Shuffles of copulas

Abstract

We show that every copula that is a shuffle of Min is a special push-forward of the doubly stochastic measure induced by the copula M. This fact allows to generalize the notion of shuffle by replacing the measure induced by M with an arbitrary doubly stochastic measure, and, hence, the copula M by any copula C .

Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis

“…With the suitable specification of marginal probabilities and dependence structures, the Gaussian copula can be derived with the principle of maximum entropy [29]. The entropy copula can also be interpreted as the approximation of the copula, for which other types of copula approximation schemes exist [95,97,134,135], such as shuffle of min copula [136,137], Bernstein copula [138,139], checkerboard copula [31,134] or others based on splines or kernels [140][141][142]. Moreover, the entropy based bivariate copula can also be integrated in the vine structure to derive the vine copula [32], which is particularly attractive in modeling the flexible dependence in higher dimensions when parametric copulas fall short in this case [88,132].…”

Defects and transformations of quasi-copulas

“…Observe that the function Q visualized in Figure 7 (left) is a copula (actually, it turns out to be a shuffle of the Fréchet-Hoeffding upper bound M ). For more details about shuffles of M see [12,14,27]. In analogy, the functions Q , Q , Q , Q M , and Q O are shuffles of M and, therefore, copulas (for a visualization of their supports see Figure 8).…”

Measure-Preserving Functions and the Independence Copula

“…(See [3], [4], [12], [16] or [24].) (2.10) Finally, by using the probability measure induced by a copula, an important class of copulae is provided by the Shuffles of Min (see [4], [14], or [15, p. 68], for more details).…”

“…(2.10) Finally, by using the probability measure induced by a copula, an important class of copulae is provided by the Shuffles of Min (see [4], [14], or [15, p. 68], for more details). Formally, a copula C is a shuffle of Min if there is a natural number n, two partitions 0 = s 0 < s 1 < ...s n = 1 and 0 = t 0 < t 1 < ...t n = 1 of I, and a permutation σ of {1, ..., n} such that each…”