Abstract
Vine copulas, or pair-copula constructions, have become an important tool in high-dimensional dependence modeling. Commonly, it is assumed that the data generating copula can be represented by a simplified vine copula (SVC). In this paper, we study the simplifying assumption and investigate the approximation of multivariate copulas by SVCs. We introduce the partial vine copula (PVC) which is a particular SVC where to any edge a $j$-th order partial copula is assigned. The PVC generalizes the partial correlation matrix and plays a major role in the approximation of copulas by SVCs. We investigate to what extent the PVC describes the dependence structure of the underlying copula. We show that, in general, the PVC does not minimize the Kullback-Leibler divergence from the true copula if the simplifying assumption does not hold. However, under regularity conditions, stepwise estimators of pair-copula constructions converge to the PVC irrespective of whether the simplifying assumption holds or not. Moreover, we elucidate why the PVC is often the best feasible SVC approximation in practice.
Highlights
We introduced the partial vine copula (PVC) which is a particular simplified vine copula (SVC) that coincides with the data generating copula if the simplifying assumption holds
The PVC can be regarded as a generalization of the partial correlation matrix where partial correlations are replaced by j-th order partial copulas
While a higher-order partial copula of the PVC is related to the partial copula, it does not suffer from the curse of dimensionality and can be estimated for high-dimensional data [27]
Summary
The following notation and assumptions are used throughout the paper. Dxd denote the variables of integration in fX1:d (x1:d)dx1:d. We write 11{A} = 1 if A is true, and 11{A} = 0 otherwise. We assume that all random variables are real-valued and continuous. Let d ≥ 3, if not otherwise specified, and Cd be the space of absolutely continuous d-dimensional copulas with positive density (a.e.). The distribution function of a random vector U1:d with uniform margins is denoted by F1:d = C1:d ∈ Cd. We set Ild := {(i, j) : j = l, .
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