Johanna Nešlehová

Multivariate Archimedean copulas, d-monotone functions and ℓ1-norm symmetric distributions

It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a d-dimensional copula is that the generator is a d-monotone function. The class of d-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of d-dimensional l 1 ; -norm symmetric distributions that place no point mass at the origin. The d-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189-207] in an analogous manner to the well-known Bernstein-Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the d-dimensional Kendall function and Kendalls rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.

A Primer on Copulas for Count Data

The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.

Infinite-mean models and the LDA for operational risk

Due to published statistical analyses of operational risk data, methodological approaches to the AMA modeling of operational risk can be discussed more in detail. In this paper we raise some issues concerning correlation (or diversification) effects, the use of extreme value theory and the overall quantitative risk management consequences of extremely heavy-tailed data. We especially highlight issues around infinite mean models. Besides methodological examples and simulation studies, the paper contains indications for further research.

Beyond simplified pair-copula constructions

Pair-copula constructions (PCCs) offer great flexibility in modeling multivariate dependence. For inference purposes, however, conditional pair-copulas are often assumed to depend on the conditioning variables only indirectly through the conditional margins. The authors show here that this assumption can be misleading. To assess its validity in trivariate PCCs, they propose a visual tool based on a local likelihood estimator of the conditional copula parameter which does not rely on the simplifying assumption. They establish the consistency of the estimator and assess its performance in finite samples via Monte Carlo simulations. They also provide a real data application.

A goodness-of-fit test for bivariate extreme-value copulas

Resume: It is often reasonable to assume that the dependence structure of a bivariate continuous distribution belongs to the class of extreme-value copulas. The latter are characterized by their Pickands dependence function. The talk is concerned with a procedure for testing whether this function belongs to a given parametric family. The test is based on a Cramer-von Mises statistic measuring the distance between an estimate of the parametric Pickands dependence function and either one of two nonparametric estimators thereof studied by Genest and Segers (2009). As the limiting distribution of the test statistic depends on unknown parameters, it must be estimated via a parametric bootstrap procedure, whose validity is established. Monte Carlo simulations are used to assess the power of the test, and an extension to dependence structures that are left-tail decreasing in both variables is considered.

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Neslehova (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendalls tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendalls tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.

MODELING AND GENERATING DEPENDENT RISK PROCESSES FOR IRM AND DFA

Modern Integrated Risk Management (IRM) and Dynamic Financial Analysis (DFA) rely in great part on an appropriate modeling of the stochastic behavior of the various risky assets and processes that influence the performance of the company under consideration. A major challenge here is a more substantial and realistic description and modeling of the various complex dependence structures between such risks showing up on all scales. In this presentation, we propose some approaches towards modeling and generating (simulating) dependent risk processes in the framework of collective risk theory, in particular w.r.t. dependent claim number processes of Poisson type (homogeneous and non-homogeneous), and compound Poisson processes.

Extremal behavior of Archimedean copulas

We show how the extremal behavior of d-variate Archimedean copulas can be deduced from their stochastic representation as the survival dependence structure of an ℓ1-symmetric distribution (see McNeil and Nešlehová (2009)). We show that the extremal behavior of the radial part of the representation is determined by its Williamson d-transform. This leads in turn to simple proofs and extensions of recent results characterizing the domain of attraction of Archimedean copulas, their upper and lower tail-dependence indices, as well as their associated threshold copulas. We outline some of the practical implications of their results for the construction of Archimedean models with specific tail behavior and give counterexamples of Archimedean copulas whose coefficient of lower tail dependence does not exist.

Assessing and Modeling Asymmetry in Bivariate Continuous Data

A bivariate copula is the cumulative distribution function of a pair (U, V ) of uniform random variables. This copula is said to be symmetric if and only if (V, U) and (U, V ) have the same distribution. Many standard bivariate parametric families of copulas have this property; Archimedean and meta-elliptical copulas are prime examples. In practice, however, dependence is often asymmetric. This paper revisits key aspects of this issue from a modeling perspective. Measures of asymmetry and rank-based estimators thereof are discussed, along with recently proposed tests of symmetry. Several techniques for the construction of asymmetric dependence structures are critically reviewed. A hydrological data set is used for illustration purposes.

Multivariate Archimax copulas

A multivariate extension of the bivariate class of Archimax copulas was recently proposed by Mesiar and Jagr (2013), who asked under which conditions it holds. This paper answers their question and provides a stochastic representation of multivariate Archimax copulas. A few basic properties of these copulas are explored, including their minimum and maximum domains of attraction. Several non-trivial examples of multivariate Archimax copulas are also provided.