aa r X i v : . [ ec on . E M ] S e p VECTOR COPULAS AND VECTOR SKLAR THEOREM
YANQIN FAN AND MARC HENRY
Abstract.
This paper introduces vector copulas and establishes a vector versionof Sklar’s theorem. The latter provides a theoretical justification for the use of vec-tor copulas to characterize nonlinear or rank dependence between a finite number ofrandom vectors (robust to within vector dependence), and to construct multivariatedistributions with any given non-overlapping multivariate marginals. We constructElliptical, Archimedean, and Kendall families of vector copulas and present algo-rithms to generate data from them. We introduce a concordance ordering for tworandom vectors with given within-dependence structures and generalize Spearman’srho to random vectors. Finally, we construct empirical vector copulas and show theirconsistency under mild conditions.
Keywords : Archimedean vector copulas; Elliptical vector copulas; Empirical vec-tor copulas; Kendall vector copulas; Spearman’s rho for random vectors; Vectorconcordance order; Vector quantiles and ranks.
JEL codes : C18; C46; C51.The first version is of July 16, 2020. This version is of September 15, 2020. The authors are gratefulto Guillaume Carlier, Paul Embrechts, Christian Genest, Haijun Li, L¨udger R¨uschendorf, and MarcoScarsini for helpful discussions. We thank Moyu Liao, Hyeonseok Park, and Xuetao Shi for excellentresearch assistance. Corresponding author: Yanqin Fan: [email protected]
Department of Economics,University of Washington, Box 353330, Seattle, WA 98195. Introduction
The cornerstone of copula theory, known as
Sklar’s Theorem , from Sklar (1959),states that (i) for any multivariate distribution function F on R K , with marginaldistribution functions F , ..., F K , there exists a copula function C such that F ( x , . . . , x K ) = C ( F ( x ) , . . . , F K ( x K )) , (1.1)and (ii) given any copula function C and any collection of univariate distributionfunctions F , . . . , F K , (1.1) defines a multivariate distribution function with copula C and the marginal distributions F , . . . , F K . When the marginal distributions are con-tinuous, C in part (i) of Sklar’s Theorem is the unique copula associated with F and it characterizes the dependence structure in F . Moreover, (ii) provides a generalapproach to constructing multivariate distributions from univariate ones.By virtue of Sklar’s Theorem, copulas can be used to characterize nonlinear orrank dependence between random variables, as distinct from marginal distributionalfeatures, to compute bounds on parameters of multivariate distributions in problemswith fixed marginals, and to construct parametric and semiparametric families ofmultivariate distributions from univariate ones. Applications to quantitative finance,particularly risk management, portfolio choice, derivative pricing, financial contagionand other areas, where precise measures of dependence are crucial, are well knownand extensively reviewed, see for instance Embrechts (2009) and references therein.Applications to economics, though fewer, have been no less expansive. First, thecharacterization of dependence, as distinct from marginal distributional features, hasbeen instrumental in modeling the propagation of shocks in contagion models inBaglioni and Cherubini (2013), in holding dependence fixed to measure partial distri-butional effects in Rothe (2012), in identifying private signal distributions in commonvalue auctions via stability of the copula in He (2017). Second, the copula approachto problems with fixed marginals has allowed the computation of sharp bounds onvarious relevant parameters in treatment effects models with randomized treatmentor treatment on observables, in Callaway and Li (2019), Fan and Manzanares (2017)and the many references therein. It has also been applied to problems of data com-bination, including ecological inference, see Ridder and Moffit (2007) and Fan et al. ECTOR COPULAS AND VECTOR SKLAR THEOREM 3 (2014) for instance. Third, the copula as a modelling and inference tool has been usedto discipline multiple latent variables and multiple dimensions of unobserved hetero-geneity. As such, the copula approach has been applied to sample selection models,in Smith (2003), to regime switching models in Fan and Wu (2010) and Chen et al.(2014), to simultaneous equations with binary outcomes in Han and Vytlacil (2017),as well as the modeling of earnings dynamics in Bonhomme and Robin (2009) andthe measure of intergenerational mobility in Chetty et al. (2014). Fourth, semipara-metric econometrics models including both time series and cross section models havebeen constructed using flexible parametric copulas to model contemporaneous depen-dence structures in multivariate models or time dependence in univariate time seriesmodels, see Chen and Fan (2006a; 2006b), Chen et al. (2006), Patton (2006), Beare(2010) for properties, estimation, and inference in such models. The list is surely notexhaustive.In all these applications, the need for a notion of copula that links multivariatemarginals arises naturally. In propagation models, Medovikov and Prokhorov (2017)highlight the need to distinguish within-group and between-group dependence. Mod-els of treatment effects with multivariate potential outcomes of interest fall in theclass of problems with fixed multivariate marginals. Censored and limited dependentvariables models with clustered latent variables call for hierarchical modeling, wherea copula operates on vectors of latent variables, each of which can also be modeledwith a traditional copula. In integrated risk management, modeling and measuringrisks of portfolios of several groups of risks will also benefit from a copula-like toolfor linking multivariate marginals, see Embrechts and Puccetti (2006).However, Sklar’s Theorem, as stated above, requires that all the marginals beunivariate. Indeed, Genest et al. (1995) shows that for two random vectors, if thefunction C : [0 , → [0 ,
1] is such that F ( x , x ) = C ( F ( x ) , F ( x )) definesa ( d + d )-dimensional distribution function with marginals F with support in R d and F with support in R d for all d and d such that d + d ≥
3, and for all distribu-tion functions F and F , then C ( u , u ) = u u . Hence, the only possible copulawhich works with non-overlapping multivariate marginals is the independence copula.Ressel (2019) generalizes this impossibility result to more than two random vectors. YANQIN FAN AND MARC HENRY
The objective of the present work is to circumvent this impossibility theorem. Thepaper develops a vector copula that generalizes the traditional copula to model andcharacterize nonlinear or rank dependence between a finite number of random vectorsof any finite dimensions. It relies on the combination of the theory of probability dis-tribution with given overlapping marginals, particularly Vorobev (1962) and Kellerer(1964), with the theory of optimal transport of probability distributions, particu-larly Rachev and R¨uschendorf (1990), Brenier (1991) and McCann (1995). First, weintroduce the concept of a vector copula and establish a vector version of Sklar’sTheorem using extensions of multivariate quantiles proposed in Galichon and Henry(2012), Ekeland et al. (2012) and Chernozhukov et al. (2017) as multivariate prob-ability transforms to remove marginal distributional features. Vector copulas andthe vector Sklar theorem allow the construction of distributions with any givennon overlapping multivariate marginals, thereby overcoming the weakness of tradi-tional copulas identified in Genest et al. (1995). Second, we show that vector cop-ulas are invariant to comonotonic transformations, where the multivariate notionof comonotonicity is borrowed from Galichon and Henry (2012) and Ekeland et al.(2012). Third, we construct flexible parametric families of vector copulas includingElliptical, Archimedean, and Kendall vector copulas and provide algorithms for sim-ulating from them. They reduce to the well-known Elliptical copulas, Archimedeancopulas, and copulas respectively when all the marginals are univariate. Using theVector Sklar Theorem, we construct new families of multivariate distributions withany fixed non-overlapping multivariate marginals and Elliptical, Archimedean, orKendall vector copulas. The meta-vector elliptical distributions extend the well-known meta-elliptical distributions including meta-Gaussian and meta-Student’s tdistributions in Fang et al. (2002). Fourth, we define comonotonic and countermono-tonic vector copulas extending Fr´echet extremal copulas. We define a concordanceordering for two random vectors with given within-dependence structures and gen-eralize Spearman’s rho to measure the strength of dependence between such randomvectors. Lastly, we introduce empirical vector copulas and show their consistencyunder regularity conditions.
Related literature.
Separate efforts have been carried out to develop dependencemeasures for random vectors robust to within-vector dependence on the one hand,
ECTOR COPULAS AND VECTOR SKLAR THEOREM 5 and to construct specific multivariate distributions with given multivariate marginaldistributions, on the other hand. For the former, Medovikov and Prokhorov (2017)propose a dependence measure between a finite number of random vectors that is ro-bust to within-vector dependence and apply it to the study of contagion in financialmarkets, inter alia. One potential drawback of the Medovikov and Prokhorov (2017)vector dependence measure is that it does not distinguish between negative and pos-itive dependence. Grothe et al. (2014) propose extensions of Spearman’s rho andKendall’s tau for two random vectors and show that they are invariant to increasingtransformations of each component of the random vector. As in the case of randomvariables, these global measures are insufficient to characterize the complete nonlin-ear dependence structure between random vectors for which analogues of copulas areneeded. An effort has been made in Li et al. (1996) to develop a copula-like approachfor several random vectors with given distributions . Specifically, Li et al. (1996) usethe Knothe-Rosenblatt transform (Rosenblatt (1952), Knothe (1957)) of F k for each k ≤ K and define a linkage function analogously to a copula function. Unlike copulas,no known flexible parametric families of linkage functions are available due to the useof the Knothe-Rosenblatt transform. Notation, conventions and preliminaries.
Let (Ω , A , F ) be some probabilityspace. Throughout, P denotes a class of probability distributions over R d —unlessotherwise specified, the class of all Borel probability measures on R d . Denote by S d := { x ∈ R d : k x k ≤ } the unit ball, and by S d − := { x ∈ R d : k x k = 1 } the unit sphere,in R d . Let P X stand for the distribution of the random vector X . The symbol ∂ denotes the subdifferential, ∇ the gradient and D the Jacobian. Following Villani(2003), we denote by g µ the image measure (or push-forward ) of a measure µ ∈ P bya measurable map g : R d → R d . Explicitly, for any Borel set A , g µ ( A ) := µ ( g − ( A )).Throughout the paper, we let U and Y be convex subsets of R d with non-emptyinteriors. A convex function ψ on U refers to a function ψ : U → R ∪ { + ∞} forwhich ψ ((1 − t ) x + tx ′ ) ≤ (1 − t ) ψ ( x ) + tψ ( x ′ ) for any ( x, x ′ ) such that ψ ( x ) and ψ ( x ′ )are finite and for any t ∈ (0 , In contrast, our notion of Spearman’s rho is invariant to comonotonic transformations of eachrandom vector. Related ideas are developed in R¨uschendorf (1984) and Section 1.6 of R¨uschendorf (2013).
YANQIN FAN AND MARC HENRY the convex set dom ψ := { x ∈ U : ψ ( x ) < ∞} , and differentiable Lebesgue-almosteverywhere in dom ψ . For any function ψ : U 7→ R ∪ { + ∞} , the conjugate ψ ∗ : Y 7→ R ∪ { + ∞} of ψ is defined for each y ∈ Y by ψ ∗ ( y ) := sup z ∈U [ y ⊤ z − ψ ( z )] . Theconjugate ψ ∗ of ψ is a convex lower-semi-continuous function on Y . We shall calla conjugate pair of potentials over ( U , Y ) any pair of lower-semi-continuous convexfunctions ( ψ, ψ ∗ ) that are conjugates of each other. The transpose of a matrix A isdenoted A ⊤ . Let ( d , . . . , d K ) be a finite collection of integers and for each k ≤ K ,let µ k be the uniform distribution on U k := [0 , d k . Let P denote a given distributionon R d × . . . × R d K with marginals P k on R d k , each k ≤ K . When stating generic resultsapplying to all k ≤ K such as vector quantiles and ranks, we omit the subscript k from d k , µ k , and P k unless stated otherwise.2. Vector Copulas and Vector Sklar Theorem
In order to capture nonlinear or rank dependence between random vectors, weextend copulas to vector copulas defined as multivariate distributions with uniformmultivariate marginals, see Definition 1 below. We relate a given distribution for ran-dom vectors to a vector copula that characterizes its between-vector dependence whileremoving the within-vector dependence and marginal information. This is achievedthrough our vector version of Sklar’s theorem, see Theorem 1 below. We also provideformulas to link distributions and their vector copulas, which allow the construc-tion of new distributions from given vector copula and non-overlapping multivariatemarginals.2.1.
