How simplifying and flexible is the simplifying assumption in pair-copula constructions – some analytic answers in dimension three and beyond
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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 flexibility of simplified copulas in the sense that they are dense in the family of all three-dimensional copulas with respect to the uniform metric d_∞, 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 ψ assigning each three-dimensional copula its unique partial vine copula turns out to be discontinuous with respect to d_∞ (but continuous with respect to other notions of convergence), implying a surprising sensitivity of partial vine copula approximations. The afore-mentioned main results are then extended to the general multivariate setting.
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