On the relationship between modular functions and copulas

Construction of Copulas with Hairpin Support

“…We generalize the classic result in [17] and [18] and we extend them to the general nondecreasing case without assuming bijectivity. This generalization allows us to solve the open problem posed in [5]. Furthermore, under certain conditions, we give a multidimensional method that generalizes the bivariate case and allows to construct extreme points in the set of multidimensional copulas.…”

“…´ F. Chamizo, J. Fernández-Sánchez and M. Ubeda-Flores A slightly more general problem is posed in [5], where modular functions are involved. Before tackling this problem, we recall some preliminary notions.…”

“…In [5] it is illustrated that a simple operation, such as truncating by means of a Fréchet-Hoeding bound, can transform a suitable modular function H into a copula H = min(M, H). The region where the copula H coincides with M is separated from the region where it does not by means of two curves.…”

“…On the other hand, relations between copulas and measure-preserving transformations and self-aÿne functions whose graphs are the support of a copula were established in [4], and the construction of copulas from modular functions in [5], where the copulas corresponding to C-hairpins appear as special cases and give rise to a problem that generalizes the original approach in [17,18].…”

Some Construction Methods for Pseudo-Overlaps and Pseudo-Groupings and Their Application in Group Decision Making

“…However, a key assumption in the existing definitions of overlap and grouping functions is commutativity [10]. In real-world applications, such as decision-making, criteria or experts often possess varying importance, rendering the assumption of commutativity unreasonable [11][12][13]. To address this limitation, researchers have introduced pseudooverlaps [14] and pseudo-groupings [10] as non-commutative generalizations of overlaps and groupings, respectively.…”