  Copulas are now frequently used to construct or estimate multivariate
distributions because of their ability to take into account the multivariate
dependence of the different variables while separately specifying marginal
distributions. Copula based multivariate models can often also be more
parsimonious than fitting a flexible multivariate model, such as a mixture of
normals model, directly to the data.
  However, to be effective, it is imperative that the family of copula models
considered is sufficiently flexible. Although finite mixtures of copulas have
been used to construct flexible families of copulas, their approximation
properties are not well understood and we show that natural candidates such as
mixtures of elliptical copulas and mixtures of Archimedean copulas cannot
approximate a general copula arbitrarily well. Our article develops fundamental
tools for approximating a general copula arbitrarily well by a copulas based on
finite mixtures. We show the asymptotic properties as well as illustrate the
advantages of our methodology empirically on a financial data set and on some
artificial data.
