The Multivariate Extreme Value distributions have shown their usefulness in
environmental studies, financial and insurance mathematics. The Logistic or
Gumbel-Hougaard distribution is one of the oldest multivariate extreme value
models and it has been extended to asymmetric models. In this paper we
introduce generalized logistic multivariate distributions. Our tools are
mixtures of copulas and stable mixing variables, extending approaches in Tawn
(1990), Joe and Hu (1996) and Foug\`eres et al. (2009). The parametric family
of multivariate extreme value distributions considered presents a flexible
dependence structure and we compute for it the multivariate tail dependence
coefficients considered in Li (2009)

Anne-Laure Fougères

Bandeen-Roche

Cambanis

Capéraà

Christian Genest

Clayton

Coles

de Haan

Deheuvels

Einmahl

Farlie

Frank

Galambos

Genest

Ghoudi

Hüsler

Marshall

Nelsen

Oakes

Philippe Capéraà

Pickands

Plackett

Resnick

Sibuya

Sklar

Smith

Tawn

Yanagimoto

English

arXiv

summary:We construct two pairs $(\mathscr{A}^{[1]}_{F}, \mathscr{A}^{[2]}_{F})$ and $(\mathscr{A}^{[1]}_{\psi}, \mathscr{A}^{[2]}_{\psi})$ of ordered parametric families of symmetric dependence functions. The families of the first pair are indexed by regular distribution functions $F$, and those of the second pair by elements $\psi$ of a specific function family $\mathbb\psi$. We also show that all solutions of the differential equation $\frac{{\mathrm d}y}{{\mathrm d}u}=\frac{\alpha(u)}{u(1-u)}y$ for $\alpha$ in a certain function family ${\mathbb\alpha}_{\rm s}$ are symmetric dependence functions

Wysocki, Włodzimierz

Institute of Mathematics AS CR, v. v. i.

Constructing families of symmetric dependence functions

summary:Based on a recent representation of copulas invariant under univariate conditioning, a new class of copulas linked to a distortion of the identity function is introduced and studied

Mesiar, Radko

Pekárová, Monika

Czech digital mathematics library

DUCS copulas

AbstractA new class of bivariate distributions is introduced and studied, which encompasses Archimedean copulas and extreme value distributions as special cases. Its dependence structure is described, its maximum and minimum attractors are determined, and an algorithm is given for generating observations from any member of this class. It is also shown how it is possible to construct distributions in this family with a predetermined extreme value attractor. This construction is used to study via simulation the small-sample behavior of a bivariate threshold method suggested by H. Joe, R. L. Smith, and I. Weissman (1992, J. Roy. Statist. Soc. Ser. B54, 171–183) for estimating the joint distribution of extremes of two random variates

Capéraà, Philippe

Fougères, Anne-Laure

Genest, Christian

Elsevier - Publisher Connector 

Bivariate Distributions with Given Extreme Value Attractor 

summary:The Multivariate Extreme Value distributions have shown their usefulness in environmental studies, financial and insurance mathematics. The Logistic or Gumbel-Hougaard distribution is one of the oldest multivariate extreme value models and it has been extended to asymmetric models. In this paper we introduce generalized logistic multivariate distributions. Our tools are mixtures of copulas and stable mixing variables, extending approaches in Tawn [14], Joe and Hu [6] and Fougères et al. [3]. The parametric family of multivariate extreme value distributions considered presents a flexible dependence structure and we compute for it the multivariate tail dependence coefficients considered in Li [7]

