  Copulas are popular as models for multivariate dependence because they allow
the marginal densities and the joint dependence to be modeled separately.
However, they usually require that the transformation from uniform marginals to
the marginals of the joint dependence structure is known. This can only be done
for a restricted set of copulas, e.g. a normal copula. Our article introduces
copula-type estimators for flexible multivariate density estimation which also
allow the marginal densities to be modeled separately from the joint
dependence, as in copula modeling, but overcomes the lack of flexibility of
most popular copula estimators. An iterative scheme is proposed for estimating
copula-type estimators and its usefulness is demonstrated through simulation
and real examples. The joint dependence is is modeled by mixture of normals and
mixture of normals factor analyzers models, and mixture of t and mixture of t
factor analyzers models. We develop efficient Variational Bayes algorithms for
fitting these in which model selection is performed automatically. Based on
these mixture models, we construct four classes of copula-type densities which
are far more flexible than current popular copula densities, and outperform
them in simulation and several real data sets.