Definition of Vector Copulas.
We start with the properties we require of avector copula and a useful example, namely Gaussian vector copulas, to illustrate theconcepts of vector quantiles and ranks introduced in Definition 2 and the vector Sklartheorem below.
Definition 1 (Vector Copulas) . (1) A vector copula C is defined as a joint dis-tribution function on [0 , d with uniform marginals µ k , k ≤ K , where d = d + . . . + d K . The associated probability distribution P C will also be referredto as vector copula, when there is no ambiguity. ECTOR COPULAS AND VECTOR SKLAR THEOREM 7 (2) The vector copula formally derived in Theorem 1 below from a distribution P with marginals P k on R d k , and vector quantiles T k (Definition 2), k ≤ K , willbe called a ( T , . . . , T K )- vector copula associated with P .Let C ( u , . . . , u K ) denote a vector copula as defined in Definition 1 (1), where u k = ( u k , ..., u kd k ) for k ≤ K . It is a mapping from [0 , d to [0 , C ( u , . . . , u K ) is increasing in each component of u k for all k ≤ K .(2) C (1 , . . . , , u k , , . . . ,
1) = µ k ( u k ) for all k ≤ K .(3) For all ( a , ..., a d ) , ( b , ..., b d ) ∈ [0 , d with a i ≤ b i for all i = 1 , ..., d , we have X i =1 · · · X i d =1 ( − i + ··· + i d C ( u i , · · · , u di d ) ≥ , where u j = a j and u j = b j for all j = 1 , ..., d .Any function satisfying (1)-(3) is a vector copula. Noting that (2) implies that eachunivariate marginal distribution of C ( u , . . . , u K ) is the uniform on [0 , d having thespecial feature that the K non-overlapping multivariate marginal distributions of C ( u , . . . , u K ) are µ k for k ≤ K .When d = . . . = d K = 1, the class of vector copulas is the class of copulas ofdimension K . Let ( X , . . . , X K ) ∼ F , a multivariate distribution F with continuousunivariate marginals F , . . . , F K . It follows from (1.1) that there exists a uniquecopula function of X or F . It is the distribution function of the probability integraltransforms U = F ( X ) , . . . , U K = F K ( X K ), i.e., C ( u , . . . , u K ) = Pr ( U ≤ u , . . . , U K ≤ u K )= F (cid:0) F − ( u ) , . . . , F − K ( u K ) (cid:1) , where F − ( u ) , . . . , F − K ( u K ) , denote the quantile functions of X , . . . , X K . Unique-ness results from the fact that the probability integral transform is the unique map For non-absolutely continuous marginals, one can use the distributional transform in R¨uschendorf(2009) to define a unique copula.
YANQIN FAN AND MARC HENRY pushing forward the marginals F , . . . , F K to the uniform on [0 , F is abso-lutely continuous with pdf f , the copula density function is given by c ( u , . . . , u K ) = f (cid:0) F − ( u ) , . . . , F − K ( u K ) (cid:1) K Y k =1 " f k (cid:0) F − k ( u k ) (cid:1) . (2.1)Below we illustrate the Definition 1 with the class of Gaussian vector copulas. Example 1 (Gaussian Vector Copulas) . (1) Let d , . . . , d K be a collection of in-tegers, and let Ω = I d Ω · · · Ω K Ω I d · · · Ω K ... ... . . . ...Ω K Ω K · · · I d K , (2.2)where Ω ij is a non-degenerate correlation matrix of dimension d i × d j for i, j = 1 , .., K and i = j . For u k ∈ [0 , d k , k ≤ K , let C Ga ( u , . . . , u K ; Ω) = Φ d ( ∇ ϕ ( u ) , . . . , ∇ ϕ K ( u K ) ; Ω) , (2.3)where d = d + ... + d K , Φ d ( · ; Ω) is the distribution function of the mul-tivariate normal with zero mean and variance covariance matrix Ω, and foreach k ≤ K , ∇ ϕ k ( x , . . . , x d k ) = (cid:0) Φ − ( x ) , . . . , Φ − ( x d k ) (cid:1) , where Φ is the dis-tribution function of the standard normal distribution. The map C Ga satisfiesproperties (1)-(3) above and is a vector copula by Definition 1 (1). Moreoverwhen d k = 1 for all k ≤ K , C Ga reduces to the traditional Gaussian copula.(2) We will demonstrate in the rest of this section that for suitably chosen maps T , . . . , T K , C Ga is a ( T , . . . , T K )-vector copula associated with a multivariatenormal distribution of dimension d = d + ... + d K and hence we call it aGaussian vector copula.2.2. Vector Quantiles and Vector Ranks.
We propose the use of multivariatetransformations to construct vector copulas associated with any given distribution P on R d × . . . × R d K with marginals P k on R d k , each k ≤ K . As for quantile maps in thecase of traditional copulas, the purpose of these transformation is to remove between-vector dependence and marginal information in the construction of vector copulasassociated with a distribution with given multivariate non overlapping marginals. ECTOR COPULAS AND VECTOR SKLAR THEOREM 9
The notion of multivariate transformations mapping each P k to µ k , k ≤ K , is basedon an analogue to the multivariate quantiles of Chernozhukov et al. (2017). The latterbuild on the following proposition (as stated in Chernozhukov et al. (2017)). Proposition 1 (Brenier-McCann’s Existence Result) . Let P and ν be two distribu-tions on R d . (1) If ν is absolutely continuous with respect to the Lebesgue measureon R d , with support contained in a convex set U , the following holds: there exists aconvex function ψ : U → R ∪ { + ∞} such that ∇ ψ ν = P . The function ∇ ψ existsand is unique, ν -almost everywhere. (2) If, in addition, P is absolutely continuouson R d with support contained in a convex set Y , the following holds: there exists aconvex function ψ ∗ : Y → R ∪ { + ∞} such that ∇ ψ ∗ P = ν . The function ∇ ψ ∗ exists, is unique and equal to ∇ ψ − , P -almost everywhere.As a corollary to Proposition 1, we can define vector quantiles and ranks. Definition 2 (Vector quantiles and ranks) . Let µ be the uniform distribution on [0 , d ,and let P be an arbitrary distribution on R d . Let ψ l , l ≤ L for some finite integer L ,be convex functions such that the following hold .(1) The map T := ∇ ψ L ◦ ∇ ψ L − ◦ ... ◦ ∇ ψ exists and satisfies T µ = P . Themap T is called vector quantile associated with P .(2) If P is absolutely continuous with support in a convex set V in R d , then themap T − := ∇ ψ ∗ ◦ ∇ ψ ∗ ◦ ... ◦ ∇ ψ ∗ L exists and satisfies T − P = µ . The map T − is called vector rank associated with P .The vector quantiles are the tools we use to map multivariate marginal distri-butions into multivariate uniform distribution to remove all within vector depen-dence and marginal information and concentrate on between vector dependence struc-tures. Existence of vector quantiles and ranks is guaranteed by Proposition 1. When L = 1, Definition 2 reduces to the µ -quantile notion of Galichon and Henry (2012)and Ekeland et al. (2012). However no known closed form expressions exist for µ -quantiles. By allowing L to be larger than 1 and the map T to be a composition ofgradients of convex functions ∇ ψ , ∇ ψ , ..., ∇ ψ L , we are able to derive vector quantiles Convex functions on R d are locally Lipschitz hence differentiable by Rademacher’s Theorem (seefor instance Villani (2009), Theorem 10.8(ii)) associated with general elliptical distributions and l -norm symmetric distributions inclosed form expressions and use them to construct elliptical and Archimedean vectorcopulas in Section 5.1 via the vector Sklar theorem 1. To illustrate, consider theGaussian example below. Example 2 (Gaussian Vector Quantiles and Ranks) . Let Φ d ( · ; Σ) denote the centeredmultivariate normal distribution on R d with a positive definite variance-covariancematrix Σ. No known expression exists for its µ -quantile. However, it is known that ∇ ψ ≡ Σ / is the optimal transport map between Φ d ( · ; I d ) and Φ d ( · ; Σ), where I d is the identity matrix of dimension d . Now for u = ( u , ..., u d ) ∈ (0 , d , let ∇ ψ ( u ) = (cid:0) Φ − ( u ) , . . . , Φ − ( u d ) (cid:1) and T = ∇ ψ ◦ ∇ ψ . By Definition 2, T is avector quantile associated with Φ d ( · ; Σ) and T − := ∇ ψ ∗ ◦ Σ − / is the correspondingvector rank, where ∇ ψ ∗ ( x ) = (Φ( x ) , . . . , Φ( x d )) for x = ( x , ..., x d ) ∈ R d .When d = 1, vector quantiles in Definition 2 reduce to the univariate quantilefunction for all L and all convex functions ψ l , l ≤ L .2.3. Vector Sklar Theorem.
