Preface to special issue on high-dimensional dependence and copulas

One of the biggest advances in recent years for high-dimensional copula models and applications has been the development of the vine pair-copula construction that covers continuous and discrete variables, and its extensions to include latent variables. Software has been made available in the VineCopula R package and the package that is companion to the book by Joe [6]. This special issue of the Journal of Multivariate Analysis is based on presentations by leading copula researchers at the International Workshop on High-Dimensional Dependence and Copulas: Theory, Modeling, and Applications, held at Central University of Finance & Economics (CUFE), Beijing China during January 3–5, 2014. Earlier vine copula workshops were held in Delft (Netherlands), Oslo (Norway) and Munich (Germany) in the years 2007, 2008, 2009 and 2011. Simple parametricmultivariate copula families do not have flexible dependence but there are plenty of parametric bivariate copula families that can cover a range of dependence (from countermonotonicity or independence to comonotonicity) and a range of tail behavior for the joint lower and upper tails. The vine pair-copula construction is flexible because it is built from a tree sequence of bivariate copulas, with these bivariate copulas applied to conditional distribution after the first tree (details are given in several of the articles in this issue). There is a graphical representation of vines as a sequence of trees where edges of trees indicate pairing of variables. The pair-copula construction assigns (algebraically independent) bivariate copulas to the edges of the trees. When each bivariate copula is Gaussian, the corresponding parameters are correlation parameters for the first tree and they are interpreted as partial correlation parameters for trees 2 and higher. Hence vines also provide another parametrization for the Gaussian correlation matrix. The partial correlation parametrization is exploited in order to get the copula extension of Gaussian-based factor models and structural equation models. Applications of copulamodels to joint risks involve tail inferencewhere there ismore probability in joint tails thanwould be implied by multivariate Gaussian. Hence tail properties of copulas and tail inference based on copulas have developed along with construction of copula models. Other multivariate dependence concepts have also been studied with motivation from risk analysis in finance and insurance. The word ‘vine’ for the structure was coined by Roger Cooke, and he also developed one of its graphical representations. Early research on vines was developed by Cooke, Bedford and Kurowicka at the Technische Universiteit Delft [2,3,7]. Statistical applications have been growing since Aas et al. [1] and the statistical computing for vines has simplified with the VineCopula R package, which has been developing at the Technische Universität München since 2010.