{ c ( α j , ρ ) x j } /ρ (cid:105) . This means that the components of W j are independent and such that W j j ∼ sGa { / c ( α j , ρ ) , /ρ, α j } when j (cid:44) j and W j j ∼ sGa { / c ( α j , ρ ) , /ρ, α j + ρ } . In other words, W j j ∼ Z ρ j / c ( α j , ρ ) where Z j ∼ Ga ( α j + ρ, j (cid:44) j , W j j d = Z ρ j / c ( α j , ρ ) where Z j ∼ Ga ( α j , xtremal attractors of Liouville copulas φ + j / Y j given above now allows for an easy adaptation of the algorithms in [10]. Todraw an observation from the extreme-value copula with the scaled Dirichlet stable tail dependence function (cid:96) D withparameters α > d and ρ > − min( α , . . . , α d ), ρ (cid:44)
0, one can follow Algorithms 1 and 2 below. The first procedurecorresponds to Algorithm 1 in [10] and relies on [36]; the second is an adaptation of Algorithm 2 in [10].
Algorithm 1
Exact simulations from the extreme-value copula based on spectral densities. Simulate E ∼ Exp (1) . Set Y = . while / E > min( Y , . . . , Y d ) do Simulate J from the uniform distribution on { , . . . , d } . Simulate independent Z j ∼ Ga ( α j ,
1) for j ∈ { , . . . , d } \ J and Z J ∼ Ga ( α J + ρ, Set W j ← Z ρ j / c ( α j , ρ ), j = , . . . , d . Set S ← W / (cid:107) W (cid:107) . Update Y ← max { Y , d S / E } . Simulate E ∗ ∼ Exp (1) and update E ← E + E ∗ . return U = exp( − / Y ). Algorithm 2
Exact simulations based on sequential sampling of the extremal functions. Simulate Z ∼ Ga ( α + ρ,
1) and Z j ∼ Ga ( α j , j = , . . . , d . Compute W where W j ← Z ρ j / c ( α j , ρ ), j = , . . . , d . Simulate E ∼ Exp (1). Set Y ← W / ( W E ). for k = , . . . , d do Simulate E k ∼ Exp (1). while / E k > Y k do Simulate independent Z k ∼ Ga ( α k + ρ,
1) and Z j ∼ Ga ( α j , j = , . . . , d , j (cid:44) k . Set W = ( W , . . . , W d ) where W j ← Z ρ j / c ( α j , ρ ), j = , . . . , d . if W i / ( W k E k ) < Y i for all i = , . . . , k − then Update Y ← max { Y , W / ( W k E k ) } . Simulate E ∗ ∼ Exp (1) and update E k ← E k + E ∗ . return U = exp( − / Y ). Note that S obtained in Step of Algorithm 1 has the angular distribution σ d of (cid:96) D ; see Theorem 1 in [10].Similar algorithms for drawing samples from the angular distribution of the extremal logistic and Dirichlet modelswere obtained in [3]. Algorithm 2 requires a lower number of simulations and is more e ffi cient on average, cf. [10].Both algorithms are easily implemented using the function rmev in the mev package within the R Project for StatisticalComputing [34], which returns samples of max-stable scaled extremal Dirichlet vectors with unit Fréchet margins,i.e., Y in Algorithms 1 and 2.
7. Estimation
The scaled extremal Dirichlet model can be used to model dependence between extreme events. To this end, severalschemes can be envisaged. For example, one can consider the block-maxima approach, given that max-stable distri-butions are the most natural for such data. Another option is peaks-over-threshold models. Yet another alternative,used in [13] for the Brown–Resnick model, is to approximate the conditional distribution of a random vector with unitFréchet margins given that the j th component exceeds a large threshold by the distribution of φ + j / Y j discussed inSection 6.2.Here, we focus on the multivariate tail model of [28]; see also Section 16.4 in [29]. To this end, let X , . . . , X n be arandom sample from some unknown multivariate distribution H with continuous univariate margins which is assumedto be in the maximum domain of attraction of a multivariate extreme-value distribution H . To model the tail of H , itsmargins F j , j = , . . . , d can first be approximated using the univariate peaks-over-threshold method. For all x above4 L. R. B elzile and J. G. N e ˇ slehov ´ a some high threshold u j , one then has F j ( x ) ≈ ˜ F j ( x ; η j , ξ j ) = − ν j (cid:32) + ξ j ( x − u j ) η j (cid:33) − /ξ j + , (13)where ν j = − F j ( u j ), and η j > ξ j are the parameters of the generalized Pareto distribution. Furthermore, for w su ffi ciently close to d , the copula of H can be approximated by the extreme-value copula C of H , so that, for x ≥ u , H ( x ) ≈ ˜ H ( x ) = C { ˜ F ( x ) , . . . , ˜ F d ( x d ) } . The parameters of this multivariate tail model, i.e., the parameters θ of the stable tail dependence function (cid:96) of C as well as the marginal parameters ν , η and ξ can be estimatedusing likelihood methods; this allows, e.g., for Bayesian inference, generalized additive modeling of the parametersand model selection based on likelihood-ratio tests. For a comprehensive review of likelihood inference methods forextremes, see, e.g., [22].The multivariate tail model can be fitted in low-dimensions using the censored likelihood L ( X ; ν , η , ξ , θ ) = (cid:81) ni = L i ( X i ; ν , η , ξ , θ ), where for i = , . . . , n , L i ( X i ; u , ν , η , ξ , θ ) = ∂ m i ˜ H ( y , . . . , y d ) ∂ y j · · · ∂ y j mi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = max( X i , u ) = ∂ m i exp {− (cid:96) (1 / y ) } ∂ y j · · · ∂ y j mi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = t { max( X i , u ) } m i (cid:89) k = J j k ( X i j k ) (14)In this expression, the indices j , . . . , j m i are those of the components of X i exceeding the thresholds u and for x ≥ u , t ( x ) = ( t ( x ) , . . . , t d ( x d )), where for j = , . . . , d , t j ( x j ) = − { ˜ F j ( x j ; η j , ξ j ) } , J j ( x j ) = ν j η j (cid:32) + ξ j ( x j − u j ) η j (cid:33) − /ξ j − { ˜ F j ( x j ; η j , ξ j ) } ] ˜ F j ( x j ; η j , ξ j ) . (15)The censored likelihood L ( X ; ν , η , ξ , θ ) can be maximized either over all parameters at once, or the marginal param-eters ν , η and ξ can be estimated from each univariate margin separately, so that only the estimate of θ is obtainedthrough maximizing L . When d is large, one can also maximize the likelihood in [40] that uses the tail approximation¯ H ( x ) ≈ − (cid:96) (1 / x ). In either case, (cid:96) and the higher-order partial derivatives of (cid:96) (1 / x ) need to be computed.When (cid:96) is the scaled extremal Dirichlet stable tail dependence function (cid:96) D ( · ; ρ, α ) given in Definition 1 withparameters α > ρ > − min( α , . . . , α d ), ρ (cid:44)
0, its expression is not explicit. However, (cid:96) D can be calcu-lated numerically using adaptive numerical cubature algorithms for integrals of functions defined on the simplex, asimplemented in, e.g., the R package SimplicialCubature . Given the representation in Eq. (5), (cid:96) D is also easilyapproximated using Monte Carlo methods. Instead of employing Eq. (5) directly and sampling from the Dirichletvector D α , one can use the more e ffi cient importance sampling estimator (cid:98) (cid:96) D (1 / u , ρ, α ) = B B (cid:88) i = max ≤ j ≤ d (cid:104) { c ( α j , ρ ) u j } − D ρ i j (cid:105) d (cid:80) dj = c ( α j , ρ ) − D ρ i j , where D i ∼ d − (cid:80) dj = Dir ( α + I j ρ d ) is sampled from a Dirichlet mixture.The partial derivatives of (cid:96) D can be calculated using the following result, shown in E. Proposition 6