For any multivariate distribution P on R d × . . . × R d K ,a vector copula associated with distribution P , derived formally in Theorem 1 below,is the joint distribution of the vector ranks of Y k , k ≤ K , where ( Y , . . . , Y K ) ∼ P . Theorem 1 (Vector Sklar Theorem) . For any joint distribution P on R d × . . . × R d K with marginals P k on R d k , and any vector quantile T k = ∇ ψ k,L ◦ ∇ ψ k,L − ◦ ... ◦ ∇ ψ k, associated with P k , each k ≤ K , there exists a vector copula C such that the followingproperties hold.(1) There exists a distribution on ( R d × . . . × R d K ) × ( U × . . . ×U K ) with margins P on R d × . . . × R d K , P C on U × . . . × U K , and ( Id , T k ) µ k on U k × R d k .(2) For any collection ( A , . . . , A K ) , where A k is a Borel subset of R d k , k ≤ K , P ( A × . . . × A K ) = P C ( T ∗ ( A ) × . . . × T ∗ K ( A K )) , (2.4) where T ∗ k = ∂ψ ∗ k, ◦ ∂ψ ∗ k, ◦ ... ◦ ∂ψ ∗ k,L for each k ≤ K .(3) If for each k ≤ K , P k is absolutely continuous on R d k with support in a convexset, then C is the unique vector copula, such that for all Borel sets B , . . . , B K , ECTOR COPULAS AND VECTOR SKLAR THEOREM 11 in U , . . . , U K , P C ( B × . . . × B K ) = P ( T ( B ) × . . . × T K ( B K )) . (2.5) (4) For any vector copula C defined in Definition 1 and any distributions P k on R d k with vector quantiles T k , each k ≤ K , (2.4) defines a distributionon R d × . . . × R d K with marginals P k , k ≤ K . When d = . . . = d K = 1, and A k = ( −∞ , x k ], each k ≤ K , Part (2) of Theorem 1reduces to (1.1). In other words, Theorem 1 reduces to Sklar’s theorem. As discussed,a vector copula associated with a multivariate distribution P with given marginals P k , k ≤ K , is not unique. It depends on the vector quantiles ( T , ..., T K ) used to removethe multivariate marginal information.The vector Sklar theorem plays the same role as Sklar’s Theorem for multivariatemarginals. First, it implies that the vector copula associated with P measures thebetween-dependence structure in P . To see this, let Y = ( Y , . . . , Y K ) be a randomvector with distribution P and let each Y k , k ≤ K follow the multivariate marginaldistribution P k . For each k ≤ K , let T k be a vector quantile associated with P k .Suppose that for each k ≤ K , P k is absolutely continuous on R d k with supportin a convex set. Then, from Definition 2, T − k P k = µ k for each k ≤ K . Sincethe reference measure µ k is an independence measure for each k ≤ K , the (classical)copula function of T − k ( Y k ) is the independence copula and hence the joint distributionof (cid:0) T − ( Y ) , . . . , T − K ( Y K ) (cid:1) , i.e., the vector copula associated with P , measures thebetween-dependence structure in P .Second, Part (3) of the vector Sklar theorem, or (2.5), provides a general approachto computing vector copulas of multivariate distributions. In fact, for absolutelycontinuous marginals P k with density f k , and smooth, invertible vector quantiles T k ,the Monge Amp`ere Equation (see Villani (2003), Chapter 4) gives for each k ≤ K ,det ( DT k ( u k )) = µ k ( u k ) f k ( T k ( u k )) for almost every u k ∈ [0 , d k , where µ k here denotes the density function of the uniform distribution on [0 , d k . Wetherefore obtain the following expression for the vector copula density c , in terms of the original density f : c ( u , . . . , u K ) = f ( T ( u ) , . . . , T K ( u K )) K Y k =1 det ( DT k ( u k ))= f ( T ( u ) , . . . , T K ( u K )) K Y k =1 (cid:20) µ k ( u k ) f k ( T k ( u k )) (cid:21) . (2.6)Expression (2.6) extends the copula density in (2.1) to multivariate marginals with T k replacing F − k in (2.1).Finally, Part (4) of the vector Sklar theorem provides a way of constructing distri-butions with given non-overlapping marginal distributions of any finite dimensions.Specifically, it states that for any distributions P k on R d k with vector quantiles T k ,each k ≤ K , A × . . . × A K P C ( T ∗ ( A ) × . . . × T ∗ K ( A K ))defines a distribution P on R d × . . . × R d K with marginals P k , k ≤ K , where C isany vector copula such as Gaussian vector copula in Example 1. When P C and P k for each k ≤ K are absolutely continuous and the vector quantiles T k are smooth andinvertible, the density function f associated with P is given by f ( y , . . . , y K ) = c (cid:0) T − ( y ) , . . . , T − K ( y K ) (cid:1) K Y k =1 f k ( y k ) , (2.7)which is a direct extension of the density decomposition of copula-based density func-tions in the univariate case. The above expresses the multivariate density function asthe product of the copula density function evaluated at the marginal ranks and thedensity function of K independent random vectors with marginals P , . . . , P K . Thiscan be used to construct both MLE and two-step estimators of vector copula-basedmodels in exactly the same way as copula-based models, see Fan and Patton (2014)and references therein. Remark 1.
The Knothe-Rosenblatt transform used in Li et al. (1996) to define link-age functions is not a vector quantile according to Definition 2. However, it has aunique inverse map. Inspection of the proof of the Vector Sklar Theorem below showsthat it holds for one-to-one maps as well.
ECTOR COPULAS AND VECTOR SKLAR THEOREM 13
Example 3 (Gaussian Vector Copulas (Cont’d)) . Let Φ d ( · ; Σ) denote the centeredmultivariate normal distribution on R d × . . . × R d K , where d = d + . . . + d K . It followsfrom Part (3) of the vector Sklar theorem and Gaussian vector quantiles and ranksin Example 2 applied to each of the K multivariate marginals of Φ d ( · ; Σ) that C Ga is the distribution function of (cid:0) T − ( Y ) , ..., T − K ( Y K ) (cid:1) , where T − k := ∇ ϕ ∗ k ◦ Σ − / k , Σ k is the variance-covariance matrix of Y k , and ∇ ϕ ∗ k ( x , . . . , x d k ) = (Φ( x ) , . . . , Φ( x d k ))for k ≤ K . Since (cid:16) Σ − / Y , ..., Σ − / K Y K (cid:17) ∼ Φ d ( · ; Ω), whereΩ = I d Σ − / Σ Σ − / · · · Σ − / Σ K Σ − / K Σ − / Σ Σ − / I d · · · Σ − / Σ K Σ − / K ... ... . . . ...Σ − / K Σ K Σ − / Σ − / K Σ K Σ − / · · · I d K , (2.8)we obtain that C Ga ( u , . . . , u K ; Ω) = Φ d ( ∇ ϕ ( u ) , . . . , ∇ ϕ K ( u K ) ; Ω) , (2.9)where ∇ ϕ k ( x , . . . , x d k ) = (cid:0) Φ − ( x ) , . . . , Φ − ( x d k ) (cid:1) . This is the Gaussian vector cop-ula presented in Example 1 in Section 2.1. For each k ≤ K , the (classical) copula ofΣ − / k Y k is the independence copula and the vector copula C Ga ( u , . . . , u K ; Ω) mea-sures the between-dependence structures in ( Y , . . . , Y K ).Alternatively, given T k ( u k ) = Σ / k ∇ ϕ k ( u k ), we can use (2.6) to derive the Gaussianvector copula density as c Ga ( u , . . . , u K ; Ω)= φ d (cid:16) Σ / ∇ ϕ ( u ) , . . . , Σ / K ∇ ϕ K ( u K ); Ω (cid:17) K Y k =1 µ k ( u k ) φ d k (cid:16) Σ / k ∇ ϕ k ( u k ); Σ k (cid:17) = φ d ( ∇ ϕ ( u ) , . . . , ∇ ϕ K ( u K ); Ω) K Y k =1 (cid:20) µ k ( u k ) φ d k ( ∇ ϕ k ( u k ); I d k ) (cid:21) . (2.10)4 YANQIN FAN AND MARC HENRY
Example 3 (Gaussian Vector Copulas (Cont’d)) . Let Φ d ( · ; Σ) denote the centeredmultivariate normal distribution on R d × . . . × R d K , where d = d + . . . + d K . It followsfrom Part (3) of the vector Sklar theorem and Gaussian vector quantiles and ranksin Example 2 applied to each of the K multivariate marginals of Φ d ( · ; Σ) that C Ga is the distribution function of (cid:0) T − ( Y ) , ..., T − K ( Y K ) (cid:1) , where T − k := ∇ ϕ ∗ k ◦ Σ − / k , Σ k is the variance-covariance matrix of Y k , and ∇ ϕ ∗ k ( x , . . . , x d k ) = (Φ( x ) , . . . , Φ( x d k ))for k ≤ K . Since (cid:16) Σ − / Y , ..., Σ − / K Y K (cid:17) ∼ Φ d ( · ; Ω), whereΩ = I d Σ − / Σ Σ − / · · · Σ − / Σ K Σ − / K Σ − / Σ Σ − / I d · · · Σ − / Σ K Σ − / K ... ... . . . ...Σ − / K Σ K Σ − / Σ − / K Σ K Σ − / · · · I d K , (2.8)we obtain that C Ga ( u , . . . , u K ; Ω) = Φ d ( ∇ ϕ ( u ) , . . . , ∇ ϕ K ( u K ) ; Ω) , (2.9)where ∇ ϕ k ( x , . . . , x d k ) = (cid:0) Φ − ( x ) , . . . , Φ − ( x d k ) (cid:1) . This is the Gaussian vector cop-ula presented in Example 1 in Section 2.1. For each k ≤ K , the (classical) copula ofΣ − / k Y k is the independence copula and the vector copula C Ga ( u , . . . , u K ; Ω) mea-sures the between-dependence structures in ( Y , . . . , Y K ).Alternatively, given T k ( u k ) = Σ / k ∇ ϕ k ( u k ), we can use (2.6) to derive the Gaussianvector copula density as c Ga ( u , . . . , u K ; Ω)= φ d (cid:16) Σ / ∇ ϕ ( u ) , . . . , Σ / K ∇ ϕ K ( u K ); Ω (cid:17) K Y k =1 µ k ( u k ) φ d k (cid:16) Σ / k ∇ ϕ k ( u k ); Σ k (cid:17) = φ d ( ∇ ϕ ( u ) , . . . , ∇ ϕ K ( u K ); Ω) K Y k =1 (cid:20) µ k ( u k ) φ d k ( ∇ ϕ k ( u k ); I d k ) (cid:21) . (2.10)4 YANQIN FAN AND MARC HENRY When d k = 1 for each k ≤ K , Ω reduces to the correlation matrix of ( Y , . . . , Y K ), C Ga ( u , . . . , u K ; Ω) = Φ d (cid:0) Φ − ( u ) , . . . , Φ − ( u K ) ; Ω (cid:1) , and c Ga ( u , . . . , u K ; Ω) = φ d (cid:0) Φ − ( u ) , . . . , Φ − ( u K ); Ω (cid:1) K Y k =1 (cid:20) φ (Φ − ( u k )) (cid:21) which are the (classical) Gaussian copula and its density. Example 4 (Meta-Vector Gaussian Distributions) . Consider a Gaussian vector cop-ula with density function c Ga ( u , . . . , u K ; Ω) defined in (2.10). For any set of marginalmeasures P k with density function f k and vector rank T − k for k ≤ K , (2.7) impliesthat f Ga given below f Ga ( y , . . . , y K ; Ω) = c Ga (cid:0) T − ( y ) , . . . , T − K ( y K ); Ω (cid:1) K Y k =1 " f k ( y k ) φ d k (cid:0) T − k ( y k ) (cid:1) (2.11)is the density function of a multivariate distribution with the Gaussian vector copulaand marginal distributions P k for k ≤ K . Thus we call it the density function ofthe meta-vector Gaussian distribution. When d k = 1 for all k ≤ K , the meta-vector Gaussian distribution reduces to the meta-Gaussian distribution, see Section5.1.3 in McNeil et al. (2005). Without parameterizing the marginal distributions, theabove expression results in a semiparametric multivariate distribution with the finitedimensional vector copula parameter Ω measuring the between vector dependenceand infinite dimensional marginal parameters f k , all k ≤ K .2.4. A Proof of the Vector Sklar Theorem.
For the proof of Theorem 1, we needthe following definition and result, due to Vorobev (1962) and Kellerer (1964).
