Extreme-Value Copulas

Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric context.Comment: 20 pages, 3 figures. Minor revision, typos corrected. To appear in F. Durante, W. Haerdle, P. Jaworski, and T. Rychlik (editors) "Workshop on Copula Theory and its Applications", Lecture Notes in Statistics -- Proceedings, Springer 201

Extremal attractors of Liouville copulas

Liouville copulas, which were introduced in McNeil and Neslehova (2010), are asymmetric generalizations of the ubiquitous Archimedean copula class. They are the dependence structures of scale mixtures of Dirichlet distributions, also called Liouville distributions. In this paper, the limiting extreme-value copulas of Liouville copulas and of their survival counterparts are derived. The limiting max-stable models, termed here the scaled extremal Dirichlet, are new and encompass several existing classes of multivariate max-stable distributions, including the logistic, negative logistic and extremal Dirichlet. As shown herein, the stable tail dependence function and angular density of the scaled extremal Dirichlet model have a tractable form, which in turn leads to a simple de Haan representation. The latter is used to design efficient algorithms for unconditional simulation based on the work of Dombry, Engelke and Oesting (2015) and to derive tractable formulas for maximum-likelihood inference. The scaled extremal Dirichlet model is illustrated on river flow data of the river Isar in southern Germany.Comment: 30 pages including supplementary material, 6 figure

Systemic Weather Risk and Crop Insurance: The Case of China

The supply of affordable crop insurance is hampered by the existence of systemic weather risk which results in large risk premiums. In this article, we assess the systemic nature of weather risk for 17 agricultural production regions in China and explore the possibility of spatial diversification of this risk. We simulate the buffer load of hypothetical temperature-based insurance and investigate the relation between the size of the buffer load and the size of the trading area of the insurance. The analysis makes use of a hierarchical Archimedean copula approach (HAC) which allows flexible modeling of the joint loss distribution and reveals the dependence structure of losses in different insured regions. Our results show a significant decrease of the required risk loading when the insured area expands. Nevertheless, a considerable part of undiversifiable risk remains with the insurer. We find that the spatial diversification effect depends on the type of the weather index and the strike level of the insurance. Our findings are relevant for insurers and insurance regulators as they shed light on the viability of private crop insurance in China.crop insurance, systemic weather risk, hierarchical Archimedean copulas

Clustered Archimax Copulas

When modeling multivariate phenomena, properly capturing the joint extremal behavior is often one of the many concerns. Archimax copulas appear as successful candidates in case of asymptotic dependence. In this paper, the class of Archimax copulas is extended via their stochastic representation to a clustered construction. These clustered Archimax copulas are characterized by a partition of the random variables into groups linked by a radial copula; each cluster is Archimax and therefore defined by its own Archimedean generator and stable tail dependence function. The proposed extension allows for both asymptotic dependence and independence between the clusters, a property which is sought, for example, in applications in environmental sciences and finance. The model also inherits from the ability of Archimax copulas to capture dependence between variables at pre-extreme levels. The asymptotic behavior of the model is established, leading to a rich class of stable tail dependence functions.Comment: 42 pages, 10 figure

The multivariate Piecing-Together approach revisited

The univariate Piecing-Together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. A multivariate extension was established by Aulbach et al. (2012a): The upper tail of a given copula C is cut off and replaced by a multivariate GPD-copula in a continuous manner, yielding a new copula called a PT-copula. Then each margin of this PT-copula is transformed by a given univariate distribution function. This provides a multivariate distribution function with prescribed margins, whose copula is a GPD-copula that coincides in its central part with C. In addition to Aulbach et al. (2012a), we achieve in the present paper an exact representation of the PT-copula's upper tail, giving further insight into the multivariate PT approach. A variant based on the empirical copula is also added. Furthermore our findings enable us to establish a functional PT version as well.Comment: 12 pages, 1 figure. To appear in the Journal of Multivariate Analysi