Let (cid:96) D be the scaled extremal Dirichlet stable tail dependence function with parameters α > d and − min( α , . . . , α d ) < ρ < ∞ , ρ (cid:44) . Then, for any x ∈ R d + , ∂ d (cid:96) D (1 / x ) ∂ x · · · ∂ x d = − dh D ( x ; ρ, α ) = − Γ ( ¯ α + ρ ) | ρ | d − (cid:81) di = Γ ( α i ) d (cid:88) j = (cid:110) c ( α j , ρ ) x j (cid:111) /ρ − ρ − ¯ α d (cid:89) i = { c ( α i , ρ ) } α i /ρ x α i /ρ − i , (16) where h D is as given in Proposition 5. Furthermore, for all k = , . . . , d − and x ∈ R d + , ∂ k (cid:96) D (1 / x ) ∂ x · · · ∂ x k = − (cid:90) ∞ t k k (cid:89) i = f (cid:32) x i t ; 1 c ( α i , ρ ) , ρ , α i (cid:33) d (cid:89) i = k + F (cid:32) x i t ; 1 c ( α i , ρ ) , ρ , α i (cid:33) dt , xtremal attractors of Liouville copulas where f (; a , b , c ) and F (; a , b , c ) denote, respectively the density and distribution function of the scaled Gammadistribution with parameters a , c > and b (cid:44) given in Eq. (10). Furthermore, if γ ( c , x ) = (cid:82) x t c − e − t dt de-notes the lower incomplete gamma function, then for x > , F ( x ; a , b , c ) = γ { c , ( x / a ) b } / Γ ( c ) when b > whileF ( x ; a , b , c ) = − γ { c , ( x / a ) b } / Γ ( c ) when b < . Other estimating equations could be used to circumvent the calculation of (cid:96) D (1 / x ) and its partial derivatives. Aninteresting alternative to likelihoods in the context of proper scoring functions is proposed in [8]. Specifically, theauthors advocate the use of the gradient score, adapted by them for the peaks-over-threshold framework, δ w ( x ) = d (cid:88) i = w i ( x ) ∂ w i ( x ) ∂ x i ∂ log h ( x ) ∂ x i + w i ( x ) ∂ log h ( x ) ∂ x i + (cid:40) ∂ log h ( x ) ∂ x i (cid:41) for a di ff erentiable weighting function w ( x ), unit Fréchet observations x and density h ( x ) that would correspond inthe setting of the scaled extremal Dirichlet to dh D ( x ; ρ, α ). Explicit expressions for the derivatives of log dh D may befound in E. The parameter estimates are obtained as the solution to argmax θ ∈ Θ (cid:80) ni = δ w ( x i )I R ( x i / u ) > , where θ = ( ρ, α )is the vector of parameters of the model and R is a di ff erentiable risk functional, usually the (cid:96) p norm for some p ∈ N .Although the gradient score is not asymptotically most e ffi cient, weighting functions can be designed to reproduceapproximate censoring, lending the method robustness and tractability.
8. Data illustration
In this section, we illustrate the use of the scaled extremal Dirichlet model on a trivariate sample of daily riverflow data of the river Isar in southern Germany; this dataset is a subset of the one analyzed in [1]. All the codecan be downloaded from https://github.com/lbelzile/ealc . For this analysis, we selected data measured atLenggries (upstream), Pupplinger Au (in the middle) and Munich (downstream). To ensure stationarity of the seriesand given that the most extreme events occur during the summer, we restricted our attention to the measurementsfor the months of June, July and August. Since the sites are measuring the flow of the same river, dependence atextreme levels is likely to be present, as is indeed apparent from Figure 4. Directionality of the river may further leadto asymmetry in the asymptotic dependence structure, suggesting that the scaled extremal Dirichlet model may bewell suited for these data. Furthermore, given that other well-known models like the extremal Dirichlet, logistic andnegative logistic are nested within this family, their adequacy can be assessed through likelihood ratio tests.To remove dependence between extremes over time, we decluster each series and retain only the cluster maximabased on three-day runs. Rounding of the measurements has no impact on parameter estimates and is henceforthneglected. The multivariate tail model outlined in Section 7 is next fitted to the cluster maxima. The thresholds u = ( u , u , u ) were selected to be the 92% quantiles using the parameter stability plot of [42] (not shown here).Next, set θ = ( η , ξ , α , ρ ), where η and ξ are the marginal parameters of the generalized Pareto distribution in Eq. (13)and ρ and α are the parameters of the scaled Dirichlet model. To estimate θ , the trivariate censored likelihood (14)could be used. To avoid numerical integration and because of the relative robustness to misspecification, we employedthe pairwise composite log-likelihood l C of [28] instead; the loss of e ffi ciency in this trivariate example is likely small.Specifically, we maximized l C ( θ ) = n (cid:88) i = d − (cid:88) j = d (cid:88) k = j + (cid:104) log g { t j ( x i j ) , t k ( x ik ); θ , t j ( u j ) , t k ( u k ) } + I x ij > u j log J j ( x i j ) + I x ik > u k log J k ( x ik ) (cid:105) , where g ( y j , y k ; θ , u j , u k ) = exp {− (cid:96) (1 / u j , / u k ) } , y j ≤ u j , y k ≤ u k − ∂(cid:96) (1 / y j , / u k ) /∂ y j exp {− (cid:96) (1 / y j , / u k ) } , y j > u j , y k ≤ u k − ∂(cid:96) (1 / u j , / y k ) /∂ y k exp {− (cid:96) (1 / u j , / y k ) } , y j ≤ u j , y k > u k (cid:104)(cid:110) ∂(cid:96) (1 / y j , / y k ) /∂ y j (cid:111) (cid:110) ∂(cid:96) (1 / y j , / y k ) /∂ y k (cid:111) − dh D ( y j , y k ) (cid:105) exp {− (cid:96) (1 / y j , / y k ) } , y j > u j , y k > u k where (cid:96) = (cid:96) D and for all j = , . . . , d , t j and J j are as in Eq. (15).6 L. R. B elzile and J. G. N e ˇ slehov ´ a Isar daily river flow
Muenchen
Pupplinger Au
Lenggries
Figure 4: Daily river flow of the Isar river at three sitesUncertainty assessment can be done in the same way as for general estimating equations. Specifically, let g ( θ )denote an unbiased estimating function and define the variability matrix J , the sensitivity matrix H and the Godambeinformation matrix G as J = E (cid:32) ∂ g ( θ ) ∂ θ ∂ g ( θ ) ∂ θ (cid:62) (cid:33) , H = − E (cid:32) ∂ g ( θ ) ∂ θ ∂ θ (cid:62) (cid:33) , G = HJ − H . (17)The maximum composite likelihood estimator is strongly consistent and asymptotically normal, centered at the trueparameter θ with covariance matrix given by the inverse Godambe matrix G − .Using the pairwise composite log-likelihood l C , we fitted the scaled extremal Dirichlet model as well as thelogistic and negative logistic models that correspond to the negative and positive scaled extremal Dirichlet models,respectively, and the parameter restriction α = d . The estimates of the marginal generalized Pareto parameters η and ξ are given in Table 1. As the estimates were obtained by maximizing (cid:96) C , their values depend on the fitted model; theline labeled “Marginal" corresponds to fitting the generalized Pareto distribution to threshold exceedances of each oneof the three series separately. The marginal Q-Q plots displayed in Figure 5 indicate a good fit of the model as well. η η η ξ ξ ξ Scaled Dirichlet 123.2 (7.5) 84.4 (5) 68.1 (4.2) 0.05 (0.04) -0.03 (0.04) 0.02 (0.04)Neg. logistic 117.1 (6.8) 86.2 (5.1) 70 (4.3) 0.08 (0.04) -0.05 (0.04) 0 (0.04)Logistic 117.3 (6.8) 86.6 (5.1) 70.4 (4.3) 0.08 (0.04) -0.05 (0.04) 0 (0.04)ext. Dirichlet 114.4 (6.8) 84.3 (4.9) 68.2 (4.1) 0.12 (0.04) -0.02 (0.04) 0.04 (0.04)Marginal 129.1 (14.5) 95.1 (10.6) 76 (8.7) -0.01 (0.08) -0.15 (0.08) -0.08 (0.08)Table 1: Generalized Pareto parameter estimates and standard errors (in parenthesis) for the trivariate river examplefor four di ff erent models. xtremal attractors of Liouville copulas
200 300 400 500 600 700 800 900
Q-Q plot, Muenchen
Theoretical quantiles S a m p l e qu a n til e s
200 300 400 500
Q-Q plot, Pupplinger Au
Theoretical quantiles S a m p l e qu a n til e s
100 200 300 400
Q-Q plot, Lenggries
