1. Bivariate measure-inducing quasi-copulasNik Stopar, 2024, izvirni znanstveni članek Povzetek: It is well known that every bivariate copula induces a positive measure on the Borel $\sigma$-algebra on $[0,1]^2$, but there exist bivariate quasi-copulas that do not induce a signed measure on the same $\sigma$-algebra. In this paper we show that a signed measure induced by a bivariate quasi-copula can always be expressed as an infinite combination of measures induced by copulas. With this we are able to give the first characterization of measure-inducing quasi-copulas in the bivariate setting.
Ključne besede: quasi-copulas, copula, signed measure, total variation norm, infinite series
Objavljeno v DiRROS: 19.09.2024; Ogledov: 45; Prenosov: 28
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2. Coherence and avoidance of sure loss for standardized functions and semicopulasErich Peter Klement, Damjana Kokol-Bukovšek, Blaž Mojškerc, Matjaž Omladič, Susanne Saminger, Nik Stopar, 2024, izvirni znanstveni članek Povzetek: We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, $1$-increasing functions with value $1$ at $(1, 1, \ldots , 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A \le C \le B$ for standardized $n$-variate functions $A$, $B$ and discuss methods for constructing such functions. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A \le C \le B$.
Ključne besede: copulas, quasi-copulas, semicopulas, standardized function, coherence, avoidance of sure loss, k-increasing function
Objavljeno v DiRROS: 13.03.2024; Ogledov: 378; Prenosov: 198
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3. On the exact region determined by Spearman's rho and Spearman's footruleDamjana Kokol-Bukovšek, Nik Stopar, 2024, izvirni znanstveni članek Povzetek: We determine the lower bound for possible values of Spearman’s rho of a bivariate copula given that the value of its Spearman's footrule is known and show that this bound is always attained. We also give an estimate for the exact upper bound and prove that the estimate is exact for some but not all values of Spearman's footrule. Nevertheless, we show that the estimate is quite tight.
Ključne besede: copula, dependence concepts, supremum and infimum of a set of copulas, measures of concordance, quasi-copula, local bounds
Objavljeno v DiRROS: 13.03.2024; Ogledov: 411; Prenosov: 180
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4. Quasi-copulas as linear combinations of copulasGregor Dolinar, Bojan Kuzma, Nik Stopar, 2024, izvirni znanstveni članek Povzetek: We prove that every quasi-copula can be written as a uniformly converging infinite sum of multiples of copulas. Furthermore, we characterize those quasi-copulas which can be written as a finite sum of multiples of copulas, i.e., that are a linear combination of two copulas. This generalizes a recent result of Fernández-Sánchez, Quesada-Molina, and Úbeda-Flores who considered linear combinations of discrete copulas.
Ključne besede: quasi-copulas, copulas, linear combination, affine combination, Minkowski norm
Objavljeno v DiRROS: 16.02.2024; Ogledov: 426; Prenosov: 168
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