Basic uncertain information (BUI) in the form <inline-formula><tex-math notation="LaTeX">\langle x;c \rangle </tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">x \in ...0,1</tex-math></inline-formula> is an observed datum and <inline-formula><tex-math notation="LaTeX">c \in 0,1</tex-math></inline-formula> is its reliability, is discussed and studied. The concept of BUI's aggregation is introduced and some construction methods are proposed and exemplified. The connection with interval-valued data and their aggregation is also discussed.
Our starting point are several general classes of real functions defined on the unit square satisfying some basic properties such as a boundary condition or several types of monotonicity and ...continuity. Applying to these functions some parameterized transformations and other constructions such as the transpose and flipping (which describe different aspects of symmetry) and truncation, we ask for conditions yielding (again) a bivariate copula. Some of these transformations are involutive (on one or more classes of functions), others are not even injective, and occasionally they induce additional properties, yielding, e.g., a (quasi-)copula. For several typical scenarios we identify the (not necessarily convex) sets of parameters leading to a copula and conditions imposing a minimal set of parameters.
Extending sub-quasi-copulas Kokol Bukovšek, Damjana; Košir, Tomaž; Omladič, Matjaž ...
Journal of mathematical analysis and applications,
08/2021, Letnik:
500, Številka:
1
Journal Article
Recenzirano
The main result of this paper is a method that gives all possible quasi-copulas that extend a given sub-quasi-copula to the whole domain, i.e., unit square 0,1×0,1. It is known that extending ...quasi-copulas is a deeper challenge than extending copulas, so we need to develop new techniques to do that. Perhaps surprisingly, our method is at the same time elementary and universal – two seemingly contradicting properties that none of the known methods for extending copulas seem to possesses. We also give a construction of two quasi-copulas that unveil an interesting counterexample in imprecise probability theory.
In this note we study the extremes of the mass distribution associated with a tetravariate quasi-copula and compare our results with the bi- and trivariate cases, showing the important differences ...between them.
Supports of quasi-copulas Fernández-Sánchez, Juan; Quesada-Molina, José Juan; Úbeda-Flores, Manuel
Fuzzy sets and systems,
09/2023, Letnik:
467
Journal Article
Recenzirano
Odprti dostop
It is known that for every s∈1,2 there is a copula whose support is a self-similar fractal set with Hausdorff —and box-counting— dimension s. In this paper we provide similar results for (proper) ...quasi-copulas, in both the bivariate and multivariate cases.
In this paper we present a surprisingly general extension of the main result of a paper that appeared in this journal: Montes et al. (2015) 13. The main tools we develop in order to do so are: (1) a ...theory on quasi-distributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) p-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) p-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladič and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) p-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual σ-additive probability space or on an additive space can be obtained as a copula of its margins meaning that its possible extraordinariness depends solely on its margins. This might indicate that copulas are a stronger probability concept than probability itself.
In this paper we solve in the negative the problem proposed in this journal (Montes et al. (2015) 13) whether an order interval defined by an imprecise copula contains a copula. Namely, if C is a ...nonempty set of copulas, then C_=inf{C}C∈C and C‾=sup{C}C∈C are quasi-copulas and the pair (C_,C‾) is an imprecise copula according to the definition introduced in the cited paper, following the ideas of p-boxes. We show that there is an imprecise copula (A,B) in this sense such that there is no copula C whatsoever satisfying A⩽C⩽B. So, it is questionable whether the proposed definition of the imprecise copula is in accordance with the intentions of the initiators. Our methods may be of independent interest: We upgrade the ideas of Dibala et al. (2016) 7 where possibly negative volumes of quasi-copulas as defects from being copulas were studied.
In this paper, we consider the (quasi-)copulas with a given curvilinear section, and then establish the best-possible bounds on the set of quasi-copulas with a common curvilinear section. The lower ...bound is shown to be the best-possible lower bound as well for the set of copulas with the given curvilinear section. A sufficient and necessary condition guaranteeing the upper bound to be a copula is provided, and it is shown that when the upper bound is a copula, it is the best-possible upper bound for the set of copulas with the given curvilinear section. The best-possible upper bound on the set of copulas which have a common curvilinear section and are symmetrical with respect to the given curve is also provided. Some properties of the best-possible bound copulas with the given curvilinear section are also discussed, such as the statistical characteristic, the concordance order, the nonparametric measure of association and the tail dependence.
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