Definition 3 (Decomposability) . A finite collection { S , . . . , S N } of subsets of a finiteset S is called decomposable if there exists a permutation σ of { , . . . , N } such that [ l Theorem 2 (Existence of probability measures with overlapping marginals) . Let X k :=( R , B ( R )) , for k = 1 , . . . , K . Let S := { S , . . . , S N } be an arbitrary collection of sub-sets of { , . . . , K } . For each j ≤ N , let P j be a probability measure on the productspace × k ∈ S j X i . Then there exists a probability measure on × k X k with marginal P j on × k ∈ S j X k , all j = 1 , . . . , N , if the following two conditions hold.(1) The marginals P j and P j coincide on × k ∈ S j ∩ S j X k , all j < j ≤ N .(2) The collection S is decomposable.Proof of Theorem 1. (1) For i = 1 , . . . , K X k =1 d k , define X i := ( R , B ( R )). Let S := ( , . . . , K X k =1 d k ) and for each k ≤ K , set d := 0, and define S k := k − X l =min { ,k − } d l + 1 , . . . , k X l =1 d l , K X l =1 d l + k − X l =1 d l + 1 . . . , K X l =1 d l + k X l =1 d l . We first show existence of a joint probability distribution π on × i X i withmarginals P on × i ∈ S X i and (Id , T k ) µ k on × i ∈ S k X i , each k ≤ K . For this,we only need to verify conditions (1) and (2) of Theorem 2 applied to thefamily S := { S , . . . , S N } , where N := 1 + K . Condition (1) is satisfied, sincethe marginal of (Id , T k ) µ k is T k µ k , which is equal to P k by definition of thevector quantile (Definition 2). There remains to show that the collection S isdecomposable. Take any integer m ∈ { , . . . , N } . We have [ l Much of the success of copulas as dependence functions is attributable to the factthat for strictly monotonic transformations of the random variables, they are eitherinvariant or change in predictable ways. We now show that with appropriate conceptsof multivariate monotonic transformations of random vectors, vector copulas inheritthese properties from copulas.3.1. Comonotonic Invariance. Two random variables X and Y are comonotonic ifthey are monotonic transformations of each other, or equivalently, if X = F − X ◦ F Y ( Y ),when X and Y have absolutely continuous distributions and cumulative distribu-tion functions F X and F Y respectively. Note that it is also equivalent to F X ( X ) = F Y ( Y ) = U , or X = F − X ( U ) and Y = F − Y ( U ), where U is a uniform random variableon [0 , F − X and F − Y by vector quantiles, we obtain acorresponding multivariate notion of vector comonotonicity. ECTOR COPULAS AND VECTOR SKLAR THEOREM 17 Definition 4 (Vector comonotonicity) . Let µ be the uniform distribution on [0 , d ,with d integer. Random vectors Y , . . . , Y J on R d are said to be ( T , ..., T J )- comonotonic if there exists a random vector U distributed according to µ such that Y j = T j ( U )almost surely, where T j is the vector quantile of Definition 2 associated with thedistribution of Y j , for each j ≤ J .The above definition of vector comonotonicity extends that of µ -comonotonicityin Galichon and Henry (2012) and Ekeland et al. (2012) to allow for compositions ofmaps T j for each j ≤ J . A related notion, namely c-comonotonicity, was proposed byPuccetti and Scarsini (2010). It is also based on optimal transport theory, but lackstransitivity and hence cannot be extended to more than two vectors.We now state properties of copulas that relate to vector comonotonicity. Theorem 3 (Comonotonic invariance) . Let ( d , . . . , d K ) be a finite collection of in-tegers. For each k ≤ K , let µ k be the uniform distribution on [0 , d k . Let randomvectors ( Y , . . . , Y K ) with distribution P and ( ˜ Y , . . . , ˜ Y K ) with distribution ˜ P be suchthat Y k and ˜ Y k are (cid:16) T k , ˜ T k (cid:17) -comonotonic for each k , where T k and ˜ T k denote the µ -quantiles associated with the distributions of Y k and ˜ Y k , respectively. Then, C is a ( T , ..., T K ) -vector copula associated with P if and only if it is a (cid:16) ˜ T , ..., ˜ T K (cid:17) -vectorcopula associated with ˜ P . When d k = 1 for all k ≤ K , comonotonic continuous random vectors are strictlyincreasing transformations of each other and Theorem 3 reduces to the well-knowninvariance property of copulas. When d k > 1, the comonotonic invariance property ofvector copulas depends critically on the vector quantiles. For example, let P be themultivariate normal distribution with the Gaussian vector copula C Ga ( u , . . . , u K ; Ω)defined in (2.3). Since C Ga ( u , . . . , u K ; Ω) is a ( T , ..., T K )-vector copula associatedwith P for T k ( u k ) = Σ / k ∇ ϕ k ( u k ) and ∇ ϕ k ( x , . . . , x d k ) = (cid:0) Φ − ( x ) , . . . , Φ − ( x d k ) (cid:1) , it is a (cid:16) ˜ T , ..., ˜ T K (cid:17) -vector copula associated with ˜ P if there exists a random vector U k distributed according to µ k such that Y k = Σ / k ∇ ϕ k ( U k ) and ˜ Y k = ˜ T k ( U k ) . Proof of Theorem 3. Let ( Y , . . . , Y K , U , . . . , U K ) follow a joint distribution on ( R d × . . . × R d K ) × ( U × . . . × U K ) as in Theorem 1(1). For each k ≤ K , it holds that Y k = T k ( U k ), µ k -almost surely. Since Y k and ˜ Y k are (cid:16) T k , ˜ T k (cid:17) -comonotonic, we also have˜ Y k = ˜ T k ( U k ), µ k -almost surely. Hence the joint distribution of ( ˜ Y , . . . , ˜ Y K , U , . . . , U K )satisfies the conditions that characterize a ( T , ..., T K )-vector copula associated withthe distribution ˜ P of ( ˜ Y , . . . , ˜ Y K ). (cid:3) Antitone Transformations. For two vectors, we can also entertain a notion ofcountermonotonicity as a multivariate extension of strictly decreasing transformationsof two random variables. Definition 5 (Vector Countermonotonicity) . Let µ be the uniform distribution on[0 , d , with d integer. Then random vectors Y , Y on R d are said to be ( T , T )- countermonotonic if there exists a random vector U distributed according to µ suchthat Y = T ( U ) and Y = T (1 d − U ) almost surely, where 1 d is the vector of onesand T j is the vector quantile of Definition 2 associated with the distribution of Y j , foreach j = 1 , Lemma 1. Let ( d , . . . , d K ) be a finite collection of integers. For each k ≤ K ,let µ k be the uniform distribution on R d k . Let random vectors ( Y , . . . , Y K ) withdistribution P and ( ˜ Y , . . . , ˜ Y K ) with distribution ˜ P be such that Y k and ˜ Y k are (cid:16) T k , ˜ T k (cid:17) -comonotonic for each k ≤ K and Y k and ˜ Y k are (cid:16) T k , ˜ T k (cid:17) -countermonotonicfor each k ∈ ( K , K ]. Then, the distribution of ( U , . . . , U K ) is a ( T , ..., T K )-vectorcopula associated with P if and only if the distribution of ( U , . . . , U K , d K − U K +1 , . . . , d K − U K ) is a (cid:16) ˜ T , ..., ˜ T K (cid:17) -vector copula associated with ˜ P . Proof of Lemma 1. Let the vector ( Y , . . . , Y K , U , . . . , U K ) follow a joint distributionon ( R d × . . . × R d K ) × ( U × . . . ×U K ) as in Theorem 1(1). For each k ≤ K , denote by T k and ˜ T k vector quantiles associated with the distributions of Y k and ˜ Y k , respectively.Then Y k = T k ( U k ), µ k -almost surely. Since Y k and ˜ Y k are (cid:16) T k , ˜ T k (cid:17) -comonotonic (resp.countermonotonic) for each k ≤ K (resp. k ∈ ( K , K ]), we have ˜ Y k = ˜ T k ( U k ) (resp. ECTOR COPULAS AND VECTOR SKLAR THEOREM 19 ˜ Y k = ˜ T k (1 d k − U k )) for each k ≤ K (resp. k ∈ ( K , K ]), µ k -almost surely. Hence thejoint distribution of ( U , . . . , U K , d K − U K +1 , . . . , d K − U K ) satisfies the conditionsthat characterize a (cid:16) ˜ T , ..., ˜ T K (cid:17) -vector copula associated with the distribution ˜ P of( ˜ Y , . . . , ˜ Y K ). (cid:3) Measures of Rank Dependence Vector copulas are useful tools to characterize measures of dependence betweenrandom vectors, that are invariant to the dependence within each of them.4.1. Extremal Copulas. A first step towards modeling dependence with copulas isto model extremes. We present copulas that characterize independence one on theone end, and maximal dependence on the other end. Definition 6 (Independence and Extremal Vector Copulas) . Let ( d , . . . , d K ) be afinite collection of integers. For each k ≤ K , let µ k be the uniform distributionon [0 , d k .(1) The independence vector copula has distribution µ ⊗· · ·⊗ µ K , i.e., the uniformdistribution on [0 , d , where d = d + . . . + d K .(2) Let d = . . . = d K = d . The ( T , ..., T K )-comonotonic vector copula isthe ( T , ..., T K )-vector copula with ( T , ..., T K )-comonotonic multivariate marginals.