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

Un'escursione nell'universo in alta dimensione

Ben installati nel nostro mondo in 3D, ci è difficile immaginare e descrivere un universo in quattro o più dimensioni. Solleviamo il velo esplorando le proprietà del cubo e della sfera in alta dimensione

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

Modelling extreme rain accumulation with an application to the 2011 Lake Champlain flood

A simple strategy is proposed to model total accumulation in non-overlapping clusters of extreme values from a stationary series of daily precipitation. Assuming that each cluster contains at least one value above a high threshold, the cluster sum S is expressed as the ratio S=M/P of the cluster maximum M and a random scaling factor P (0, 1]. The joint distribution for the pair (M, P) is then specified by coupling marginal distributions for M and P with a copula. Although the excess distribution of M is well approximated by a generalized Pareto distribution, it is argued that, conditionally on P<1, a scaled beta distribution may already be sufficiently rich to capture the behaviour of P . An appropriate copula for the pair (M, P) can also be selected by standard rank-based techniques.This approach is used to analyse rainfall data from Burlington, Vermont, and to estimate the return period of the spring 2011 precipitation accumulation which was a key factor in that year’s devastating flood in the RichelieuValley Basin in Qu´ebec, Canada

Un'escursione nell'universo in alta dimensione

Ben installati nel nostro mondo in 3D, ci è difficile immaginare e descrivere un universo in quattro o più dimensioni. Solleviamo il velo esplorando le proprietà del cubo e della sfera in alta dimensione

On the asymptotic covariance of the multivariate empirical copula process

Genest and Segers (2010) gave conditions under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance than the standard empirical process based on a random sample from the underlying copula. An extension of this result to the multivariate case is provided