Statistics and Computing

, Volume 28, Issue 2, pp 387–409 | Cite as

Estimating non-simplified vine copulas using penalized splines

  • Christian Schellhase
  • Fabian Spanhel


Vine copulas (or pair-copula constructions) have become an important tool for high-dimensional dependence modeling. Typically, so-called simplified vine copula models are estimated where bivariate conditional copulas are approximated by bivariate unconditional copulas. We present the first nonparametric estimator of a non-simplified vine copula that allows for varying conditional copulas using penalized hierarchical B-splines. Throughout the vine copula, we test for the simplifying assumption in each edge, establishing a data-driven non-simplified vine copula estimator. To overcome the curse of dimensionality, we approximate conditional copulas with more than one conditioning argument by a conditional copula with the first principal component as conditioning argument. An extensive simulation study is conducted, showing a substantial improvement in the out-of-sample Kullback–Leibler divergence if the null hypothesis of a simplified vine copula can be rejected. We apply our method to the famous uranium data and present a classification of an eye state data set, demonstrating the potential benefit that can be achieved when conditional copulas are modeled.


Vine Pair-copula Simplifying assumption Conditional copula Penalized spline 



We thank Göran Kauermann (LMU Munich) for discussions at the beginning of this project.

Supplementary material

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Supplementary material 1 (pdf 1301 KB)
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Supplementary material 2 (pdf 96 KB)


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Business Administration and Economics, Centre for StatisticsBielefeld UniversityBielefeldGermany
  2. 2.Department of StatisticsLudwig-Maximilians-Universität MünchenMunichGermany

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