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.

翻译：Copula函数因其能够在分别指定边际分布的同时考虑不同变量之间的多元依赖关系，如今被广泛用于构建或估计多元分布。基于Copula的多元模型通常比直接对数据拟合灵活多元模型（如正态混合模型）更为精简。然而，要实现有效应用，所考虑的Copula函数族必须具备足够的灵活性。尽管有限混合Copula已被用于构建灵活Copula族，但其逼近特性尚未得到充分理解。我们证明，椭圆型Copula混合与阿基米德Copula混合等自然候选方法无法任意逼近一般Copula函数。本文发展了基于有限混合Copula任意逼近一般Copula函数的基础工具。我们展示了渐近性质，并通过金融数据集和人工数据实证验证了该方法的优势。