Theoretical quantiles S a m p l e qu a n til e s Figure 5: Marginal Q-Q plots for the three sites based on pairwise composite likelihood estimates for the scale andshape parameters obtained from the scaled Dirichlet model, retaining marginal exceedances of the 92% quantiles. Thepointwise confidence intervals were obtained from the transformed Beta quantiles of the order statistics. α α α ρ Scaled Dirichlet 0.76 (0.3) 1.65 (0.82) 2.03 (1.15) − − Negative logistic w = 1 w = 1 w = 1 Logistic w = 1 w = 1 w = 1 scaledDirichlet w = 1 w = 1 w = 1 Figure 6: Angular density plots for the three models, the negative logistic (left), logistic (middle) and scaled Dirichlet(right). The colours correspond to log density values and range from red (high density) to blue (low density).The estimates of the dependence parameters α and ρ are given in Table 2. The last line displays the maximum gra-dient score estimates were obtained from the raw data, i.e., ignoring the clustering, after transforming the observationsto the standard Fréchet scale using the probability integral transform. We retained only the 10% largest values basedon the (cid:96) p norm with p =
20; this risk functional is essentially a di ff erentiable approximation of (cid:96) ∞ . We selected theweight function w ( x , u ) = x [1 − exp {− ( (cid:107) x (cid:107) p / u − } ] based on [8] to reproduce approximate censoring. The estimatesare similar to the composite maximum likelihood estimators, though not e ffi cient.8 L. R. B elzile and J. G. N e ˇ slehov ´ a The angular densities of the fitted logistic, negative logistic and scaled extremal Dirichlet models are displayedin Figure 6. The right panel of this figure shows asymmetry caused by a few extreme events that only happeneddownstream. Whether this asymmetry is significant can be assessed through composite likelihood ratio tests; recallthat the logistic model, the negative logistic model and the extremal Dirichlet model of [7] are all nested withinthe scaled extremal Dirichlet model. To this end, consider a partition of θ = ( ψ , λ ) into a q dimensional parameter ofinterest ψ and a 3 d + − q dimensional nuisance parameter λ , and the corresponding partitions of the matrices H , J and G . Let (cid:98) θ C = ( (cid:98) ψ C , (cid:98) λ C ) denote the maximum composite likelihood parameter estimates and (cid:98) θ = ( ψ , (cid:98) λ ) the restrictedparameter estimates under the null hypothesis that the simpler model is adequate. The asymptotic distribution ofthe composite likelihood ratio test statistic 2 { log l C ( (cid:98) θ C ) − log l C ( (cid:98) θ ) } is equal to (cid:80) qi = c i Z i where Z i are independent χ variables and c i are the eigenvalues of the q × q matrix ( H ψψ − H ψλ H − λλ H λψ ) G − ψψ ; see [26]. We estimated theinverse Godambe information matrix, G − , by the empirical covariance of B nonparametric bootstrap replicates. Thesensitivity matrix H was obtained from the Hessian matrix at the maximum composite likelihood estimate and thevariability matrix J from Eq. (17). Since the Coles–Tawn extremal Dirichlet, negative logistic and logistic models arenested within the scaled Dirichlet family, we test for a restriction to these simpler models; the respective approximate P -values were 0.003, 0.74 and 0.78. These values suggest that while the Coles–Tawn extremal Dirichlet model isclearly not suitable, there is not su ffi cient evidence to discard the logistic and negative logistic models. The e ff ects ofpossible model misspecification are also visible for the Coles–Tawn extremal Dirichlet model, as the parameter valuesof α , α and α are very large (viz. Table 2) and this induces negative bias in the shape parameter estimates, as canbe seen from Table 1.
9. Discussion
In this article, we have identified extremal attractors of copulas and survival copulas of Liouville random vectors R D α , where D α has a Dirichlet distribution on the unit simplex with parameters α , and R is a strictly positive randomvariable independent of D α . The limiting stable tail dependence functions can be embedded in a single family,which can capture asymmetry and provides a valid model in dimension d . The latter is novel and termed here thescaled extremal Dirichlet; it includes the well-known logistic, negative logistic as well as the Coles–Tawn extremalDirichlet models as special cases. In particular, therefore, this paper is first to provide an example of a random vectorattracted to the Coles–Tawn extremal Dirichlet model, which was derived by enforcing moment constraints on asimplex distribution rather than as the limiting distribution of a random vector.A scaled extremal Dirichlet stable tail dependence function (cid:96) D has d + ρ and α . The parametervector α is inherited from D α and induces asymmetry in (cid:96) D . The parameter ρ comes from the regular variation of R at zero and infinity, respectively; this is reminiscent of the extremal attractors of elliptical distributions [33]. Themagnitude of ρ has impact on the strength of dependence while its sign changes the overall shape of (cid:96) D . Having d + ffi ciently rich to account for spatial dependence,unlike the Hüsler–Reiss or the extremal Student- t models, which have one parameter for each pair of variables andare thus easily combined with distances. Also, it is less flexible than Dirichlet mixtures [4], which are however hardto estimate in high dimensions and require sophisticated machinery. To achieve greater flexibility, the scaled extremalDirichlet model could perhaps be extended by working with more general scale mixtures, such as of the weightedDirichlet distributions considered, e.g., in [18].Nonetheless, the scaled extremal Dirichlet model may naturally find applications whenever asymmetric extremaldependence is suspected; the latter may be caused, e.g., by causal relationships between the variables [15]. Thestochastic structure of the scaled extremal Dirichlet model has several major advantages, that make the model easyto interpret, estimate and simulate from. Its angular density has a simple form; in contrast to the asymmetric gen-eralizations of the logistic and negative logistic models, this model does not place any mass on the vertices andlower-dimensional facets of the unit simplex. Another plus is the tractability of the de Haan representation and ofthe extremal functions, both expressible in terms of independent scaled Gamma variables; this allows for feasibleinference and stochastic simulation. While the scaled extremal Dirichlet stable tail dependence function (cid:96) D does nothave a closed form in general, closed-form algebraic expressions exist when α is integer-valued and in the bivariatecase. Model selection for well-known families of extreme-value distributions can be performed through likelihoodratio tests. Another potentially useful feature is that ρ ∈ ( −∞ , ∞ ) can be allowed, with the convention that all variableswhose indices i are such that − ρ ≤ − α i are independent. xtremal attractors of Liouville copulas Acknowledgment
Funding in partial support of this work was provided by the Natural Sciences and Engineering Research Council(RGPIN-2015-06801, CGSD3-459751-2014), the Canadian Statistical Sciences Institute, and the Fonds de recherchedu Québec – Nature et technologies (2015–PR–183236). We thank the acting Editor-in-Chief, Richard Lockhart, theAssociate Editor and two anonymous referees for their valuable suggestions.
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Duke Math. J. , 23:189–207, 1956. xtremal attractors of Liouville copulas Appendix A. : Proofs from Section 2
Proof of Proposition 2.
