New constructions of diagonal patchwork copulas

On the relationship between modular functions and copulas

“…Although for continuous triangular norms this operation leads to an important representation theorem (see [25]), for copulas many more opportunities arise. These have been laid bare by some of the present authors [3] under the general name of orthogonal grid constructions of copulas (making use of so-called background and foreground copulas) and have been further investigated by Durante et al [12,13]. For instance, the W -ordinal sum (see also [27]) is obtained by linearly rescaling given copulas to given squares along the opposite diagonal of the unit square and taking the value of W outside these squares.…”

“…A comprehensive overview of basic results in the theory of copulas is given in [28]. In order to increase the modelling potential of copulas in realworld problems, new constructions methods for copulas are being proposed continuously: block-wise methods along the line of ordinal sums [3,6,12,13,26], construction methods starting from given sections [2,9,14,23], interpolative methods [5,10,20,21], perturbation methods [7], construction methods for higher-dimensional copulas [6,8], etc.…”

Directional dependence of random vectors

“…For this purpose, the concept of copula has appeared to be very useful for capturing the dependence properties of a continuous random vector apart from its marginal behavior (see [7,8,22]). …”

On the Classes of Copulas and Quasi-Copulas With a Given Diagonal Section

“…Specifically, given a suitable δ: I → I, (hereafter called diagonal ), the class of (quasi-)copulas having diagonal section equal to δ has been the object of several investigations. [10][11][12][13][14][15][16] Although a number of investigations has been conducted in this direction, it is worth of further consideration to clarify some aspects about the classes C δ and Q δ formed, respectively, by all copulas and quasi-copulas whose diagonal section is δ.…”