Constructing generalized FGM copulas by means of certain univariate distributions

In this paper we focus on specific generalized Fairlie-Gumbel-Morgenstern (or Sarmanov) copulas which are generated by a single function (so-called generator or generator function) defined on the unit interval. In particular, we introduce a class of generators based on density-quantile functions of certain univariate distributions. Many of the generator functions from the literature are recovered as special cases. Moreover, two new generators are suggested, implying to new copulas. Finally, the opposite way around, it is shown how to calculate the univariate distribution which belongs to a given copula generator function.