To prove parts (a) and (b), recall that 1 / X i is distributed as 1 / ( RD i ), where D i ∼ Beta ( α i , ¯ α − α i )is independent of R . Furthermore, it is easy to show that 1 / D i ∈ M ( Φ α i ), which implies that E(1 / D β i ) < ∞ for any β < α i . The extremal behavior of 1 / X i will thus be determined by the extremal behavior of either 1 / R or 1 / D i ,depending on which one has a heavier tail. Indeed, Breiman’s Lemma [5] implies that 1 / X i ∈ M ( Φ ρ ) if 1 / R ∈ M ( Φ ρ )for some ρ < α i and that 1 / X i ∈ M ( Φ α i ) if E(1 / R α i + ε ) < ∞ for some ε >
0. Finally, the fact that 1 / X i ∈ M ( Φ α i ) when1 / R ∈ M ( Φ α i ) follows directly from the Corollary to Theorem 3 in [11].The following lemma is a side result of Proposition 2, which is needed in the subsequent proofs. Lemma 1
Suppose that X = R D α . If / R ∈ M ( Φ α i ) for some i ∈ { , . . . , d } , then lim x →∞ Pr (1 / R > x )Pr (1 / X i > x ) = . Proof of Lemma 1.
Because 1 / R ∈ M ( Φ α i ), Pr(1 / R > x ) is regularly varying with index − α i . In particular, for any b ∈ (0 , / R > xb ) / Pr(1 / R > x ) → b − α i as x → ∞ . An application of Fatou’s lemma thus giveslim inf x →∞ Pr(1 / X i > x )Pr(1 / R > x ) = lim inf x →∞ (cid:90) Pr(1 / R > xb )Pr(1 / R > x ) f D i ( b ) d b ≥ (cid:90) b − (1 − b ) ¯ α − α i − B( α i , ¯ α − α i ) d b = ∞ and hence the result. Appendix B. : Proofs from Section 3
First recall the following property of the Dirichlet distribution, which is easily shown using the transformation formulafor Lebesgue densities.
Lemma 2
Let D α be a Dirichlet random vector with parameters α . Then for any ≤ k ≤ d and any collection ofdistinct indices ≤ i < · · · < i k ≤ d, ( D i , . . . , D i k ) d = B i ,..., i k × D ( α i ,...,α ik ) , where B i ,..., i k ∼ Beta ( α i + · · · + α i k , ¯ α − ( α i + · · · + α i k )) is independent of the k-variate Dirichlet vector D ( α i ,...,α ik ) with parameters ( α i , . . . , α i k ) .Proof of Theorem 1. In order to prove part (a), recall that (cid:107) X (cid:107) d = R is independent of X / (cid:107) X (cid:107) d = D α . Because R ∈ M ( Φ ρ ), there exists a sequence ( b n ) of constants in (0 , ∞ ) such that, for any Borel set B ⊆ S d and any r > n →∞ n Pr (cid:32) (cid:107) X (cid:107) > b n r , X (cid:107) X (cid:107) ∈ B (cid:33) = lim n →∞ n Pr( R > b n r ) Pr( D α ∈ B ) = r − ρ Pr( D α ∈ B ) . By Corollary 5.18 in [35], X ∈ M ( H ) where for all x ∈ R d + , H ( x ) = exp − E max D ρ x ρ , . . . , D ρ d x ρ d . Let B( · , · ) denote the Beta function. The univariate margins of H are given, for all i = , . . . , d and x >
0, by F i ( x ) = exp (cid:110) − x − ρ E( D ρ i ) (cid:111) = exp (cid:40) − x − ρ B( ρ + α i , ¯ α − α i )B( α i , ¯ α − α i ) (cid:41) = exp (cid:40) − x − ρ Γ ( α i + ρ ) Γ ( ¯ α ) Γ ( ¯ α + ρ ) Γ ( α i ) (cid:41) for i ∈ { , . . . , d } . The copula of H then satisfies, for all u ∈ [0 , d , C ( u ) = H { F − ( u ) , . . . , F − d ( u d ) } = exp (cid:32) − Γ ( ¯ α + ρ ) Γ ( ¯ α ) E (cid:34) max ≤ i ≤ d (cid:40) ( − log u i ) Γ ( α i ) D ρ i Γ ( α i + ρ ) (cid:41)(cid:35)(cid:33) . elzile and J. G. N e ˇ slehov ´ a By Eq. (2), the stable tail dependence function of C thus indeed equals, for all x ∈ R d + , (cid:96) ( x ) = Γ ( ¯ α + ρ ) Γ ( ¯ α ) E max x Γ ( α ) D ρ Γ ( α + ρ ) , . . . , x d Γ ( α d ) D ρ d Γ ( α d + ρ ) . The part (b) follows directly from Proposition 2.2 in [19] upon setting p = i = , . . . , d and j = , . . . , d , λ i j = i = j and λ i j = i = , . . . , d , X i ∈ M ( Ψ ρ + ¯ α − α i ) and hence there existsequences ( a ni ) ∈ (0 , ∞ ), ( b ni ) ∈ R , such that for all x ∈ R ,lim n →∞ n Pr( X i > a ni x + b ni ) = − log { Ψ ρ + ¯ α − α i ( x ) } . Next, observe that as in the proof of Proposition 5.27 in [35], X ∈ M ( H ) follows if for all 1 ≤ i < j ≤ d and x k suchthat Ψ ρ + ¯ α − α k ( x k ) > k = i , j , lim n →∞ n Pr( X i > a ni x i + b ni , X j > a n j x j + b n j ) = . (B.1)To prove that (B.1) indeed holds, it su ffi ces to assume that d =
2. This is because for arbitrary indices 1 ≤ i < j ≤ d , Lemma 2 implies that ( X i , X j ) d = R ∗ ( B , − B ), where B ∼ Beta( α i , α j ), R ∗ d = RY is independent of B and Y ∼ Beta( α i + α j , ¯ α − α i − α j ) is independent of B and R . Because Pr( R ∗ ≤ =
0, Theorem 4.5 in [20] implies that R ∗ ∈ M ( Ψ ρ + ¯ α − α i − α j ) when R ∈ M ( Ψ ρ ) for some ρ >
0. Thus suppose that d = D , D ) ≡ ( B , − B ),where B ∼ Beta ( α , α ). Fix arbitrary x , x ∈ R are such that Ψ ρ + ¯ α − α i ( x i ) > i = ,
2. Then because for any a , c > b ∈ (0 , { a / b , c / (1 − b ) } ≥ a + c , one has0 ≤ Pr( X > a n x + b n , X > a n x + b n ) = Pr (cid:40) R > max (cid:32) a n x + b n B , a n x + b n − B (cid:33)(cid:41) ≤ Pr( R > a n x + b n + a n x + b n ) . In order to prove Eq. (B.1), it thus su ffi ces to show thatlim n →∞ n Pr( R > a n x + b n + a n x + b n ) = . (B.2)This however follows immediately from the fact that if R ∈ M ( Ψ ρ ) for some ρ >
0, the upper end-point r of R ,viz. r = sup { x : Pr( R ≤ x ) < } , is finite. Because for i = , , r is also the upper endpoint of X i , a ni x i + b ni → r as n → ∞ . This means that there exists n ∈ N so that for all n ≥ n , a n x + b n + a n x + b n > r and Pr( R > a n x + b n + a n x + b n ) =
0. This proves Eq. (B.1) and hence also Theorem 1 (c). Note that alternatively, part (c)could be proved using Theorem 2.1 in [19] similarly to the proof of Proposition 2.2 therein.The proof of Theorem 2 requires the following technical lemma.
Lemma 3
Suppose that D α = ( D , . . . , D d ) is a Dirichlet random vector with parameters α . Further let R be apositive random variable independent of D α such that Pr( R ≤ = , and let X = R D α . Then for any ≤ i < j ≤ dand any x i , x j ∈ (0 , ∞ ) , lim n →∞ n Pr (cid:32) X i > a ni x i , X j > a n j x j (cid:33) = if either:(i) / R ∈ M ( Φ ρ ) with ρ ∈ [ α i ∧ α j , α ∨ α ] , and for k = i , j, ( a nk ) is a sequence of positive constants such thatn Pr(1 / X k > a nk x k ) → x − ( α k ∧ ρ ) k as n → ∞ ;(ii) E (1 / R β ) < ∞ for some β > α i ∨ α j and for k = i , j, ( a nk ) is a sequence of positive constants such thatn Pr(1 / X k > a nk x k ) → x − α k k as n → ∞ . xtremal attractors of Liouville copulas Proof of Lemma 3.
