Inspired by the notion of quasi-homogenous copulas, we introduce a new class of functions with a given curved section. The convexity of curved sections plays a key role in characterizing the ...corresponding copulas. This class of copulas generalizes the class of quasi-homogenous copulas.
In this paper we prove that the set of all irreducible discrete quasi-copulas coincides with the set of all discrete quasi-copulas with minimal range. We also provide answers, throughout some ...counterexamples, to two questions posed in Aguiló et al. 2 regarding both discrete quasi-copulas with minimal range and irreducible discrete quasi-copulas. Some additional results concerning discrete (quasi-)copulas are also given.
In this paper we find pointwise best-possible bounds on the set of copulas with a given value of the Spearman's footrule coefficient. We show that the lower bound is always a copula but, unlike the ...bounds on sets of copulas with a given value of other measures, such as Kendall's tau, Spearman's rho and Blonqvist's beta, the upper bound can be a copula or a proper quasi-copula. We characterised both of these cases.
Copulas are becoming an essential tool in analyzing data thus encouraging interest in related questions. In the early stage of exploratory data analysis, say, it is helpful to know local copula ...bounds with a fixed value of a given measure of association. These bounds have been computed for Spearman’s rho, Kendall’s tau, and Blomqvist’s beta. The importance of another two measures of association, Spearman’s footrule and Gini’s gamma, has been reconfirmed recently. It is the main purpose of this paper to fill in the gap and present the mentioned local bounds for these two measures as well. It turns out that this is a quite non-trivial endeavor as the bounds are quasi-copulas that are not copulas for certain values of the two measures. We also give relations between these two measures of association and Blomqvist’s beta.
Residual implications derived from conjunctors are one of the most important fuzzy implications in fuzzy logic which possess interesting theoretical and practical properties. Copulas are a special ...kind of conjunctors that come from probability theory and statistics that have important applications in many fields. We mainly discuss residual implications derived from copulas in this article. We further specify the domain of variables in a characterization of residual implications derived from quasi-copulas. A characterization of residual implications derived from proper quasi-copulas is obtained, which provides a characterization of residual implications derived from copulas. Since that characterization is not intuitive enough, we give another characterization of residual implications derived from copulas by level sets, which provides a geometrical insight.
In this paper, we focus on constructions of (quasi-)copulas with a given curvilinear section by the curvilinear patchwork operation. It is shown that the curvilinear patchwork of any two ...quasi-copulas with a common curvilinear section is always a quasi-copula, and some sufficient conditions for the curvilinear patchwork of two copulas with a common curvilinear section to be a copula are also provided. The density, the singular component and tail dependence coefficients of copulas constructed by the curvilinear patchwork operation are discussed. The concept of a simple curvilinear section is introduced, and an elementary way of creating copulas with simple curvilinear sections is developed by the curvilinear patchwork construction. Best-possible bounds for the set of copulas with a common simple curvilinear section are given. By the curvilinear patchwork construction, two generalized bound copulas with given curvilinear sections and a new extension of the Farlie-Gumbel-Morgenstern family of copulas are also proposed.
Following a historical overview of the development of ordinal sums, a presentation of the most relevant results for ordinal sums of triangular norms and copulas is given (including gluing of copulas, ...orthogonal grid constructions and patchwork operators). The ordinal sums of copulas considered here are constructed not only by means of the comonotonic copula, but also by using the lower Fréchet-Hoeffding bound and the independence copula. We provide alternative proofs to some results on ordinal sums, elaborate properties common to all or just some of the ordinal sums discussed. Also included are a discussion of the relationship between ordinal sums of copulas and the Markov product and an overview of ordinal sums of multivariate copulas, illustrating aspects to be considered when extending concepts for ordinal sums of bivariate copulas to the multivariate case.
The omnipotent instrument for modeling multivariate dependence of random variables in standard probability theory has become copulas discovered by A. Sklar in 1959. Only recently Omladič and Stopar ...prove that in the bivariate case an analogous role is played by exactly the same copulas for random variables coming from finitely additive probability spaces. One of the main results of this paper is that this is true also in the general multivariate case. The extension to n dimensions requires a better understanding of quasi-copulas, the lattice closure of copulas with respect to the pointwise order. We need to develop a new equivalent definition of this notion that should be useful in other applications as well. Another tool we introduce and seems to be new even in the standard probability approach, is multivariate quasi-distributions. We also expand the coherence theory for quasi-copulas and quasi-distributions to the multivariate situation. Finally, our main result is a multivariate Sklar type theorem in the imprecise setting.
This paper presents a review of the concept of an n-quasi-copula, starting from the characterization of functions that can be derived from operations on random variables. We recall several other ...characterizations and properties that have been proven in the literature, highlighting the differences between the bivariate case and the higher-dimensional case. Additionally, we recall several applications of n-quasi-copulas such as their role in the study of bounds on sets of n-copulas. We also recall various classes of n-quasi-copulas that have been studied in the literature. Finally, we present some questions that have not yet been answered in the literature.
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