(3) Let K = 2, d = d = d . The ( T , T )-countermonotonic vector copula isthe ( T , T )-vector copula with ( T , T )-countermonotonic multivariate marginals.We will denote C I , C and C the independence, ( T , ..., T K )-comonotonic and ( T , T )-countermonotonic vector copulas respectively. As we show in the following lemma,they do not depend on the vector quantiles T , ..., T K . Lemma 2 (Comonotonic Vector Copula) . Let d be an integer, and µ be the uniformdistribution on [0 , d . Let U be any random vector with distribution µ . Then, theprobability distribution associated with the ( T , ..., T K )-comonotonic vector copula isthe distribution of ( U, . . . , U ). In addition, the probability distribution associatedwith the ( T , T )-countermonotonic vector copula is the distribution of ( U, d − U ). Proof of Lemma 2. Let U , . . . , U K be ( T , ..., T K )-comonotonic vectors. Then, by def-inition of ( T , ..., T K )-comonotonicity, in view of the fact that U has distribution µ , U k = T k ( U ), where T k is the vector quantile of U k for each k ∈ [2 , K ]. Since U k alsohas distribution µ , it follows that T k = Id, for each k ∈ [2 , K ]. Hence, U = · · · = U K .The conclusion on ( T , ..., T K )-comonotonic copulas follows. Let U and U be dis-tributed according to µ and be ( T , T )-countermonotonic. Then U = T (1 d − U ),where T is the vector quantile of U . Since U also has distribution µ , it fol-lows that T = Id. Hence, U = 1 d − U . The characterization of the ( T , T )-countermonotonic copulas follows. (cid:3) Hence, for any collection of Borel measurable subsets A k ⊂ [0 , d , k ≤ K, theprobability distribution associated with the comonotonic vector copula takes val-ues P C ( A × . . . × A K ) = µ ( ∩ k ≤ K A k ) and the countermonotonic vector copula takesvalues P C ( A × A ) = µ ( A ∩ (1 d − A )) . Letting A k = (0 , u k ] with u k ∈ [0 , d foreach k ≤ K , we obtain C ( u , . . . , u K ) = µ (cid:18) (0 , min k ≤ K u k ] (cid:19) = Π dj =1 (cid:18) min k ≤ K u jk (cid:19) . (4.1)When d = 1, C ( u , . . . , u K ) = min k ≤ K u k , the Fr´echet upper bound copula. How-ever C differs from the Fr´echet upper bound when d > 1. Since it is a copula, C ( u , . . . , u K ) ≤ min j ≤ d,k ≤ K u jk and strict inequality holds for some ( u , . . . , u K ) ∈ [0 , dK . Similarly, when d = 1, C ( u , u ) = Pr ( U ≤ u , − U ≤ u )= Pr (1 − u ≤ U ≤ u )= max ( u + u − , C differs from the Fr´echet lower boundwhen d > 1. In fact, C ( u , u ) is still a copula although the the Fr´echet lower boundis not a copula when d > ECTOR COPULAS AND VECTOR SKLAR THEOREM 21 It follows from the vector Sklar theorem that for any collection of distributions P k on R d k with vector quantiles T k , each k ≤ K , the expression below defines a distri-bution P on R d × . . . × R d K with marginals P k , k ≤ K : P ( A × . . . × A K ) = P C (cid:0) T − ( A ) × . . . × T − K ( A K ) (cid:1) . (4.2)The distribution P characterizes ( T , ..., T K )-comonotonic random vectors with ab-solutely continuous marginals P k , k ≤ K . To show this, first let ( Y , . . . , Y K ) ∼ P .Then (cid:0) T − ( Y ) , . . . , T − K ( Y K ) (cid:1) ∼ P C and hence there exists a random vector U ∼ µ such that T − k ( Y k ) = U for all k ≤ K . Second, let ( Y , . . . , Y K ) be comonotonic ran-dom vectors with absolutely continuous marginals P k , k ≤ K . By definition, thereexists a random vector U ∼ µ such that T − k ( Y k ) = U for all k ≤ K . HencePr ( Y ∈ A , . . . , Y K ∈ A k ) = Pr ( T ( U ) ∈ A , . . . , T K ( U ) ∈ A K )= Pr (cid:0) U ∈ T − ( A ) , . . . , U ∈ T − K ( A K ) (cid:1) = P C (cid:0) T − ( A ) × . . . × T − K ( A K ) (cid:1) = µ (cid:0) ∩ k ≤ K T − k ( A k ) (cid:1) . (4.3)Specifically for d = 1 , A k = ( −∞ , x k ] with x k ∈ R for each k ≤ K , we have F ( x , . . . , x K ) = µ (cid:18) min ≤ k ≤ K F k ( x k ) (cid:19) = min ≤ k ≤ K F k ( x k ) , which is the well-known upper Fr´echet bound distribution. However, it is well knownthat for d ≥ 2, the upper Fr´echet bound min ≤ k ≤ K F k ( x k ) is not a distribution functionexcept for very specific marginals F k ( x k ), k ≤ K , see Proposition 5.3 in R¨uschendorf(2013). In sharp contrast, the comonotonic vector copula C always defines a distri-bution function for any marginals through (4.3). Remark 2. The comonotonic vector copula maximizes the risk of the combinedportfolio Y + Y measured by Tr[Var C ( Y + Y )] for all random vectors with µ -vector copula P C ∈ M ( µ, µ ). Indeed, when K = 2, we have C = arg max P C ∈M ( µ,µ ) E C (cid:0) Y T Y (cid:1) = arg max P C ∈M ( µ,µ ) E C (cid:0) k Y + Y k (cid:1) = arg max P C ∈M ( µ,µ ) Tr [Var C ( Y + Y )] . Hence, Tr[Var C ( Y + Y )] ≤ Tr [Var C ( Y + Y )] for all P C ∈ M ( µ, µ ) . Moreover, take any pair ( Y , Y ) of random vectors, and any measure of risk R thatis convex and µ -comonotonic additive (see Definitions 3.1 and 3.10 of Ekeland et al.(2012)). Then R ( Y + Y ) is maximized when Y and Y are µ -comonotonic, or equiv-alently, when the µ -vector copula of ( Y , Y ) is the comonotonic vector copula.4.2. Vector Concordance Order and Vector Dependence Measure. Considertwo random vectors, i.e., K = 2. When d = d = 1, the Fr´echet upper bound copuladenoted as C + ( u , u ) = min ( u , u ) is the maximal concordance copula and gener-ates proper bivariate distributions with any univariate marginals. It is not very usefulfor multivariate marginals due to the fact that C + ( F ( y ) , F ( y )) is in general nota proper distribution for multivariate marginals F , F . Instead the comonotonic vec-tor copula C generates proper multivariate distributions from any given multivariatemarginals by the vector Sklar theorem. Below we extend the concept of concordanceorder to random vectors with identical multivariate marginals referred to as vectorconcordance order and show that the comonotonic vector copula C is the uniquemaximal vector concordance copula. Definition 7 (Vector Concordance Order) . Let X = ( X , X ) and Y = ( Y , Y ) betwo random vectors on R d × R d such that X ∼ Y and X ∼ Y . Let U = ( U , U )and V = ( V , V ) be distributed according to the ( T , T )-vector copulas associatedwith X and Y respectively. The vector Y is said to dominate X in the ( T , T )-vectorconcordance order, denoted X c Y , if E (cid:2) ∇ c ( U ) ⊤ ∇ c ( U ) (cid:3) ≤ E (cid:2) ∇ c ( V ) ⊤ ∇ c ( V ) (cid:3) (4.4) Our definition of vector concordance ordering departs from Definition A.1.2 in Joe (1990) for randomvectors in that we require vectors X and Y to have identical multivariate marginals. ECTOR COPULAS AND VECTOR SKLAR THEOREM 23 for any pair of differentiable convex functions c and c on [0 , d such that ∇ c ◦ ∇ c − is the gradient of some convex function .By construction, vector concordance is a property of the vector copula only. Itfollows from Definitions 4 and 7 that the ( T , T )-comonotonic vector copula C isthe unique maximal copula for the ( T , T )-vector concordance order in the sensethat X c Y for any X and Y in Definition 4 such that Y and Y are ( T , T )-comonotonic. Indeed, when Y and Y are ( T , T )-comonotonic, V = V . Hence,given that ∇ c ◦∇ c − is the gradient of a convex function, it follows from Theorem 4.7of Puccetti and Scarsini (2010) that V and ∇ c ◦ ∇ c − ( V ) = ∇ c ◦ ∇ c − ( V ) arec-comonotonic. Therefore, E (cid:2) ∇ c ( U ) ⊤ ∇ c ( U ) (cid:3) = E (cid:2) U ⊤ (cid:0) ∇ c ◦ ∇ c − ( U ) (cid:1)(cid:3) ≤ E (cid:2) V ⊤ (cid:0) ∇ c ◦ ∇ c − ( V ) (cid:1)(cid:3) = E (cid:2) ∇ c ( V ) ⊤ ∇ c ( V ) (cid:3) , by Theorem 4.3 of Puccetti and Scarsini (2010), as desired.Suppose that the marginal distributions of ( X , X ) are absolutely continuous, weobtain that U j = T − j ( X j ) and V j = T − j ( Y j ), for j = 1 , 2, where T − j is the vector rankcommon to X j and Y j , j = 1 , , since X and Y have identical marginals. Hence, theinequality in (4.4) becomes E (cid:2) ∇ c ( T − ( X )) ⊤ ∇ c ( T − ( X )) (cid:3) ≤ E (cid:2) ∇ c ( T − ( Y )) ⊤ ∇ c ( T − ( Y )) (cid:3) for any pair of differentiable convex functions c and c on [0 , d . When d = 1, theabove condition reduces to one of the equivalent definitions of concordance betweenbivariate random vectors, i.e., E [ h ( X ) h ( X )] ≤ E [ h ( Y ) h ( Y ))]for any pair of monotone increasing functions h and h on R , see Theorem 3.8.2 (iii)in M¨uller and Stoyan (2002). Boissard et al. (2015) call a family of one-to-one maps T j , j ∈ J such that T i ◦ T − j , each ( i, j ) ∈ J ,an admissible family of deformations . Examples of such families include location-scale families andradial distortions, by Proposition 4.1 in Boissard et al. (2015).4 YANQIN FAN AND MARC HENRY 2, where T − j is the vector rankcommon to X j and Y j , j = 1 , , since X and Y have identical marginals. Hence, theinequality in (4.4) becomes E (cid:2) ∇ c ( T − ( X )) ⊤ ∇ c ( T − ( X )) (cid:3) ≤ E (cid:2) ∇ c ( T − ( Y )) ⊤ ∇ c ( T − ( Y )) (cid:3) for any pair of differentiable convex functions c and c on [0 , d . When d = 1, theabove condition reduces to one of the equivalent definitions of concordance betweenbivariate random vectors, i.e., E [ h ( X ) h ( X )] ≤ E [ h ( Y ) h ( Y ))]for any pair of monotone increasing functions h and h on R , see Theorem 3.8.2 (iii)in M¨uller and Stoyan (2002). Boissard et al. (2015) call a family of one-to-one maps T j , j ∈ J such that T i ◦ T − j , each ( i, j ) ∈ J ,an admissible family of deformations . Examples of such families include location-scale families andradial distortions, by Proposition 4.1 in Boissard et al. (2015).4 YANQIN FAN AND MARC HENRY Now that we have a vector concordance ordering at hand, we list desirable prop-erties of measures of rank dependence between vectors, going back to R´enyi (1959)and Scarsini (1984):W: The dependence measure is a function of the vector copula only (hence invari-ant to comonotonic transformations of the multivariate marginals).S: The dependence measure is symmetric (invariant to permutation of the tworandom vectors).N: Dependence of the independence copula is zero, and dependence of the comono-tonic copula is 1.C: The measure of dependence is increasing relative to the vector concordanceorder of Definition 4.Spearman’s rho measures the linear correlation between ranks of the original ran-dom variables. This can be extended to rank dependence between random vectorswith the linear correlation between vector ranks from Definition 2. Let Y and Y betwo random vectors in R d , and let U := T − ( Y ) and U := T − ( Y ) be their vectorranks. We can extend Spearman’s rho with the linear correlation between the ranksusing the vector correlation measure proposed by Escouffier (1973), Grothe et al.