Observe first that when d >
2, Lemma 2 implies that ( X i , X j ) d = R ∗ ( B , − B ), where R ∗ ⊥⊥ B , B ∼ Beta ( α i , α j ) and R ∗ = RY , with Y ⊥⊥ R and Y ∼ Beta ( α i + α j , ¯ α − α i − α j ). Now note that 1 / Y ∈ M ( Φ α i + α j ). Thusif 1 / R ∈ M ( Φ ρ ) for ρ ∈ [ α i ∧ α j , α ∨ α ], ρ < α i + α j and Breiman’s Lemma implies that 1 / R ∗ ∈ M ( Φ ρ ). Further, ifE(1 / R β ) < ∞ for some β ∈ ( α i ∨ α j , α i + α j ), E { / ( R ∗ ) β } < ∞ given that E(1 / Y β ) < ∞ . We can thus assume withoutloss of generality that d = α ≤ α ; we shall also write ( D , D ) ≡ ( B , − B ), where B ∼ Beta ( α , α ).To prove part (i), note first that the existence of the sequences ( a nk ), k = i , j , follows from Proposition 2, by which1 / X k ∈ M ( Φ ρ ∧ α k ) for k = i , j , and the Poisson approximation [12, Proposition 3.1.1]. Next, observe that for anyconstants a , c > b ∈ (0 , aca + c ≤ ab ∨ c (1 − b ) < a ∨ c . (B.3)Indeed, when b < c / ( a + c ), ab ∨ c (1 − b ) = c (1 − b ) and c (1 − b ) ∈ ( ac / ( a + c ) , c ), while when b ≥ c / ( a + c ), ab ∨ c (1 − b ) = ab and ab ∈ [ ac / ( a + c ) , a ). To show the claim in part (i), distinguish the cases below: Case I. α = α . Here, ρ = α = α and X d = X , so that a n / a n → ≤ n Pr (cid:32) X > a n x , X > a n x (cid:33) = n Pr (cid:34) R > max { a n x B , a n x (1 − B ) } (cid:35) ≤ n Pr (cid:32) R > a n a n x x a n x + a n x (cid:33) . (B.4)Because ( x x ) / { ( a n / a n ) x + x } → ( x x ) / ( x + x ) as n → ∞ ,lim n →∞ n Pr (cid:32) X > a n a n x x a n x + a n x (cid:33) = (cid:32) x x x + x (cid:33) − α . Furthermore, by Lemma 1, given that ( a n a n x x ) / ( a n x + a n x ) → ∞ as n → ∞ ,lim n →∞ Pr (cid:16) / R > a n a n x x a n x + a n x (cid:17) Pr (cid:16) / X > a n a n x x a n x + a n x (cid:17) = , so that the right-hand side in Eq. (B.4) tends to 0 as n → ∞ , and this implies the claim. Case II. α < α and ρ = α . Then for i = ,
2, there exists a slowly varying function L i such that a ni = n /α i L i ( n ).Hence a n / a n → x x ) / { x + x ( a n / a n ) } → x as n → ∞ . Consequently,lim n →∞ n Pr (cid:32) X > a n a n x x a n x + a n x (cid:33) = x − α . Moreover, by Lemma 1, given that ( a n a n x x ) / ( a n x + a n x ) → ∞ as n → ∞ ,lim n →∞ Pr (cid:16) / R > a n a n x x a n x + a n x (cid:17) Pr (cid:16) / X > a n a n x x a n x + a n x (cid:17) = , so that again the right-hand side in Eq. (B.4) tends to 0 as n → ∞ . Case III. α < α and ρ ∈ [ α , α ). In this case, 1 / X ∈ M ( Φ α ) and 1 / X ∈ M ( Φ ρ ). Therefore, either directlywhen ρ > α or by Lemma 1, one can easily deduce thatlim x →∞ Pr(1 / R > x )Pr(1 / X > x ) = . (B.5)At the same time, Breiman’s Lemma [5] implies thatlim x →∞ Pr(1 / R > x )Pr(1 / X > x ) = { / (1 − B ) ρ } = B( α , α )B( α , α − ρ ) . (B.6)4 L. R. B elzile and J. G. N e ˇ slehov ´ a Hence, for any b ∈ (0 , n Pr { / R > a n x (1 − b ) } as n → ∞ equalslim n →∞ n Pr { / X > a n x (1 − b ) } Pr { / R > a n x (1 − b ) } Pr { / X > a n x (1 − b ) } = { x (1 − b ) } − ρ B( α , α )B( α , α − ρ )so thatlim n →∞ (cid:90) n Pr { / R > a n x (1 − b ) } b α − (1 − b ) α − B( α , α ) d b = n Pr(1 / X > a n x ) = x − ρ = x − ρ (cid:90) b α − (1 − b ) α − ρ − B( α , α − ρ ) d b = (cid:90) lim n →∞ n Pr { / R > a n x (1 − b ) } b α − (1 − b ) α − B( α , α ) d b . (B.7)Given that for any b ∈ (0 , { / R > a n x b , / R > a n x (1 − b ) } ≤ Pr { / R > a n x (1 − b ) } , (cid:90) lim inf n →∞ (cid:0) n (cid:2) Pr { / R > a n x (1 − b ) } − Pr { / R > a n x b , / R > a n x (1 − b ) } (cid:3)(cid:1) b α − (1 − b ) α − B( α , α ) d b ≤ lim inf n →∞ (cid:90) n (cid:2) Pr { / R > a n x (1 − b ) } − Pr { / R > a n x b , / R > a n x (1 − b ) } (cid:3) b α − (1 − b ) α − B( α , α ) d b by Fatou’s Lemma. Because of Eq. (B.7), this inequality simplifies to x − ρ − (cid:90) lim sup n →∞ (cid:2) n Pr { / R > a n x b , / R > a n x (1 − b ) } (cid:3) b α − (1 − b ) α − B( α , α ) d b ≤ x − ρ − lim sup n →∞ (cid:90) n Pr { / R > a n x b , / R > a n x (1 − b ) } b α − (1 − b ) α − B( α , α ) d b and hence0 ≤ lim sup n →∞ (cid:8) n Pr(1 / X > a n x , / X > a n x ) (cid:9) ≤ (cid:90) lim sup n →∞ (cid:2) n Pr { / R > a n x b , / R > a n x (1 − b ) } (cid:3) b α − (1 − b ) α − B( α , α ) d b . To show the desired claim, it thus su ffi ces to show that for arbitrary b ∈ (0 , n →∞ n Pr { / R > a n x b , / R > a n x (1 − b ) } = . (B.8)To this end, fix b ∈ (0 ,
1) and observe that a n / a n → ∞ . Indeed, if ρ > α , this follows directly from the factthat a n = n /α L ( n ) and a n = n /ρ L ( n ) for some slowly varying functions L , L . When ρ = α , suppose thatlim inf n →∞ a n / a n were finite. Then there exists a subsequence a n k / a n k such that a n k / a n k → a as k → ∞ for some a ∈ [0 , ∞ ). Hence, for a fixed ε > k ≥ k , a n k / a n k ≤ a + ε . Using the latter observation and Eq. (B.6),lim k →∞ n k Pr(1 / R > a n k ) ≥ lim k →∞ n k Pr { / R > a n k ( a + ε ) } = lim k →∞ n k Pr { / X > a n k ( a + ε ) } Pr { / R > a n k ( a + ε ) } Pr { / X > a n k ( a + ε ) } = ( a + ε ) − ρ B( α , α )B( α , α − ρ ) > . At the same time, by Eq. (B.5),lim k →∞ n k Pr(1 / R > a n k ) = lim k →∞ n k Pr(1 / X > a n k ) Pr(1 / R > a n k )Pr(1 / X > a n k ) = n →∞ a n / a n = ∞ and hence a n / a n → ∞ as n → ∞ . Because a n b > a n (1 − b ) if and only if b > a n / ( a n + a n ) and a n / ( a n + a n ) → n → ∞ , there exists n such that for all n ≥ n , n Pr { / R > a n x b , / R > a n x (1 − b ) } = n Pr(1 / R > a n x b ) = n Pr(1 / X > a n x b ) Pr(1 / R > a n x b )Pr(1 / X > a n x b ) . xtremal attractors of Liouville copulas n → ∞ by Eq. (B.5) and hence Eq. (B.8) indeed holds.To prove part (ii), first recall that by Proposition 2 (b), 1 / X i ∈ M ( Φ α i ), i = ,
2, and hence the scaling sequences( a n ) and ( a n ) indeed exist. Recall that for i = , a ni = n /α i L i ( n ) for some slowly varying function L i . As inthe proof of part (i), n Pr(1 / X > a n x , / X > a n x ) can be bounded above by the right-hand side in Eq. (B.4).Markov’s inequality further implies that for β ∈ ( α , α + α ) such that E(1 / R β ) < ∞ , n Pr (cid:32) R > a n a n x x a n x + a n x (cid:33) ≤ n E (cid:16) / R β (cid:17) ( a n x + a n x ) β ( a n a n x x ) β = E (cid:16) / R β (cid:17) ( x x ) β (cid:40) x n /α − /β L ( n ) + x n /α − /β L ( n ) (cid:41) β The right-most expression tends to 0 as n → ∞ because for any i = , ρ > n ρ L i ( n ) → ∞ . Proof of Theorem 2.
