A class of bivariate copula mappings

In this thesis we analyze one particular mapping in the set of bivariate copulas, which allows flexible construction of new copulas and families of copulas. In particular, we suggest mapping which depends on a univariate function. We provide necessary and sufficient conditions on this univariate function so that output of the mapping is a copula for any bivariate copula. Then we discuss several important bivariate copula properties preserved or not by this mapping. The considered bivariate copula construction unifies many examples found in the literature and opens a qualitatively new way of obtaining new copulas by simply using a univariate function which must satisfy a few easily verifiable properties. In next chapter of the thesis, we focus our attention on conditionally eligible functions, in particular, the case of independence copula, i.e., we try to characterize all univariate functions such that output of the mapping is a copula for the independence copula. We provide a complete characterization for the two cases: (i) when output of the mapping is, in addition, totally positive of order 2 and (ii) when univariate function is twice continuously differentiable. In general, this function need only be twice differentiable Lebesgue almost everywhere, as shown by investigation of necessary conditions. This chapter also contains numerous examples illustrating obtained results and connections to known facts from the literature