(2014) and Puccetti (2019). Definition 8 (Spearman’s rho for vectors) . Let µ be the uniform distribution on [0 , d ,and let P and P be arbitrary distributions on R d . Let Y and Y be random vec-tors with distributions P and P , respectively. The ( T , T )-linear rank correlationcoefficient (Spearman’s rho) between Y and Y is defined as ρ ( T ,T ) := (cid:0) E C [ U T U ] − E I [ U T U ] (cid:1) − (cid:0) E C [ U T U ] − E I [ U T U ] (cid:1) , where E Q indicates expectation with respect to Q ∈ { C, I, C } , C is the ( T , T )-vectorcopula associated with the distribution of ( Y , Y ), and I and C are the independenceand comonotonic vector copulas, respectively.Spearman’s rho satisfies Properties W, S and N by construction. Since the innerproduct ( u , u ) u ⊤ u = ∇ c ( u ) ⊤ ∇ c ( u ) with c ( u ) = c ( u ) := u ⊤ u convex,Spearman’s rho is increasing with respect to the vector concordance ordering of Def-inition 4 (Property C). Moreover, by Lemma 3 below, Spearman’s rho lies in [ − , ECTOR COPULAS AND VECTOR SKLAR THEOREM 25 Lemma 3 (Extremal couplings) . Let µ be the Gaussian or uniform on [0 , d , andLet Y and Y be random vectors in R d . Spearman’s rho from Definition 8 is max-imized (resp. minimized) when Y and Y are ( T , T )-comonotonic (resp. counter-monotonic). Hence, ρ ∈ [ − , Proof of Lemma 3. Random vectors Y and Y are comonotonic if and only if theirvector ranks U and U are equal, hence trivially c-comonotonic. Therefore, by The-orem 4.3 of Puccetti and Scarsini (2010), they maximize E C [ U ⊤ U ] among all copu-las C . Random vectors Y and Y are countermonotonic if and only if their vectorranks U and U satisfy U = 1 d − U , hence U and − U are c-comonotonic. Theresult follows again by Theorem 4.3 of Puccetti and Scarsini (2010). (cid:3) When d = 1, the comonotonic vector copula C coincides with the Fr´echet upperbound copula and Definition 8 reproduces Spearman’s rho for random variables (seeNelsen (2006), Chapter 5). Since E I [ U T U ] = d E C [ U T U ] = d , we obtain ρ ( T ,T ) = 12 d (cid:18) E C [ U T U ] − d (cid:19) = 12 d E C [ U T U ] − , which reduces to Theorem 5.1.6 in Nelsen (2006) when d = 1 . Example 5. Let ( Y , Y ) follow the meta-vector Gaussian distribution with densityfunction f Ga ( y , y ; Ω) defined in (2.11), whereΩ = I d Σ − / Σ Σ − / Σ − / Σ Σ − / I d ! Then ρ G = 1 d (cid:18) Σ dj =1 (cid:20) E C [ U j U j ] − (cid:21)(cid:19) = 1 d (cid:18) Σ dj =1 π arcsin 12 ρ j (cid:19) , where ρ j is the j -th diagonal element of Σ − / Σ Σ − / for j = 1 , ..., d . Parametric Vector Copula Families This section introduces three general classes of vector copulas. Similar to Gaussianvector copulas constructed in the previous section, the first two classes of copulasare associated with elliptical distributions and l -norm symmetric or simplicially con-toured distributions. They are derived from the vector Sklar theorem and are calledelliptical vector copulas and Archimedean vector copulas respectively. The third classof vector copulas called Kendall vector copulas are hierarchical Kendall copulas withindependence copulas as cluster copulas, see Brechmann (2014) for the definition ofhierarchical Kendall copulas.5.1. Elliptical Vector Quantiles and Copulas. The vector Sklar theorem presentsa general approach to constructing vector copulas of a multivariate distribution P withabsolutely continuous marginals P k on R d k , k ≤ K . A critical step in this approachis to derive vector quantiles of the marginal distributions P k . Below we give closedform solutions for vector copulas associated with elliptical distributions. Definition 9 (Elliptical Distributions) . A (regular) elliptical distribution on R d isthe distribution of a random vector R Σ / U ( d ) , where R is a random variable, Σ is afull rank d × d scale matrix, U ( d ) is uniform on the unit sphere S d − , and R and U ( d ) are mutually independent.Examples of elliptical distributions include the following:(1) The spherical uniform distribution. Following Chernozhukov et al. (2017), wedefine the spherical uniform distribution as the distribution of a random vector e RU ( d ) , where e R is uniform on [0 , U ( d ) is uniform on the unit sphere S d − ,and e R and U ( d ) are mutually independent.(2) The centered multivariate Gaussian distribution N (0 , Σ), i.e., the distributionof a random vector R Σ / U ( d ) , where R follows a chi distribution with d degreesof freedom, Σ is a full rank d × d variance-covariance matrix, U ( d ) is uniformon the unit sphere S d − , and R and U ( d ) are mutually independent. ECTOR COPULAS AND VECTOR SKLAR THEOREM 27 (3) Student’s t distribution: the multivariate t ν, Σ with degrees of freedom ν andscale Σ is the distribution of a random vector R Σ / U ( d ) , where R /d is a ran-dom variable with an F ν,d distribution, U ( d ) is uniform on the unit sphere S d − ,and R and U ( d ) are mutually independent (see Ghaffari and Walker (2018)).The remainder of this section is devoted to the definition, characterization andclosed form expressions for vector copulas associated with elliptical distributions re-ferred to as elliptical vector copulas . Definition 10 (Elliptical Vector Copulas) . The ( T , ..., T K )-vector copula associatedwith an elliptical distribution P on R d × . . . × R d K is called elliptical vector copulaassociated with P . The vector Sklar theorem 1 guarantees that the elliptical copulaassociated with a specific elliptical distribution P exists and is uniquely defined.Closed form expressions for elliptical vector copulas rely on closed form expressionsfor the vector quantiles ( T , ..., T K ). To derive these quantiles, we restate some resultsfrom Cuesta-Albertos et al. (1993) in the following lemma. Lemma 4 (Cuesta-Albertos et al. (1993)) . Let A be a positive definite matrix. Define ϕ ( x ) = α ( k x k ) k x k x and ϕ A ( x ) = α ( k x k A ) k x k A Ax, where α is a non-decreasing function and k x k A = ( x ⊤ Ax ) / . Then ( X, AX ), ( X, ϕ ( X )),and ( X, ϕ A ( X )) are all optimal couplings with respect to quadratic transportationcost. Lemma 5 (Elliptical Vector Quantiles) . Let ν be the elliptical distribution of thevector R Σ / ν U ( d ) and let P be the elliptical distribution of the vector ˜ R Σ / U ( d ) ,where R and ˜ R are random variables with absolutely continuous distributions, Σ andΣ ν are full rank d × d scale matrices, U ( d ) is uniform on the unit sphere S d − , inde-pendent of R and of ˜ R . Let T = ∇ ψ ◦ ∇ ϕ , where ∇ ϕ ( x ) = (cid:0) Φ − ( x ) , . . . , Φ − ( x d ) (cid:1) in which Φ is the cumulative distribution function of a standard normal distribution,and ∇ ψ ( u ) = F − R ◦ F R (cid:16) Σ − / ν k u k (cid:17)(cid:13)(cid:13)(cid:13) Σ − / ν u (cid:13)(cid:13)(cid:13) Σ − / ν (cid:0) Σ / ν ΣΣ / ν (cid:1) / Σ − / ν u. (5.1)Then T is a vector quantile associated with P . Proof. Let A = Σ / ν and B = Σ / . Note that ∇ ψ ( u ) = α ( k A − u k ) k A − u k A − ( ABBA ) / A − u = A − ( ABBA ) / α ( k A − u k ) k A − u k A − u = T ◦ T ◦ T ◦ T ( u ) , where T ( u ) = A − u , T ( u ) = α ( k u k ) k u k u , T ( u ) = ( ABBA ) / u , and T ( u ) = A − u .By Lemma 4, T , T , T , T are each gradients of convex functions. Since ∇ ϕ is alsothe gradient of a convex function, we obtain the stated result. (cid:3) Lemma 5 and the vector Sklar theorem 1 are now used to characterize ellipticalvector copulas. Lemma 6 (Characterization of Elliptical Vector Copulas) . For each k ≤ K , let ν k bethe multivariate standard normal distribution with dimension d k , i.e., the distributionof R Σ / ν k U ( d k ) , where Σ ν k is the identity, R is a standard normal random variable,and U ( d k ) is uniform on the unit sphere S d k − , independent of R . The ( T , ..., T K )-vector copula associated with the elliptical distribution P of a vector ˜ R Σ / U ( d ) ischaracterized by P C E ( B × . . . × B K ) = P ( ∇ ψ ◦ ∇ ϕ ( B ) × . . . × ∇ ψ K ◦ ∇ ϕ K ( B K ))for all Borel subsets B k of [0 , d k , all k ≤ K , where for each k ≤ K, T k = ∇ ψ k ◦ ∇ ϕ k in which ∇ ψ k and ∇ ϕ k are given by ∇ ψ k ( u ) = F − R ◦ Φ ( k u k ) Σ / k u k u k and ∇ ϕ k ( x , . . . , x d k ) = (cid:0) Φ − ( x ) , . . . , Φ − ( x d k ) (cid:1) , where Σ k denotes the k -th diagonal block of Σ for all k ≤ K . Equivalently, for anyrandom vector ( Y , . . . , Y K ) with distribution P,P C E ( B × . . . × B K ) = Pr ( ∇ ϕ ∗ ◦ ∇ ψ ∗ ( Y ) ∈ B , . . . , ∇ ϕ ∗ K ◦ ∇ ψ ∗ K ( Y K ) ∈ B K ) . (5.2) ECTOR COPULAS AND VECTOR SKLAR THEOREM 29 Proof. Noting that P k is the distribution of ˜ R Σ / k U ( d k ) , k ≤ K , Lemma 6 followsfrom Theorem 1(2) and Lemma 5. (cid:3) The vector quantiles and ranks associated with elliptical distributions P k , whenthe references measures are also elliptical, can be derived from Lemma 5 and thecorresponding elliptical copulas can be obtained from Lemma 6 and Formula (5.2).We present several examples of elliptical vector copulas below. Example 6 (Gaussian vector copulas) . For any integer d , a Gaussian distribution on R d is the distribution of a randomvector e R Σ / U ( d ) , where e R ∼ χ [ d ] , U ( d ) is uniform on the unit sphere S d − , and e R and U ( d ) are mutually independent. Thus, the Gaussian vector copula presented inExample 1 in Section 2.1 can also be constructed from (5.2).We now provide an algorithm for simulation of Gaussian vector copula. It gener-alizes Algorithm 5.9 in McNeil et al. (2005) for the simulation of Gaussian copulas.Step 1. Compute Ω in (2.8) and perform a Cholesky decomposition of Ω to obtainthe Cholesky factor Ω / ;Step 2. Generate a d -dimensional vector Z of independent standard normal randomvariates and set Y = Ω / Z ;Step 3. Let U = ( ∇ ϕ ∗ ( Y ) , . . . , ∇ ϕ ∗ K ( Y K )). The random vector U follows distri-bution C Ga ( u , . . . , u K ; Ω). Example 7 (Student’s t vector copulas) . A zero mean Student t distribution with ν degrees of freedom and scale matrix Σon R d is characterized by Q Σ / U ( d ) , where Q /d follows an F distribution with ( d, ν )degrees of freedom. For k ≤ K , let T k = ∇ ψ k ◦ ∇ ϕ k , where ∇ ψ k ( u k ) = F − Q k ◦ F R k ( k u k k ) k u k k Σ / k u k and ∇ ϕ k ( u k ) = Φ − ( u k ) , in which Q k /d k has an F d k ,ν distribution and R k follows a X [ d k ] distribution. It followsfrom (5.2) that the ( T , ..., T K )-vector copula density associated with the centeredStudent t distribution with ν degrees of freedom and scale matrix Σ on R d , where d = d + . . . + d K , is given by c t ( u , . . . , u K ; Σ , ν ) = t d ( T ( u ) , . . . , T K ( u K ) ; Σ , ν ) K Y k =1 (cid:20) µ k ( u k ) t d k ( T k ( u k ) ; Σ k , ν ) (cid:21) , where