First note that a positive random vector Y is in the maximum domain of attraction of a multivariateextreme-value distribution H with Fréchet margins if and only if there exist sequences of positive constants ( a ni ) ∈ (0 , ∞ ), i = , . . . , d , so that, for all y ∈ R d + ,lim n →∞ n { − Pr( Y ≤ a n y , . . . , Y d ≤ a nd y d ) } = lim n →∞ n d (cid:88) k = (cid:88) ≤ i < ··· < i k ≤ d ( − k + Pr( Y i > a ni y i , . . . , Y i k > a ni k y i k ) = − log H ( y ) . (B.9)This multivariate version of the Poisson approximation holds by the same argument as in the univariate case [12,Proposition 3.1.1].To prove part (a), suppose that 1 / R ∈ M ( Φ ρ ) for some ρ ∈ (0 , α M ]. By Proposition 2, one then has that forany i ∈ I , 1 / ( RD i ) ∈ M ( Φ α i ). For any i ∈ I , let ( a ni ) be a sequence of positive constants such that, for all x > n Pr { / ( RD i ) > a ni x } → x − α i as n → ∞ ; such a sequence exists by the univariate Poisson approximation [12,Proposition 3.1.1]. The same result also guarantees the existence of a sequence ( a n ) of positive constants such that,for all x > n Pr(1 / R > a n x ) → x − ρ as n → ∞ . Now set, for any i ∈ I , b i = E (cid:16) D − ρ i (cid:17) = Γ ( α i − ρ ) Γ ( ¯ α − α i ) Γ ( ¯ α − ρ ) × Γ ( ¯ α ) Γ ( α i ) Γ ( ¯ α − α i ) = Γ ( ¯ α ) / Γ ( ¯ α − ρ ) Γ ( α i ) / Γ ( α i − ρ ) , (B.10)and define, for any i ∈ I and n ∈ N , a ni = b /ρ i a n . As detailed in the proof of Proposition 2 (a), Breiman’s Lemmathen implies that, for all i ∈ I and x > n →∞ n Pr (cid:40) RD i > a ni x (cid:41) = lim n →∞ n Pr (cid:40) R > a n ( b /ρ i x ) (cid:41) Pr (cid:110) RD i > a n ( b /ρ i x ) (cid:111) Pr (cid:110) R > a n ( b /ρ i x ) (cid:111) = x − ρ b − i b i = x − ρ , given that for all i ∈ I , D i ∼ Beta ( α i , ¯ α − α i ).Next, fix an arbitrary x ∈ (0 , ∞ ) d , k ∈ { , . . . , d } and indices 1 ≤ i < · · · < i k ≤ d . To calculate the limit of n Pr(1 / ( RD i ) > a ni x i , . . . , / ( RD i k ) > a ni k x i k ), two cases must be distinguished: Case I. { i , . . . , i k } ∩ I (cid:44) ∅ . In this case, suppose, without loss of generality, that i ∈ I . Then0 ≤ n Pr (cid:32) RD i > a ni x i , . . . , RD i k > a ni k x i k (cid:33) ≤ n Pr (cid:32) RD i > a ni x i , RD i > a ni x i (cid:33) . Now either i ∈ I , in which case ρ ≥ α i ∨ α i , or i ∈ I , so that α i ≤ ρ < α i . Either way, Lemma 3 implies thatlim n →∞ n Pr (cid:32) RD i > a ni x i , RD i > a ni x i (cid:33) = n Pr { / ( RD i ) > a ni x i , . . . , / ( RD i k ) > a ni k x i k } → n → ∞ .6 L. R. B elzile and J. G. N e ˇ slehov ´ a Case II. { i , . . . , i k } ∩ I = ∅ . In this case, let Z i ,..., i k = max( x i ( b i ) /ρ D i , . . . , x i k ( b i k ) /ρ D i k ) and observe that forany ε > ρ + ε < min( α , . . . , α d ),E Z ρ + ε i ,..., i k ≤ x − ρ − ε i b − ( ρ + ε ) /ρ i E D ρ + ε i < ∞ . Therefore, by Breiman’s Lemma,lim n →∞ n Pr (cid:32) RD i > a ni x i , . . . , RD i k > a ni k x i k (cid:33) = lim n →∞ n Pr (cid:32) RZ i ,..., i k > a n (cid:33) = E (cid:16) Z − ρ i ,..., i k (cid:17) = E (cid:34)(cid:40) max ≤ j ≤ k (cid:0) x i j b /ρ i j D i j (cid:1)(cid:41) − ρ (cid:35) = E min ≤ j ≤ k (cid:16) x i j D i j (cid:17) − ρ b i j . Putting the above calculations together, one then has, for any x ∈ R d + ,lim n →∞ n (cid:40) − Pr (cid:32) RD ≤ a n x , . . . , RD d ≤ a nd x d (cid:33)(cid:41) = (cid:88) i ∈ I x − α i i + | I | (cid:88) k = (cid:88) { i ,..., i k }⊆ I i < ··· < i k ( − k + E min ≤ j ≤ k (cid:16) x i j D i j (cid:17) − ρ b i j Furthermore, one can readily establish by induction that for any t ∈ R d , | I | (cid:88) k = (cid:88) { i ,..., i k }⊆ I i < ··· < i k ( − k + min( t i , . . . , t i k ) = max i ∈ I ( t i ) . Hence, for any x ∈ R d + ,lim n →∞ n (cid:40) − Pr (cid:32) RD ≤ a n x , . . . , RD d ≤ a nd x d (cid:33)(cid:41) = (cid:88) i ∈ I x − α i i + E (cid:34) max i ∈ I (cid:40) ( x i D i ) − ρ b i (cid:41)(cid:35) . By the multivariate Poisson approximation (B.9), 1 / X ∈ M ( H ), where for all x ∈ R d + , H ( x ) = exp − (cid:88) i ∈ I x − α i i − E (cid:34) max i ∈ I (cid:32) ( x i D i ) − ρ b i (cid:41)(cid:35) . The univariate margins of H are given, for all i ∈ I , by F i ( x ) = x − α i and for all i ∈ I , F i ( x ) = exp( − x − ρ ). BySklar’s Theorem, the unique copula of H is given, for all u ∈ [0 , d , by (2), where for all x ∈ R d + , (cid:96) ( x ) = (cid:88) i ∈ I x i + E max i ∈ I x i D − ρ i b i . The first expression for (cid:96) follows immediately from Eq. (B.10). The second expression is readily verified usingLemma 2, given the fact that if B ∼ Beta( ¯ α , ¯ α − ¯ α ), E( B − ρ ) = Γ ( ¯ α − ρ ) Γ ( ¯ α ) / Γ ( ¯ α − ρ ) Γ ( ¯ α ).To prove part (b), recall that by Proposition 2 (b), 1 / X i ∈ M ( Φ α i ). Hence, there exist sequences of positiveconstants ( a ni ), i = , . . . , d , such that for all i = , . . . , d and all x > n Pr(1 / ( RD i ) > a ni x ) → x − α i as n → ∞ . ByLemma 2 (ii), it also follows that for arbitrary x ∈ (0 , ∞ ) d , k ∈ { , . . . , d } and indices 1 ≤ i < · · · < i k ≤ d ,0 ≤ lim n →∞ n Pr (cid:32) RD i > a ni x i , . . . , RD i k > a ni k x i k (cid:33) ≤ lim n →∞ n Pr (cid:32) RD i > a ni x i , RD i > a ni x i (cid:33) = . Thus, by Eq. (B.9), 1 / X is in the domain of attraction of the multivariate extreme-value distribution given, for all x ∈ R d + , by H ( x ) = exp( − x − α − · · · − x − α d d ), as was to be showed. xtremal attractors of Liouville copulas Appendix C. : Proofs from Section 4
Proof of Proposition 3.