Ferreira, Helena

Pereira, Luisa

Generalized logistic model and its orthant tail dependence

KybernetikaHelena Ferreira; Luisa PereiraGeneralized logistic model and its orthant tail dependenceKybernetika, Vol. 47 (2011), No. 5, 732--739Persistent URL: http://dml.cz/dmlcz/141688Terms of use:© Institute of Information Theory and Automation AS CR, 2011Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czKYB ERNET IK A — VO LUME 4 7 ( 2 0 1 1 ) , NUMBER 5 , PAGES 7 3 2 – 7 3 9GENERALIZED LOGISTIC MODELAND ITS ORTHANT TAIL DEPENDENCEHelena Ferreira and Luisa PereiraThe Multivariate Extreme Value distributions have shown their usefulness in environ-mental studies, financial and insurance mathematics. The Logistic or Gumbel–Hougaarddistribution is one of the oldest multivariate extreme value models and it has been extendedto asymmetric models. In this paper we introduce generalized logistic multivariate distribu-tions. Our tools are mixtures of copulas and stable mixing variables, extending approachesin Tawn [14], Joe and Hu [6] and Fougères et al. [3]. The parametric family of multivari-ate extreme value distributions considered presents a flexible dependence structure and wecompute for it the multivariate tail dependence coefficients considered in Li [7].Keywords: multivariate extreme value distribution, tail dependence, logistic model, mix-tureClassification: 60G701. INTRODUCTIONIn multivariate extreme value theory, the main interest has been in building mul-tivariate extreme value distributions. Joe [5] and McNeil et al. [9], among others,provide a source of multivariate dependence models turned into its copulas func-tions. Extreme value copulas arise naturally in the extreme value theory but theyare also itself a suitable choice to model dependence structures (Capéraà et al. [1]).Recently several authors proposed construction schemes of d-variate copulas andsome of them can be seen as transformations of given copulas, enlarging the numberof parameters and allowing tail asymmetries. The probabilistic representations of thetransformations, not always available, allows the understanding and interpretationof the dependence structure and can suggest a sampling strategy for the new copulas.In this paper we start from a q-variate random vector S which margins Sj arestandard positive α-stable variables and q d-variate random vectors Xj , j = 1, . . . , q,that conditionally on each Sj have dependence structure regulated by given copulasCj . We then study the componentwise maxima model from the Xj ’s. From thismodel we derive a new family of copulas and analyze its orthant tail dependence bycomputing the multivariate tail dependence coefficients considered in Li [7].In section 2 we introduce the model and the resulting copula as the main resultand discuss several special cases. It is shown how several copulas of the literatureGeneralized logistic model 733arise as special cases of the present construction. In section 3 we investigate howthe proposed construction affects the multivariate tail dependence. It allows taildependence and different dependence coefficients between each variable pair. Finallywe apply the results to the particular case of Cj being the copula arising from thedistribution of the variables in a M4 process (Smith and Weissman [13]).2. THE MODELLet L (Z|W ) denotes the conditional distribution of a random variable or vector Zgiven another random variable or vector W . For the vectors Xj = (Xj,1, . . . , Xj,d),j = 1, . . . , q, and S = (S1, . . . , Sq), defined on the same probability space, we shallassume that:(a) L ((X1, . . . ,Xq) |S) =∏qj=1 L (Xj |S),(b) L (Xj |S) = L (Xj |Sj),(c) P(⋂di=1 Xji ≤ xi|Sj)= Cj(e−“x1βj1”−1/αjSj, . . . , e−“xdβjd”−1/αjSj), xj > 0,j = 1, . . . , q, where Cj ’s are max-stable copulas and {βji, j = 1, . . . , q, i = 1, . . . , d}are non-negative constants such that∑qj=1 βji = 1, i = 1, . . . , d,(d) E(e−tSj)= e−tαj , t ≥ 0, j = 1, . . . , q, where αj ’s are constants in (0, 1]and(e) L (S) =∏qj=1 L (Sj).Thus every Xji is a scale mixture with mixing variable βjiSαjj and Xj , j =1, . . . , q, are conditionally independent given S.Scale mixtures have been studied and used in a variety of applications (Marshalland Olkin [8], Joe and Hu [6] and Fougères et al. [3], Li [7]).We shall consider here a componentwise maxima model from the Xj ’s and wepresent in the next result its distribution.Theorem 2.1. If the random vectors Xj , j = 1, . . . , q, and S satisfy the condi-tions (a) – (e) then Y = (Y1, . . . , Yd) defined by Yi =∨qj=1 Xji, i = 1, . . . , d, hasmultivariate extreme value distribution with unit Fréchet margins and copulaCY (u1, . . . , ud) (1)= exp−q∑j=1(− lnCj(e−(−βj1 ln u1)1/αj, . . . , e−(−βjd ln ud)1/αj))αj .P r o o f . To obtain CY we just apply the conditional independence of the Xj ’s734 H. FERREIRA AND L. PEREIRAfollowed by the max-stability of Cj ’s and the αj-stability of each Sj , as follows:P(d⋂i=1{Yi ≤ xi})=∫P q⋂j=1d⋂i=1{Xjieqxi} |S = sdS (s1, . . . , sq)=∫ q∏j=1Cj(e−“x1βj1”−1/αjsj, . . . , e−“xdβjd”−1/αjsj)dS (s1, . . . , sq)=q∏j=1exp{−(− lnCj(e−“x1βj1”−1/αj, . . . , e−“xdβjd”−1/αj))αj}.For each j and i, Xji is a positive αj-stable size mixture of a Fréchet distributionwith location βji, scale βjiαj and shape parameter αj and has itself Fréchet distri-bution with same location and the same right end point, but scale βji and shapeparameter 1. Since∑qj=1 βji = 1, i = 1, . . . , d, each Yi has unit Fréchet distribution.The max-stability of CY follows from its expression and the max-stability of theCj ’s. We now discuss some particular cases of (1) that has been explored.(I) If q = 1 then β1i = 1, i = 1, . . . d, andCY (u1, . . . , ud) = exp{−(− lnC1(e−(− ln u1)1/α1, . . . , e−(− ln ud)1/α1))α1}is a generalisation of the Archimedean copula (Joe [5]), which for the particular caseof the product copula C1 = Π leads to the Gumbel–Hougaard or logistic copula.The above copula is a particular case of the copula Cϕ considered in Morillas [10]with ϕ(x) = exp(−(− lnx)1/α1).The dependence properties of the special case ofC1(u1, . . . , ud) =∏1≤s<t≤dC{s,t}(upss , uptt )d∏i=1upiνii ,where C{s,t}, 1 ≤ s < t ≤ d, are bivariate copulas and (d − 1)pi + piνi = 1,i = 1, . . . , d, were analysed in Joe and Hu [6].(II) If Cj = Π, j = 1, . . . , q, thenq∏j=1exp{−(− lnCj(e−“x1βj1”−1/αj, . . . , e−“xdβjd”−1/αj))αj}= exp−q∑j=1(d∑i=1(xiβji)−1/αj)αj ,which leads to an asymmetric logistic copulaCY (u1, . . . , ud) = exp−q∑j=1(d∑i=1(−βji lnui)−1/αj)αj . (2)Generalized logistic model 735In (2), if we take αj = α, j = 1, . . . , q ≤ +∞, we find an analogous mixture of ex-treme value distributions to those considered in Fougeres et al. [3] by departing justfrom a random vector X = (X1, . . . , Xd) satisfying L (Xj |S) =∏di=1 L (Xi|S) andP (Xi ≤ x|S) = exp{−(∑qj=1 cjiSj)(1 + γi x−µiσi)−1/γi}, i = 1, . . . , d. In otherwords, in this different approach, conditionally on S, the vector X has independentmargins and each margin is a power mixture of an extreme value distribution withmixing variable∑qj=1 cjiSj , where the cji are non-negative constants.(III) Assume now, in (2), that each j corresponds to an element A of the set S,the class of all nonempty subsets of D = {1, . . . , d}. If βAi = 0 for each i 6∈ A thenthe copula (2) becomesCY (u1, . . . , ud) = exp{−∑A⊂S(d∑i∈A(−βAi lnui)−1/αA)αA}, (3)with∑A⊂S βAi = 1, i = 1, . . . , d. This is the asymmetric logistic model consideredin Tawn [14], by following a different probabilistic approach. More generally, byapplying the same interpretation of the constants βji in (1), we obtainCY (u1, . . . , ud) (4)= exp{∑A⊂S(− lnCA(e−(−βAi1(A) ln u1)1/αA, . . . , e−(−βAis(A) ln ud)1/αA))αA},where CA’s are copulas with different dimensions and we denote by (i1(A), . . . , is(A))the sub-vector of (1, . . . , d) corresponding to indices in A. In particular, if we beginwith one copula Cj = C, j = 1, . . . , q, then CA, A ⊂ S, are all the sub-copulas of C.