t d ( · ; Σ , ν ) denotes the Student t density on R d with scale matrix Σ and degreeof freedom ν .When d k = 1 for each k ≤ K , it holds that Q k /d k = Q k ∼ F ,ν and R k ∼ X [1] .Then for Z ∼ N (0 , T k ( u k ) = ∇ ψ k (cid:0) Φ − ( u k ) (cid:1) = F − Q k ◦ F R k ( | Φ − ( u k ) | ) | Φ − ( u k ) | Φ − ( u k )= F − Q k ◦ Pr (cid:0) | Z | ≤ (cid:12)(cid:12) Φ − ( u k ) (cid:12)(cid:12)(cid:1) Φ − ( u k ) | Φ − ( u k ) | = F − Q k ◦ Pr (cid:0) − (cid:12)(cid:12) Φ − ( u k ) (cid:12)(cid:12) ≤ Z ≤ (cid:12)(cid:12) Φ − ( u k ) (cid:12)(cid:12)(cid:1) Φ − ( u k ) | Φ − ( u k ) | . Let S k follow the Student’s t distribution with ν degrees of freedom. By the relationbetween the Student’s t distribution and the F distribution, it holds that S k ∼ F ,ν .Since Q k ∼ F ,ν , Q k can be characterized as | S k | . Thus, we have that for q k ≥ F Q k ( q k ) = Pr (0 ≤ Q k ≤ q k ) = Pr (0 ≤ | S k | ≤ q k )= Pr ( − q k ≤ S k ≤ q k ) = 2 ( F S k ( q k ) − / , where F S k ( · ) is the distribution function of S k . For u k ≥ / 2, it holds that T k ( u k ) = F − Q k ◦ Pr (cid:0) − Φ − ( u k ) ≤ Z ≤ Φ − ( u k ) (cid:1) = F − Q k (2 ( u k − / F − S k ( u k ) , and for u k < / 2, it holds that T k ( u k ) = − F − Q k ◦ Pr (cid:0) Φ − ( u k ) ≤ Z ≤ − Φ − ( u k ) (cid:1) = − F − Q k (2 (1 / − u k )) = F − S k ( u k ) . Thus, T k ( u k ) = T − ( u k ; ν ) for T − ( · ; ν ) being the quantile function of the Student’s t distribution with ν degrees of freedom. Let t ( · ; ν ) denote the density function ofthe Student’s t distribution with ν degrees of freedom. The vector copula density is ECTOR COPULAS AND VECTOR SKLAR THEOREM 31 given by c t ( u , . . . , u K ; Σ , ν ) = t d (cid:0) T − ( u ; ν ) , . . . , T − ( u K ; ν ) ; Σ , ν (cid:1) K Y k =1 (cid:20) t ( T − ( u k ; ν ) ; ν ) (cid:21) , which is the same as the density of the classical t copula.We now provide an algorithm for simulation of Student’s t vector copula. It gen-eralizes Algorithm 5.10 in McNeil et al. (2005) for simulation of Student’s t copulas.Step 1. Generate Z ∼ N d (0 , Σ);Step 2. Generate a variable W ∼ Ig (cid:16) ν , ν (cid:17) independantly and let Y = √ W Z ;Step 3. Let U = ( T ∗ ( Y ) , . . . , T ∗ K ( Y K )), where for k ≤ K , T ∗ k = ∇ ϕ ∗ k ◦ ∇ ψ ∗ k . Therandom vector U follows distribution C t ( u , . . . , u K ; Σ , ν ). Example 8 (Spherical uniform vector copulas) . A Spherical uniform measure is thedistribution of a random vector e RU ( d ) , where e R is uniform on [0 , U ( d ) is uniformon the unit sphere S d − , and e R and U ( d ) are mutually independent. It follows from(2.6) that the spherical copula density is given by c S ( u , . . . , u K ) = f ( T ( u ) , . . . , T K ( u K )) K Y k =1 (cid:20) µ k ( u k ) f k ( T k ( u k )) (cid:21) , where T k ( u k ) = Φ ( k∇ ϕ k ( u k ) k ) k∇ ϕ k ( u k ) k ∇ ϕ k ( u k ) , where f is the density function of the spherical distribution on S d − , and f , . . . , f K are the K non-overlapping marginal density functions of f of dimensions d , . . . , d K respectively.5.2. Archimedean Vector Copulas. In this section, we construct vector copulasassociated with l -norm symmetric or simplicially contoured distributions via the vec-tor Sklar theorem. Because survival copulas of l -norm symmetric distributions areArchimedean, we refer to vector copulas associated with l -norm symmetric distribu-tions as Archimedean vector copulas . Definition 11 ( l -norm symmetric distributions) . An l -norm symmetric distribu-tion on R d + = [0 , ∞ ) d is the distribution of a random vector RU ( d ) , where R is a nonnegative random variable, U ( d ) is uniform on the unit simplex, and R and U ( d ) are mutually independent.The following push-forward maps for l -norm symmetric distributions will be usedin the application of the vector Sklar theorem to l -norm symmetric distributions. Lemma 7 (Archimedean Push-Forward) . Let ν be the l -norm symmetric distri-bution of the vector RU ( d ) and let P be the l -norm symmetric distribution of thevector ˜ RU ( d ) , where R and ˜ R are nonnegative random variables with absolutely con-tinuous distributions, U ( d ) is uniform on the unit simplex, independent of R and of ˜ R .Then ∇ ψ defined as ∇ ψ ( u ) = F − R ◦ F R ( k u k ) k u k u. (5.3)pushes forward ν to P , i.e. ∇ ψ ν = P . Proof. Let α ( r ) = F − R ( F R ( r )), X ∼ ν , and T ( X ) = α ( k X k ) k X k X. Then (cid:18) k X k , X ′ k X k (cid:19) ′ has the same distribution as ( R, U ′ X ) ′ , where R and U X areindependent. This implies that (cid:18) α ( k X k ) , X ′ k X k (cid:19) ′ has the same distribution as (cid:16) ˜ R, U ′ Y (cid:17) ′ . As a result, we obtain that T ( X ) has the same distribution as α ( k X k ) X k X k which has the same distribution as SU Y ∼ P . (cid:3) Lemma 8 (Characterization of Archimedean Vector Copulas) . For each k ≤ K ,let ν k be the multivariate distribution of a random vector with d k independent andidentically Exp (1) distributed components. Let T k = ∇ ψ k ◦ ∇ ϕ k for each k ≤ K ,where ∇ ψ k is given in the proof and ∇ ϕ k ( x , . . . , x d k ) = ( − ln(1 − x ) , . . . , − ln(1 − x d k )) . (1) The Archimedean vector copula associated with P , the l -norm symmetricdistribution of the vector ˜ RU ( d ) , with multivariate marginals P k , k ≤ K, is ECTOR COPULAS AND VECTOR SKLAR THEOREM 33 given by P C A ( B × . . . × B K ) = Pr (cid:0) T − ( Y ) ∈ B , . . . , T − K ( Y K ) ∈ B K (cid:1) , for any random vector ( Y , . . . , Y K ) with distribution P and for all Borel sub-sets B k of [0 , d k , all k ≤ K , where T − k = ∇ ϕ − k ◦ ∇ ψ − k with ∇ ψ − k ( u ) given inthe proof below and ∇ ϕ − k ( x , . . . , x d k ) = (1 − exp ( − x ) , ..., − exp ( − x d k )) . (2) When d k = 1 for all k ≤ K , it holds that T k = F Y k and the vector copula C A defined in part 1 is the Archimedean copula of P . Proof. (1) It follows from Fang and Fang (1988) that for each k ≤ K , the multi-variate marginal P k is the distribution of W k ˜ RU ( d k ) , where W k ∼ Beta ( d k , d − d k ) and is independent of ˜ R , U ( d k ) . Let ˜ R k = W k ˜ R . Noting that ν k is the l -norm symmetric distribution of R k U ( d k ) , where R k ∼ Γ ( d k , ∇ ψ k ( u ) = F − R k ◦ F R k ( k u k ) k u k u and ∇ ψ − k ( u ) = F − R k ◦ F e R k ( k u k ) k u k u are push-forward maps between P k and ν k and are unique inverses of eachother. The claimed results follow from Definition 1 and the distribution andquantile functions of Exp (1).(2) When d k = 1, R k ∼ Γ (1 , 1) so F R k ( x ) = 1 − exp( − x ) and ∇ ϕ − k ( x , . . . , x d k ) = ( F R ( x ) , ..., F R K ( x K )) . Since u ∈ R + , we obtain that∆ ψ − k ( u ) = F − R k ( F ˜ R k ( k u k )) k u k u = F − R k ( F ˜ R k ( | u | )) | u | u = F − R k ( F ˜ R k ( u )) u u = F − R k ( F ˜ R k ( u )) . By definition, T − k = ∇ ϕ − k ◦∇ ψ − k = F ˜ R k . Now observe that Y k = ˜ R k U ( d k ) ∈ R d k + and k U ( d k ) k = 1. This implies that when d k = 1, U ( d k ) = k U ( d k ) k = 1 and4 YANQIN FAN AND MARC HENRY 1) so F R k ( x ) = 1 − exp( − x ) and ∇ ϕ − k ( x , . . . , x d k ) = ( F R ( x ) , ..., F R K ( x K )) . Since u ∈ R + , we obtain that∆ ψ − k ( u ) = F − R k ( F ˜ R k ( k u k )) k u k u = F − R k ( F ˜ R k ( | u | )) | u | u = F − R k ( F ˜ R k ( u )) u u = F − R k ( F ˜ R k ( u )) . By definition, T − k = ∇ ϕ − k ◦∇ ψ − k = F ˜ R k . Now observe that Y k = ˜ R k U ( d k ) ∈ R d k + and k U ( d k ) k = 1. This implies that when d k = 1, U ( d k ) = k U ( d k ) k = 1 and4 YANQIN FAN AND MARC HENRY Y k = ˜ R k . We can conclude that when d k = 1, T − k = F Y k and T −− k = F − Y k .Taking B k = ( x k , ∞ ) for each k ≤ K , we obtain Archimedean copula of P . (cid:3) Example 9 (Clayton Vector Copulas) . Let P denote the probability measure as-sociated with the l -norm symmetric distribution of the vector ˜ RU ( d ) , where ˜ R ∼ θdF (2 d, /θ ) for θ > 0. It is known that the distribution P has Archimedean sur-vival copula generated by ψ θ ( s ) = (1 + θs ) − /θ and univariate marginal distributiongiven by 1 − ψ θ ( s ), where the copula is the Clayton copula of the form: C θ ( u , ..., u d ) = d X j =1 u − θj − d + 1 ! − /θ . Let d = d + ... + d K . Then Lemma 5 implies that the vector copula of P satisfies P C Cl ( B × . . . × B K ) = Pr (cid:0) ∇ ϕ − ◦ ∇ ψ − ( Y ) ∈ B , . . . , ∇ ϕ − K ◦ ∇ ψ − K ( Y K ) ∈ B K (cid:1) , for all Borel subsets B k of [0 , d k , all k ≤ K , where for each k ≤ K, ∇ ϕ − k ( x , . . . , x d k ) = (1 − exp ( − x ) , ..., − exp ( − x d k )) and∆ ψ − k ( u ) = F − R k ( F ˜ R k ( k u k )) k u k u in which ˜ R k = W k ˜ R , W k ∼ Beta ( d k , d − d k ) and is independent of ˜ R , and R k ∼ Γ ( d k , c Cl ( u , . . . , u K ; θ ) = f ( T ( u ) , . . . , T K ( u K ) ; θ ) K Y k =1 (cid:20) µ k ( u k ) f k ( T k ( u k ) ; θ ) (cid:21) , where f ( y , ..., y K ; θ ) and f k ( y k ; θ ) are the density functions of F ( y , ..., y K ; θ ) and F k ( y k ; θ ) respectively, F ( y , ..., y K ; θ ) = C θ (cid:0) − ψ θ ( y ) , ..., − ψ θ ( y d ) , ..., − ψ θ ( y K ) , ..., − ψ θ ( y Kd K ) (cid:1) , F k ( y k ; θ ) = C θ (cid:0) − ψ θ ( y k ) , ..., − ψ θ ( y kd k ) (cid:1) for each k ≤ K. Below is an algorithm to simulate a Clayton vector copula with parameter θ > ECTOR COPULAS AND VECTOR SKLAR THEOREM 35 Step 1. Generate a random vector U ( d ) uniformly distributed on the d -dimensionalunit simplex as follows. Generate a vector of i.i.d. standard exponential variates Z = ( Z , ..., Z d ) ′ and let U ( d ) = ( Z / k Z k , ..., Z d / k Z k ) ′ .Step 2. Generate a univariate random variable R and let Y = RU ( d ) .Step 3. Let U = ( T ∗ ( Y ) , . . . , T ∗ K ( Y K )), where for k ≤ K , T ∗ k = ∇ ϕ ∗ k ◦ ∇ ψ ∗ k . Therandom vector U follows the Clayton vector copula distribution.5.3. Kendall Vector Copulas. Brechmann (2014) studies properties and inferenceof the family of hierarchical Kendall copulas introduced for a large number of randomvariables. A subclass of hierarchical Kendall copulas, namely, those with indepen-dence cluster copulas, turns out to be vector copulas by Definition 1 (1). We thuscall them Kendall vector copulas. To avoid burdening the readers with unnecessarynotations, we refer interested readers to Brechmann (2014) for the definition of hi-erarchical Kendall copulas. Below we give a definition and two characterizations ofKendall vector copulas. Definition 12 (Kendall Vector Copulas) . Let U be a random vector of dimension d distributed as C KV ( · ) with marginals µ k for k ≤ K , where d = d + ... + d K . Wecall C KV a Kendall vector copula with nesting copula C : [0 , K → [0 , 1] if it is ahierarchical Kendall copula with nesting copula C and independence cluster copulas C k ( u ) = µ k ( u ) for k ≤ K .