New Prospects on Vines

In this paper, we present a new methodology based on vine copulas to estimate multivariate distributions in high dimensions, taking advantage of the diversity of vine copulas. Considering the huge number of vine copulas in dimension n, we introduce an efficient selection algorithm to build and select vine copulas with respect to any test T. Our methodology offers a great flexibility to practitioners to compute VaR associated to a portfolio in high dimension.Vine copulas, multivariate copulas, model selection, VaR.

Bounds on Integrals with Respect to Multivariate Copulas

Finding upper and lower bounds to integrals with respect to copulas is a quite prominent problem in applied probability. In their 2014 paper, Hofer and Iaco showed how particular two dimensional copulas are related to optimal solutions of the two dimensional assignment problem. Using this, they managed to approximate integrals with respect to two dimensional copulas. In this paper, we will further illuminate this connection, extend it to d-dimensional copulas and therefore generalize the method from Hofer and Iaco to arbitrary dimensions. We also provide convergence statements. As an example, we consider three dimensional dependence measures

Bivariate copulas defined from matrices

We propose a semiparametric family of copulas based on a set of orthonormal functions and a matrix. This new copula permits to reach values of Spearman's Rho arbitrarily close to one without introducing a singular component. Moreover, it encompasses several extensions of FGM copulas as well as copulas based on partition of unity such as Bernstein or checkerboard copulas. Finally, it is also shown that projection of arbitrary densities of copulas onto tensor product bases can enter our framework

Characterizations of bivariate conic, extreme value, and Archimax copulas

Based on a general construction method by means of bivariate ultramodular copulas we construct, for particular settings, special bivariate conic, extreme value, and Archimax copulas. We also show that the sets of copulas obtained in this way are dense in the sets of all conic, extreme value, and Archimax copulas, respectively

Some results on weak and strong tail dependence coefficients for means of copulas

Copulas represent the dependence structure of multivariate distributions in a natural way. In order to generate new copulas from given ones, several proposals found its way into statistical literature. One simple approach is to consider convex-combinations (i.e. weighted arithmetic means) of two or more copulas. Similarly, one might consider weighted geometric means. Consider, for instance, the Spearman copula, defined as the geometric mean of the maximum and the independence copula. In general, it is not known whether weighted geometric means of copulas produce copulas, again. However, applying a recent result of Liebscher (2006), we show that every weighted geometric mean of extreme-value copulas produces again an extreme-value copula. The second contribution of this paper is to calculate extremal dependence measures (e.g. weak and strong tail dependence coe±cients) for (weighted) geometric and arithmetic means of two copulas. --Tail Dependence,Extreme-value copulas,arithmetic and geometric mean

On approximating copulas by finite mixtures

Copulas are now frequently used to approximate or estimate multivariate distributions because of their ability to take into account the multivariate dependence of the variables while controlling the approximation properties of the marginal densities. Copula based multivariate models can often also be more parsimonious than fitting a flexible multivariate model, such as a mixture of normals model, directly to the data. However, to be effective, it is imperative that the family of copula models considered is sufficiently flexible. Although finite mixtures of copulas have been used to construct flexible families of copulas, their approximation properties are not well understood and we show that natural candidates such as mixtures of elliptical copulas and mixtures of Archimedean copulas cannot approximate a general copula arbitrarily well. Our article develops fundamental tools for approximating a general copula arbitrarily well by a mixture and proposes a family of finite mixtures that can do so. We illustrate empirically on a financial data set that our approach for estimating a copula can be much more parsimonious and results in a better fit than approximating the copula by a mixture of normal copulas.Comment: 26 pages and 1 figure and 2 table