Motivated by the increasing popularity and the seemingly broad applicability of pair-copula constructions underlined by numerous publications in the last decade, in this contribution we tackle the unavoidable question on how flexible and simplifying the commonly used `simplifying assumption' is from an analytic perspective and provide answers to two related open questions posed by Nagler and Czado in 2016. Aiming at a simplest possible setup for deriving the main results we first focus on the three-dimensional setting. We prove that the family of simplified copulas is flexible in the sense that it is dense in the set of all three-dimensional co\-pulas with respect to the uniform metric $d_\infty$ - considering stronger notions of convergence like the one induced by the metric $D_1$, by weak conditional convergence, by total variation, or by Kullback-Leibler divergence, however, the family even turn out to be nowhere dense and hence insufficient for any kind of flexible approximation. Furthermore, returning to $d_\infty$ we show that the partial vine copula is never the optimal simplified copula approximation of a given, non-simplified copula $C$, and derive examples illustrating that the corresponding approximation error can be strikingly large and extend to more than 28\% of the diameter of the metric space. Moreover, the mapping $\psi$ assigning each three-dimensional copula its unique partial vine copula turns out to be discontinuous with respect to $d_\infty$ (but continuous with respect to $D_1$ and to weak conditional convergence), implying a surprising sensitivity of partial vine copula approximations. The afore-mentioned main results concerning $d_\infty$ are then extended to the general multivariate setting.
翻译:在过去十年中,许多出版物都强调,成衣造型的多层铜板越来越受欢迎,而且似乎具有广泛适用性,为此,我们从分析角度着手处理一个不可避免的问题,即如何灵活和简化常用“简化假设”这一常用的“简化假设”从分析角度出发,并回答Nagler和Czado在2016年提出的两个相关的开放问题。然而,为了为得出我们最初关注的三维环境的主要结果,我们首先着眼于一个最简单的设置。我们证明,简化的多层铜板的组合是灵活的,因为它是所有三维的co-co-pulas组合中与统一标准“美元”的堆积的堆积。我们从考虑更强烈的趋汇概念,如以公吨$1美元、条件趋汇差、完全变差或Kullback-Lebeler差为导。然而,家族甚至退缩到任何灵活的缩缩略图。 此外,我们证明,部分的葡萄制铜板从来不是最优化的三维基价。