(IV) Finally, let us suppose that βji = βj , i = 1, . . . , d, in (1). ThenCY (u1, . . . , ud) =q∏j=1exp{−(lnCj(e−(− ln u1)1/αj, . . . , e−(− ln ud)1/αj))αjβj}, (5)with∑qj=1 βj = 1, that is, CY is a geometric mean of mixtures of powers of multi-variate extreme value distributions. The particular case of the weighted geometricmean CY(u1, u2) = (u1 ∧ u2)β1(u1u2)1−β1 is due to Cuadras and Augé [2].3. ORTHANT TAIL DEPENDENCEFor a random vector Y = (Y1, . . . , Yd) with continuos margins F1, . . . , Fd and copulaC, the bivariate (upper) tail dependence coefficients are defined byλ(Y){s,t} ≡ λ(C){s,t} = limu↑1P(Fs (Ys) > u|Ft (Yt) > u), 1 ≤ s < t ≤ d. (6)The tail dependence is a copula based measure and it holdsλ(C){s,t} = 2− limu↑1lnC{s,t} (u, u)lnu, (7)736 H. FERREIRA AND L. PEREIRAwhere C{s,t} is the copula of the sub-vector (Ys, Yt) (Joe [5], Nelsen [11]).To characterise the relative strength of extremal dependence with respect to aparticular subset of random variables of Y one can use conditional orthant tail prob-abilities of Y given that the components with indices in the subset J are extreme.The tail dependence of bivariate copulas can be extended as done in Schmid andSchmidt [12] and Li [7].For ∅ 6= J ⊂ D = {1, . . . , d}, letλ(Y)J ≡ λ(C)J = limu↑1P⋂j /∈J{Fj (Yj) > u} |⋂j∈J{Fj (Yj) > u} . (8)If for some ∅ 6= J ⊂ {1, . . . , d} the coefficient λ(C)J exists and is positive then wesay that Y is (upper) orthant tail dependent.We have λ(C)J =λ(C){s}λ(CJ ){s}, if λ(CJ ){s} 6= 0 and the relation (7) between the tail depen-dence coefficient and the bivariate copula can also be generalized byλ(C)J = limu↑1∑∅6=A⊂D(−1)|A|−1 lnCA (uA)∑∅6=A⊂J(−1)|A|−1 lnCA (uA), (9)where CA denotes the sub-copula of C corresponding to margins with indices in Aand and uA the |A|-dimensional vector (u, . . . , u). By applying this relation and themax-stability of the copulas Cj , we get the following result.Theorem 3.1. For a copula C defined by (1), it holds(a)λ(C)J =q∑j=1∑∅6=A⊂D(−1)|A|−1(− lnCj,A(e−β1/αjj1 , . . . , e−β1/αjjd)A)αjq∑j=1∑∅6=A⊂J(−1)|A|−1(− lnCj,A(e−β1/αjj1 , . . . , e−β1/αjjd)A)αj , (10)where Cj,A denotes the sub-copula of Cj corresponding to the margins withindices in A.(b) If Cj = Π, for each j = 1, . . . , q, thenλ(C)J =q∑j=1∑∅6=A⊂D(−1)|A|−1(∑i∈Aβ1/αjji)αjq∑j=1∑∅6=A⊂J(−1)|A|−1(∑i∈Aβ1/αjji)αj . (11)Generalized logistic model 737The tail dependence result in (10) depends on the mixing variables through theparameters αj , even for the case of q = 1, that is the global dependence added bythe mixing variables doesn’t vanish in extremes of maxima. This contrast with theresult in Li [7], where the scale mixture of MEV distributions (RX1, . . . , RXd) isconsidered with the mixing variable R satisfying E(e−ctR)E(e−tR)→ c−α, as t tends to ∞,and c ≥ 1, α > 0. In this case the upper tail dependence coefficients are exactly thesame as the coefficients of the MEV distribution without mixing.We remark that, for βji = βj , i = 1, . . . , d, the numerator in (10) is, for eachA ⊂ D,λ(CA){s} =q∑j=1βjλ(Cj,A){s}that is, the tail dependence coefficient λ(CA){s} is a linear convex combination of the cor-responding tail dependence coefficients for the sub-copulas Cj,A of Cj , j = 1, . . . , q.The result in (11) leads toλ(C){s,t} = 2−q∑j=1(β1/αjjs + β1/αjjt)αj,extending the the known resultλ(C){s,t} = 2− 2α, (12)corresponding to q = 1 (Joe [5], Nelsen [11]). The result in (10) enables to extendthe equation (12) for other copulae C1 than the product copula