(1) For k ≤ K , let V k = K k (cid:16) Π d k j =1 U kj (cid:17) , where K k ( u ) = u d k − X i =0 log i (1 /u ) i !is the Kendall distribution of the independence copula C k . Suppose ( V , ..., V K )is distributed according to C ( v , ..., v K ). Then C KV is the Kendall vectorcopula with nesting copula C and cluster copulas C k for k ≤ K .(2) C KV ( · ) is the distribution function of − (cid:16) ln (cid:16) R U ( d )1 (cid:17) , ..., ln (cid:16) R K U ( d K ) K (cid:17)(cid:17) ′ , where R k = exp (cid:0) − K − k ( V k ) (cid:1) , U ( d k ) k is independent of R k and is uniformlydistributed on the unit simplex on R d k for k ≤ K . Proof. By Remark 5 in Brechmann (2014), under parts (1) or (2), C KV is the hierar-chical Kendall copula with nesting copula C and cluster copulas C k for k ≤ K andis thus a Kendall vector copula by Definition 12. (cid:3) The nesting copula C characterizes the between-vector dependence and can be any K -dimensional copula such as Archimedean or elliptical copula. Below we present analgorithm to simulate from a Kendall vector copula with nesting copula C . It isbased on Algorithms 14 and 20 in Brechmann (2014). Step 1. Generate a random sample from ( V , ..., V K ) from C . Step 2. Let Z k = K − k ( V k ) for k ≤ K .Step 3. Generate U ( d k ) k from the uniform distribution on the unit simplex on R d k for k ≤ K .Step 4. Let U = ( U , ..., U K ) , where U k = ( U k , . . . , U kd k ), where U kj = − ln (cid:16) U ( d k ) kj exp ( − Z k ) (cid:17) for j = 1 , ..., d k and k ≤ K . The random vector U follows the Kendall vector copulawith nesting copula C . Multivariate Distributions With Given Multivariate Marginals. Ellipti-cal vector copulas in Lemma 6, Archimedean vector copulas in Lemma 8, and Kendallvector copulas in Example 2 in Section 3.3 facilitate the construction of multivari-ate distributions with given non-overlapping multivariate marginals via the vectorSklar theorem. Indeed, Theorem 1 implies that given an elliptical or Archimedean orKendall vector copula C , and given a set of multivariate marginal distributions P k on R d k with associated push-forward maps T k as in Theorem 1, the distribution P defined for Borel sets A , . . . , A K , by P ( A × . . . × A K ) = P C (cid:0) T − ( A ) × . . . × T − K ( A K ) (cid:1) is a multivariate distribution with the elliptical/Archimedean/Kendall vector cop-ula C and non-overlapping marginals P k . Furthermore, for absolutely continuousmarginals P k with density functions f k , k ≤ K , we can use the expression in (2.6) ECTOR COPULAS AND VECTOR SKLAR THEOREM 37 to compute the density function associated with P as in Example 4 for the meta-vector Gaussian distributions. Below we present density functions of the meta-vector t distributions and the meta-vector distributions with Clayton vector copulas. Example 10 (Distributions with Student’s t vector copula) . Let c t ( u , . . . , u K ; Σ , ν )denote the density function of a Student t vector copula. Then f t ( y , . . . , y K ) = c t (cid:0) T − ( y ) , . . . , T − K ( y K ); Σ , ν (cid:1) K Y k =1 " f k ( y k ) φ d k (cid:0) T − k ( y k ) (cid:1) is the density function of the meta-vector t distribution with the Student’s t vectorcopula and marginal measures P k , where T − k is a vector rank of P k for k ≤ K .Without parameterizing the marginal distributions, the above expression results ina semiparametric multivariate distribution with the finite dimensional vector copulaparameter (Σ , ν ) characterizing between vector dependence and infinite dimensionalmarginal parameters f k , all k ≤ K . Example 11 (Distributions with Clayton vector copula) . Let c Cl ( u , . . . , u K ; θ ) de-note the density function of Clayton vector copula with parameter θ . Then f ( y , . . . , y K ) = c Cl (cid:0) T − ( y ) , . . . , T − K ( y K ); θ (cid:1) K Y k =1 " f k ( y k ) φ d k (cid:0) T − k ( y k ) (cid:1) is the density function of the meta distribution with the Clayton vector copula andmarginal measures P k , where T − k is a vector rank of P k for k ≤ K . Without parame-terizing the marginal distributions, the above expression results in a semiparametricmultivariate distribution with the finite dimensional vector copula parameter θ be-tween vector dependence and infinite dimensional marginal parameters f k , all k ≤ K .Importantly, the copula approach can be used to construct multivariate marginals P k such as semiparametric distributions P k further reducing the dimensionality ofthe model in all three examples above. In general the combination of the copulaapproach and the new vector copula approach we develop in this paper leads to aflexible hierarchical approach to multivariate modeling which we’ll explore in futurework.To compute vector copulas or to construct multivariate distributions with givennon-overlapping multivariate marginal distributions using vector copulas, we need to compute push-forward maps as in the vector Sklar theorem. One important classof such maps are µ -vector quantiles and µ -vector ranks of the given marginals P k .Although they don’t have closed form expressions, they can be computed quicklythanks to the recent developments in computational OT. We refer interested readersto Peyr´e and Cuturi (2019). Once the µ -vector quantiles and µ -vector ranks are com-puted, we can easily simulate from vector copulas and the multivariate distributionsconstructed from vector copulas on the basis of the vector Sklar theorem.6. Empirical Vector Copulas Definition from Chernozhukov et al. (2017) with the randomization device of Ghosal and Sen(2019). We concentrate on the case of ( T , ..., T K )-vector copulas for L = 1, wherethere are typically no closed form solutions for ( T , ..., T K ). Definition 13 (Empirical Ranks) . Let µ be the uniform distribution on [0 , d . Let P be an arbitrary distribution on R d . Let P n be the empirical distribution associatedwith a sample ( Y , . . . , Y n ) of n independent draws from a distribution P on R d . Forall i ≤ n , the empirical rank R n ( Y i ) of Y i are defined by R n ( Y i ) | ( Y , . . . , Y n ) ∼ Uniform( { u : ∂ψ n ( u ) = { Y i }} ) , where ψ n is a convex function defined in Proposition 1. Remark 3. The Kantorovich potential ψ n is piecewise affine convex, hence can bewritten ψ n ( u ) = max i =1 ,...,n { u T Y i + h i } for some vector h = ( h , . . . , h n ). The subdiffer-ential of ψ n has closed form ∂ψ n ( u ) = Conv( { Y i : u T Y i + h i = ψ n ( u ) } ).The following is Lemma 3.6 of Ghosal and Sen (2019). Lemma 9 (Distribution-Free Ranks) . Let P be an absolutely continuous distributionon R d , and µ the uniform distribution on [0 , d . The empirical ranks of Definition 13have distribution µ . Definition 14 (Empirical Vector Copula) . Let ( d , . . . , d K ) be a collection of integers.Let P be a distribution over R d × . . . × R d K with absolutely continuous marginaldistributions P , . . . , P K . Let ( Y i , . . . , Y iK ) ≤ i ≤ n be a random sample drawn from ECTOR COPULAS AND VECTOR SKLAR THEOREM 39 distribution P , and P n the associated empirical distribution. The empirical copula C n associated with P n is defined as the empirical distribution of the sample of empiricalranks ( R n ( Y i ) , . . . , R n ( Y iK )) ≤ i ≤ n . Theorem 4 (Consistency) . Let ( d , . . . , d K ) be a finite collection of integers. Foreach k ≤ K , let µ k be the uniform distribution on [0 , d k and let P k be an absolutelycontinuous distribution with support on a convex set Y k ⊆ R d k and let the µ k -vectorquantile ∇ ψ k associated with P k be a homeomorphism from Int ([0 , d k ) to Int ( Y k ) .Let ( Y i , . . . , Y iK ) ≤ i ≤ n be a random sample drawn from distribution P , with associatedvector copula C and marginals ( P , . . . , P K ) and let C n be the associated empiricalvector copula. For any collection of compact sets A k ⊂ Int ([0 , d k ) , j ≤ K , C n ( A , . . . , A K ) a.s. −−→ C ( A , . . . , A K ) . Proof of Theorem 4. Corollary of Theorem 4.1 in Ghosal and Sen (2019). (cid:3) Discussion We have developed vector copulas as a tool to characterize between vector nonlinearor rank dependence as distinct from within vector dependence, and to develop newparametric and semiparametric models for distributions with given non overlappingmultivariate marginals. We anticipate many applications of this new tool, that mirrorapplications of traditional copulas in quantitative finance and econometrics. However,an important aspect of our vector copula notion is the multiplicity of possible vectorcopula choices, driven by the multiplicity of transport based multivariate probabilitytransforms we entertain. This is an advantage, as it allows us to develop severalparametric families of vector copulas, some of which are given in closed form. Itis also a drawback in the sense that the between versus within vector dependencedistinction is governed by the choice of vector copula, so that there is no unique notionof between-vector dependence. This feature is typical of multivariate extensions ofnotions, such as quantiles, comonotonicity and copulas, beyond the natural orderingof the real line. 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