In view of Corollary 1 and Theorem 2 in [27], it only remains to derive the explicit expressionfor (cid:96) nD . Because 1 − ψ (1 / · ) ∈ R − ρ , there exists a slowly varying function L such that for all x >
0, 1 − ψ (1 / x ) = x − ρ L ( x ).Given that the distribution function ψ (1 / · ) is in the domain of attraction of Φ ρ , the Poisson approximation implies thatthere exists a sequence ( a n ) of positive constants such that, for all x > n →∞ n (cid:2) − ψ { / ( a n x ) } (cid:3) = lim n →∞ n ( a n x ) − ρ L ( a n x ) = x − ρ . (C.1)Furthermore, by Equation (A6) in the proof of Theorem 2 (a) in [27], one has, for any j = , . . . , ¯ α − x →∞ ( − j x − j ψ ( j ) (1 / x ) κ j x − ρ L ( x ) = , (C.2)where κ j = ρ Γ ( j − ρ ) / Γ (1 − ρ ). Now for all i = , . . . , d , Eq. (7) yields, for any x > n Pr (cid:32) X i > a n x (cid:33) = n (cid:40) − ¯ H i (cid:32) a n x (cid:33)(cid:41) = n ( a n x ) − ρ L ( a n x ) − α i − (cid:88) j = ( − j ( a n x ) − j ψ ( j ) (1 / a n x ) j !( a n x ) − ρ L ( a n x ) . Given that a n → ∞ as n → ∞ , the last expression converges by Equations (C.1) and (C.2) as n → ∞ to x − ρ − α i − (cid:88) j = κ j j ! = x − ρ − ρ α i − (cid:88) j = Γ ( j − ρ ) Γ ( j + Γ (1 − ρ ) = x − ρ c ( α i , − ρ ) Γ (1 − ρ ) . The Poisson approximation thus implies that, as n → ∞ , for all i = , . . . , d and x > H ni (cid:32) a n x (cid:33) → exp (cid:40) − x − ρ c ( α i , − ρ ) Γ (1 − ρ ) (cid:41) . (C.3)For any x ∈ (0 , ∞ ) d , let 1 / ( a n x ) = { / ( a n x ) , . . . , / ( a n x d ) } and denote by ¯ x H the harmonic mean of x , viz. ¯ x H = d / (1 / x + · · · + / x d ). From Eq. (6) one then has n (cid:40) − ¯ H (cid:32) a n x (cid:33)(cid:41) = n (cid:18) a n ¯ x H d (cid:19) − ρ L (cid:18) a n ¯ x H d (cid:19) − (cid:88) ( j ,..., j d ) ∈ I α ( j ,..., j d ) (cid:44) d ( − j + ··· + j d (cid:16) a n ¯ x H d (cid:17) − j −···− j d ψ ( j + ··· + j d ) (cid:16) da n ¯ x H (cid:17) j ! · · · j d ! (cid:16) a n ¯ x H d (cid:17) − ρ L (cid:16) a n ¯ x H d (cid:17) d (cid:89) i = (cid:32) ¯ x H dx i (cid:33) j i By Eq. (C.2), the right most expression in the curly brackets converges, as n → ∞ , to1 − ρ (cid:88) ( j ,..., j d ) ∈ I α ( j ,..., j d ) (cid:44) d Γ ( j + · · · + j d − ρ ) Γ (1 − ρ ) j ! · · · j d ! d (cid:89) i = (cid:32) ¯ x H dx i (cid:33) j i = − ρ (cid:88) ( j ,..., j d ) ∈ I α ( j ,..., j d ) (cid:44) d Γ ( j + · · · + j d − ρ ) Γ (1 − ρ ) d (cid:89) i = Γ ( j i + (cid:32) / x i / x + · · · + / x d (cid:33) j i . Furthermore, Eq. (C.1) implies that, as n → ∞ , n (cid:18) a n ¯ x H d (cid:19) − ρ L (cid:18) a n ¯ x H d (cid:19) → (cid:32) x + · · · + x d (cid:33) ρ . Consequently, as n → ∞ , n { − ¯ H (1 / a n x ) } → − log H ( x ), where − log H ( x ) = (cid:32) x + · · · + x d (cid:33) ρ − ρ (cid:88) ( j ,..., j d ) ∈ I α ( j ,..., j d ) (cid:44) d Γ ( j + · · · + j d − ρ ) Γ (1 − ρ ) d (cid:89) i = Γ ( j i + / x i (cid:80) dj = x j j i . elzile and J. G. N e ˇ slehov ´ a By Eq. (B.9), 1 / X ∈ M ( H ). From Eq. (C.3), the univariate margins of H are scaled Fréchet, and Sklar’s theoremimplies that the unique copula of H is of the form (2) with stable tail dependence function as in Proposition 3. Proof of Proposition 4.
In view of Corollary 2 and Theorem 1 in [27], it only remains to compute the expression for (cid:96) pD given in part (a). Suppose that ψ ∈ R − ρ for some ρ >
0. This means that there exists a slowly varying functionsuch that for all x > ψ ( x ) = x − ρ L ( x ). Because ψ is itself a survival function, ψ ∈ M ( Φ ρ ) and by the univariatePoisson approximation, there exists a sequence ( a n ) of strictly positive constants such that, for all x > n →∞ n ψ ( a n x ) = x − ρ . (C.4)Furthermore, by Equation (A1) in the proof of Theorem 1 (a) in [27], one has, for any j = , . . . , ¯ α − x →∞ ( − j x j ψ ( j ) ( x ) ψ ( x ) = c ( j , ρ ) . (C.5)Now let X be the Dirichlet random vector with parameters α and radial part R whose Williamson ¯ α -transform is ψ .Denote the distribution function of X by H and its univariate margins by F i , i = , . . . , d . Then for all i = , . . . , d ,Equations (C.4) and (C.5) imply thatlim n →∞ n ¯ F i ( a n x ) = lim n →∞ n α i − (cid:88) j = ( − j ( a n x ) j ψ ( j ) ( a n x ) j ! = x − ρα i − (cid:88) j = Γ ( j + ρ ) Γ ( ρ ) Γ ( j + = x − ρ c ( α i , ρ ) Γ ( ρ +
1) (C.6)and hence, by the Poisson approximation, F ni ( x ) → exp {− x − ρ c ( α i , ρ ) / Γ ( ρ + } as n → ∞ .Next, for arbitrary k = , . . . , d and 1 ≤ i < · · · < i k ≤ d , let I ( α i ,...,α ik ) = { , . . . , α i − } × · · · × { , . . . , α i k − } .For any x ∈ (0 , ∞ ) d , Equations (6), (C.4) and (C.5) imply thatlim n →∞ n Pr( X i > x i , . . . , X i k > x i k ) = lim n →∞ n (cid:88) ( j ,..., j k ) ∈ I ( α i ,...,α ik ) ( − j + ··· + j k ψ ( j + ··· + j k ) { a n ( x i + · · · + x i k ) } j ! · · · j k ! k (cid:89) m = ( a n x i m ) j m = ( x i + · · · + x i k ) − ρ (cid:88) ( j ,..., j k ) ∈ I ( α i ,...,α ik ) Γ ( j + · · · + j k + ρ ) Γ ( ρ ) j ! · · · j k ! k (cid:89) m = (cid:32) x i m x i + · · · + x i k (cid:33) j m . Therefore, for any x ∈ (0 , ∞ ) d ,lim n →∞ n d (cid:88) k = (cid:88) ≤ i < ··· < i k ≤ d ( − k + Pr( X i > a n x i , . . . , X i k > a n x i k ) = − log H ( x ) , where − log H ( x ) = d (cid:88) k = (cid:88) ≤ i < ··· < i k ≤ d ( − k + ( x i + · · · + x i k ) − ρ (cid:88) ( j ,..., j k ) ∈ I ( α i ,...,α ik ) Γ ( j + · · · + j k + ρ ) Γ ( ρ ) j ! · · · j k ! k (cid:89) m = (cid:32) x i m x i + · · · + x i k (cid:33) j m . By Eq. (B.9), X ∈ M ( H ). As argued above, the univariate margins of H are given, for all i = , . . . , d and x >
0, byexp {− x − ρ c ( α i , ρ ) / Γ ( ρ + } . Sklar’s theorem thus implies that the unique copula of H is of the form (2) with stabletail dependence function indeed as given by the expression in part (a). Appendix D. : Proofs from Section 5
Proof of Proposition 5.
