Simplified Pair Copula Constructions --- Limits and Extensions

So called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep them tractable for inference and model selection, the simplifying assumption that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on is popular. In this paper, we show for which classes of distributions such a simplification is applicable, significantly extending the discussion of Hob{\ae}k Haff et al. (2010). In particular, we show that the only Archimedean copula in dimension d \geq 4 which is of the simplified type is that based on the gamma Laplace transform or its extension, while the Student-t copula is the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on

On the simplified pair-copula construction -- Simply useful or too simplistic?

Due to their high flexibility, yet simple structure, pair-copula constructions (PCCs) are becoming increasingly popular for constructing continuous multivariate distributions. However, inference requires the simplifying assumption that all the pair-copulae depend on the conditioning variables merely through the two conditional distribution functions that constitute their arguments, and not directly. In terms of standard measures of dependence, we express conditions under which a specific pair-copula decomposition of a multivariate distribution is of this simplified form. Moreover, we show that the simplified PCC in fact is a rather good approximation, even when the simplifying assumption is far from being fulfilled by the actual model.Copulae Vines Multivariate distributions Hierarchical structures

Nonparametric estimation of pair-copula constructions with the empirical pair-copula

A pair-copula construction is a decomposition of a multivariate copula into a structured system, called regular vine, of bivariate copulae or pair-copulae. The standard practice is to model these pair-copulae parametrically, which comes at the cost of a large model risk, with errors propagating throughout the vine structure. The empirical pair-copula proposed in the paper provides a nonparametric alternative still achieving the parametric convergence rate. It can be used as a basis for inference on dependence measures, for selecting and pruning the vine structure, and for hypothesis tests concerning the form of the pair-copulae.Comment: 23 pages, 7 figure