Matrix representation of discrete quasi-copulas

Aguiló, Isabel; Suòer, Jaume; Torrens, Joan

doi:10.1016/j.fss.2007.10.004

Cited by 22 publications

(8 citation statements)

References 10 publications

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“…We have studied discrete copulas from a lattice-theoretic point of view as a continuation of recent work on matrix representations [1,9,12,14]. The main idea was to introduce a new class of copulas, meet-irreducible copulas.…”

Section: Discussionmentioning

confidence: 99%

Matrix representation of meet-irreducible discrete copulas

Kobayashi

2014

Fuzzy Sets and Systems

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Following Aguiló-Suñer-Torrens ( 2008), Kolesárová-Mesiar-Mordelová-Sempi (2006) and Mayor-Suñer-Torrens (2005), we continue to develop a theory of matrix representation for discrete copulas. To be more precise, we give characterizations of meet-irreducible discrete copulas from an order-theoretical aspect: we show that the set of all irreducible discrete copulas is a lattice in analogy with Nelsen andÚbeda-Flores (2005). Moreover, we clarify its lattice structure related to Kendall's τ and Spearman's ρ borrowing ideas from Coxeter groups.

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Section: Discussionmentioning

confidence: 99%

Matrix representation of meet-irreducible discrete copulas

Kobayashi

2014

Fuzzy Sets and Systems

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“…This is a beautiful result since, as explained in [15], the (p × p)−discrete copulas of minimal range correspond to the so-called empirical copulas, which are the foundation of rank-based (non-parametric) copula approaches [23,24]. The case of (p × q)-discrete (quasi-) copulas is considered in [1,2,16]. There, a first characterization of the set of irreducible discrete copulas when q = pk is also given.…”

Section: Introductionmentioning

confidence: 97%

“…. , F X d (x d )), (1) where the function C : [0, 1] d → [0, 1] is a ddimensional copula and F X 1 , . .…”

Section: Introductionmentioning

confidence: 99%

“…On the mathematical side, discrete copulas (and discrete quasi-copulas) have been extensively studied in terms of their matrix representations [1,2,9,12,14,15,16,21]. In [18], the authors highlight that the space of discrete copulas with fixed marginal distributions of finite support of size p and q, respectively, corresponds to a polytope known as the generalized Birkhoff polytope [6].…”

Section: Introductionmentioning

confidence: 99%

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Extreme Points of Polytopes of Discrete Copulas

Perrone¹,

Durante²

2021

Atlantis Studies in Uncertainty Modelling

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Discrete copulas are useful tools in statistics to represent the joint distribution of discrete random vectors. Furthermore, they are fascinating mathematical objects that admit a representation as a convex polytope. In this work, we analyze the set of extreme points of convex polytopes of discrete copulas. We focus on the general class of discrete copulas defined on arbitrary grid domains of the unit square, thereby providing a complete characterization of extremal discrete copulas in the most general setting. To do so, we exploit well-known results on the extreme points of transportation polytopes. Finally, we show that the characterization presented here generalizes previous work on the topic.

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“…The space of discrete copulas in the bivariate setting (i.e., d = 2) was studied by Aguiló et al (2008Aguiló et al ( , 2010; Kolesárová et al (2006); Mayor et al (2005); Mesiar (2005); Mordelová and Kolesárová (2007). The results of these papers collectively demonstrate that the space of bivariate discrete copulas constructed from marginal distributions with finite state spaces of sizes p and q correspond to the points within a special polytope known as the generalized Birkhoff polytope (Ziegler, 1995).…”

Section: Introductionmentioning

confidence: 99%

Geometry of discrete copulas

Perrone

Solus

Uhler

2019

Journal of Multivariate Analysis

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Multivariate distributions are fundamental to modeling. Discrete copulas can be used to construct diverse multivariate joint distributions over random variables from estimated univariate marginals. The space of discrete copulas admits a representation as a convex polytope which can be exploited in entropy-copula methods relevant to hydrology and climatology. To allow for an extensive use of such methods in a wide range of applied fields, it is important to have a geometric representation of discrete copulas with desirable stochastic properties. In this paper, we show that the families of ultramodular discrete copulas and their generalization to convex discrete quasi-copulas admit representations as polytopes. We draw connections to the prominent Birkhoff polytope, alternating sign matrix polytope, and their most extensive generalizations in the discrete geometry literature. In doing so, we generalize some well-known results on these polytopes from both the statistics literature and the discrete geometry literature.

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Matrix representation of discrete quasi-copulas

Cited by 22 publications

References 10 publications

Matrix representation of meet-irreducible discrete copulas

Matrix representation of meet-irreducible discrete copulas

Extreme Points of Polytopes of Discrete Copulas

Geometry of discrete copulas

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