Multivariate dependence modeling using copulas

In this contribution we review models for construction of higher dimensional dependence that have arisen recent years. In particular we focus on specific generalized Farlie Gumbel (or Sarmanov) copulas which are generated by a single function (so-called generator or generator function) defined on the unit interval. An alternative approach to generalize the FGM family of copulas is to consider the semi-parametric family of symmetric copulas. This family is generated by an univariate function, determining the symmetry (radial symmetry, joint symmetry) and dependence property (quadrant dependence, total positivity) of copulas. A multivariate data set, which exhibit complex patterns of dependence, particularly in the tails, can be modeled using a cascade of lower-dimensional copulas. Moreover, there exist necessary and sufficient conditions on the generating functions of the FGM family, in order to obtain various dependence properties. So we present multivariate generalizations of this class studying symmetry and dependence concepts, measuring the dependence among the components of each class and providing several examples.