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Meeting-ID: 280 519 2121
Passwort: 584190

Abstract:

The relevance of copulas in dependence modeling originates from the fact that, due to Sklar's theorem, for (continuous) multivariate distributions the modeling of its univariate marginals and the dependence structure can be separated, where the latter can be represented by a copula. A copula may be seen as a multivariate distribution function with all univariate margins being uniformly distributed on [0,1]. Hence, if C is a copula, then it is the distribution function of a vector of dependent U(0,1) random variables; in case of independence, the corresponding copula being the product.

In literature one may find quite some different classes and families of copulas either deduced, e.g., from multivariate distributions or following different construction approaches like, e.g., Archimedean copulas or some patchwork and gluing techniques; as such also examplifying the many views one can have on this class of functions, its analytical, probabilistic, measure-theoretic, or also algebraic properties, as well as different fields of applications.

In this talk we will focus on bivariate copulas. The Fréchet-Hoeffding bounds W(x,y)=max{x+y-1,0} and M(x,y)=min{x,y} allowing, as extremal cases, to model the counter- or comonotone behaviour of a pair of continuous random variables, modeling types of complete dependence. We are interested in perturbing some basic dependence behaviour by some parametrized transformations and study under which conditions we obtain again copulas. An example of this type is the well-known family of Eyraud-Farley-Gumble-Morgenstern copulas as a perturbation of the product copula. We will discuss obtained families based on the Fréchet-Hoeffindg bounds and their properties. In a second part, we will turn to another class of perturbations of the product copula, namely polynomial bivariate copulas. We will present a nice characterization result for polynomial bivariate copulas of degree five and discuss some of the related dependence properties.