asλ(C){s,t} = 2− (2− λ(C1){s,t})α. (13)The above results include a large number of possibilities for the tail dependence.We now illustrate the results with an example.Example 3.2. We will suppose that Cj = C, j = 1, . . . , d, withC(u1, . . . , ud) =∞∏l=1∞∏k=−∞(d∧i=1ualkii), uj ∈ [0, 1], j = 1, . . . , d,where {alkj , l ≥ 1, −∞ < k < ∞, 1 ≤ j ≤ d}, are nonnegative constants satisfying∞∑l=1∞∑k=−∞alkj = 1 for j = 1, . . . , d.That copula arises from the common distribution of the variables of an M4 process(Smith and Weissman [13]).738 H. FERREIRA AND L. PEREIRAThen the copula in (1) becomesCY (u1, . . . , ud) = exp−q∑j=1( ∞∑l=1∞∑k=−∞d∨i=1(−βjiaαjlki lnui)1/αj)αj . (14)By applying the result in Proposition 2.1. (a), we obtain for the numerator in (10)λ(CY){s} =q∑j=1∑∅6=A⊂D(−1)|A|−1( ∞∑l=1∞∑k=−∞∨i∈A(alkiβ1/αjji))αj.For the bivariate tail dependence it holdsλ(CY){s,t} = 2−∞∑l=1∞∑k=−∞q∑j=1(alksβ1/αjjs ∨ alktβ1/αjjt)αj,which, for the case q = 1 leads to the result λ(CY){s,t} = 2−∑∞l=1∑∞k=−∞ (alks ∨ alkt)in Heffernan et al. [4].ACKNOWLEDGEMENTThe authors gratefully acknowledge the referee for the suggestions and corrections. Thisresearch is partially supported by FCT/POCI2010/FEDER/CMUBI (project EVT) andFCT/PTDC/MAT108575/2008.(Received May 17, 2011)R E FER E NCE S[1] P. Capéraà, A. L. Fougères, and C. Genest: Bivariate distributions with given extremevalue attractor. J. Multivariate Anal. 72 (2000), 30–49.[2] C. M. Cuadras and J. Augé: A continuous general multivariate distribution and itsproperties. Comm. Statist. A - Theory Methods 10 (1981), 339–353.[3] A.-L. Fougères, J. P. Nolan, and H. Rootzén: Models for dependent extremes usingscale mixtures. Scand. J. Statist. 36 (2009), 42–59.[4] J. E. Heffernan, J. A. Tawn, and Z. Zhang: Asymptotically (in)dependent multivari-ate maxima of moving maxima processes. Extremes 10 (2007), 57–82.[5] H. Joe: Multivariate Models and Dependence Concepts. Chapman & Hall, London1997.[6] H. Joe and T. Hu: Multivariate distributions from mixtures of max-infinitely divisibledistributions. J. Multivariate Anal. 57 (1996), 240–265.[7] H. Li: Orthant tail dependence of multivariate extreme value distributions. J. Mul-tivariate Anal. 100 (2009), 243–256.[8] A. W. Marshall and I. Olkin: Families of multivariate distributions. J. Amer. Statist.Assoc. 83 (1988), 834–841.Generalized logistic model 739[9] A. J. McNeil, R. Frey, and P. Embrechts: Quantitative Risk Management: Concepts,Techniques and Tools. Princeton University Press, Princeton 2005.[10] P. M. Morillas: A method to obtain new copulas from a given one. Metrika 61 (2005),169–184.[11] R. B. Nelsen: An Introduction to Copulas. Springer, New York 1999.[12] F. Schmid and R. Schmidt: Multivariate conditional versions of Spearman’s rho andrelated measures of tail dependence. J. Multivariate Anal. 98 (2007), 1123–1140.[13] R. L. Smith and I. Weissman: Characterization and Estimation of the MultivariateExtremal Index. Technical Report, Univ. North Carolina 1996.[14] J. Tawn: Modelling multivariate extreme value distributions. Biometrika 77 (1990),2, 245–253.Helena Ferreira, Department of Mathematics, University of Beira Interior, 6200 Covilhã.Portugal.e-mail: helenaf@ubi.ptLuisa Pereira, Department of Mathematics, University of Beira Interior, 6200 Covilhã.Portugal.e-mail: lpereira@ubi.pt

summary:The aim of this paper is to open a new way of modelling non-exchangeable random variables with a class of Archimax copulas. We investigate a connection between powers of generators and dependence functions, and propose some construction methods for dependence functions. Application to different hydrological data is given

Bacigál, Tomáš

Jágr, Vladimír

Non-exchangeable random variables, Archimax copulas and their fitting to real data

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