First, we show that for any ρ > − min( α , . . . , α d ), ρ (cid:44) (cid:34) max ≤ i ≤ d (cid:40) x i D ρ i c ( α i , ρ ) (cid:41)(cid:35) = Γ ( ¯ α ) | ρ | d − (cid:81) di = Γ ( α i ) (cid:90) S d max( x i t i ) d (cid:88) i = { c ( α i , ρ ) t i } /ρ − ρ − ¯ α d (cid:89) i = { c ( α i , ρ ) } α i /ρ ( t i ) α i /ρ − d t . (D.1) xtremal attractors of Liouville copulas D , . . . , D d ) d = Z / (cid:107) Z (cid:107) , where Z i ∼ Ga ( α i , i = , . . . , d are independent,E (cid:34) max ≤ i ≤ d (cid:40) x i D ρ i c ( α i , ρ ) (cid:41)(cid:35) = (cid:90) R d + max ≤ i ≤ d (cid:40) x i z ρ i c ( α i , ρ ) (cid:41) ( z + · · · + z d ) − ρ d (cid:89) i = e − z i z α i − i Γ ( α i ) d z . Make a change of variable t i = { z ρ i / c ( α i , ρ ) } / (cid:80) dj = z ρ j / c ( α j , ρ ) for i = , . . . , d − w = (cid:80) dj = z ρ j / c ( α j , ρ ). For ease ofnotation, set also t d = − (cid:80) d − i = t i . Then, for i = , . . . , d , z i = { c ( α i , ρ ) t i w } /ρ and the absolute value of the Jacobian is | J | = | ρ | d w d /ρ − d (cid:89) i = c ( α i , ρ ) /ρ t /ρ − i . Therefore,E (cid:34) max ≤ i ≤ d (cid:40) x i D ρ i c ( α i , ρ ) (cid:41)(cid:35) = | ρ | d (cid:81) di = Γ ( α i ) (cid:90) S d max ≤ i ≤ d ( x i t i ) d (cid:88) i = { c ( α i , ρ ) t i } /ρ − ρ d (cid:89) i = c ( α i , ρ ) α i /ρ t α i /ρ − i × (cid:90) ∞ w ¯ α/ρ − e − w /ρ (cid:80) di = { c ( α i ,ρ ) t i } /ρ d w d t . Eq. (D.1) now follows from the fact that (cid:90) ∞ w ¯ α/ρ − e − w /ρ (cid:80) di = { c ( α i ,ρ ) t i } /ρ d w = | ρ | Γ ( ¯ α ) d (cid:88) i = { c ( α i , ρ ) t i } /ρ − ¯ α . The expression for h D now follows directly from Eqs. (3) and (D.1), while the formulas for h pD and h nD obtain uponsetting ρ = ρ and ρ = − ρ , respectively. Appendix E. : Proofs from Section 7
Proof of Proposition 6.
For k = , . . . , d , the formula for the k th order mixed partial derivatives of (cid:96) D (1 / x ) canbe established from Eq. (12). Indeed, if V denotes a random vector with independent scaled Gamma components V i ∼ sGa { / c ( α i , ρ ) , /ρ, α i } , then the point process representation Eq. (12) implies that, for all x ∈ R d + , (cid:96) D (1 / x ) = (cid:90) ∞ Pr (cid:18) V i t > x i for at least one i ∈ { , . . . , d } (cid:19) d t = (cid:90) ∞ − d (cid:89) i = F (cid:40) x i t ; 1 c ( α i , ρ ) , ρ , α i (cid:41) d t . (18)For any k = , . . . , d , the expression on the right-hand side of Eq. (18) can be di ff erentiated with respect to x , . . . , x k under the integral sign. This gives the formulas for ∂(cid:96) D (1 / x ) /∂ x . . . ∂ x k . When k = d , Eq. (18) implies that ∂ d (cid:96) D (1 / x ) ∂ x · · · ∂ x d = − (cid:90) ∞ t d d (cid:89) i = f (cid:40) x i t ; 1 c ( α i , ρ ) , ρ , α i (cid:41) d t = ρ d d (cid:89) j = c ( α j , ρ ) { c ( α j , ρ ) x j } α j /ρ − Γ ( α j ) (cid:90) ∞ t α j /ρ exp − t /ρ d (cid:88) j = { c ( α j , ρ ) x j } /ρ d t = Γ ( ¯ α + ρ ) ρ d − (cid:104)(cid:80) dj = { c ( α j , ρ ) x j } /ρ (cid:105) ¯ α + ρ d (cid:89) j = c ( α j , ρ ) { c ( α j , ρ ) x j } α j /ρ − Γ ( α j ) , where the last equality follows upon making the change of variable u = (cid:80) dj = { c ( α j , ρ ) x j } /ρ t /ρ . Alternatively, Theo-rem 1 in [7] implies that that the d th order mixed partial derivative of (cid:96) D (1 / x ) equals − d (cid:107) x (cid:107) − d − h D ( x / (cid:107) x (cid:107) ; ρ, α ), whichindeed simplifies to − dh D ( x ; ρ, α ) given that h D ( x / (cid:107) x (cid:107) ; ρ, α ) = (cid:107) x (cid:107) d + h D ( x ; ρ, α ).Finally, the formulas for F ( x ; a , b , c ) follow immediately from the fact that the scaled Gamma distribution is alsothe distribution of the random variable aZ / b , where Z is Gamma with shape c and unit scaling.0 L. R. B elzile and J. G. N e ˇ slehov ´ a Derivation of the gradient score.
Straightforward calculations show that ∂ log dh D ( x ) ∂ x i = − ( ¯ α + ρ ) c ( α i , ρ ) /ρ x /ρ − i ρ (cid:80) dj = { c ( α j , ρ ) x j } /ρ + (cid:32) α i ρ − (cid:33) x i ∂ log dh D ( x ) ∂ x i ∂ x k = − ( ¯ α + ρ ) c ( α i , ρ ) /ρ x /ρ − i ρ (cid:80) dj = { c ( α j , ρ ) x j } /ρ (cid:32) ρ − (cid:33) I ik x i − c ( α k , ρ ) /ρ x /ρ − k ρ (cid:80) dj = { c ( α j , ρ ) x j } /ρ − (cid:32) α i ρ − (cid:33) I ik x ii