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Abstract

In this paper, we provide a tutorial on multivariate extreme value methods which allows to estimate the risk associated with rare events occurring jointly. We draw particular attention to issues related to extremal dependence and we insist on the asymptotic independence feature. We apply the multivariate extreme value theory on two data sets related to hydrology and meteorology: first, the joint flooding of two rivers, which puts at risk the facilities lying downstream the confluence; then the joint occurrence of high speed wind and low air temperatures, which might affect overhead lines.

Abstract

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.

Abstract

We show that each infinite exchangeable sequence τ1, τ2, . . . of random variables of the generalised Marshall–Olkin kind can be uniquely linked to an additive subordinator via its deFinetti representation. This is useful for simulation, model estimation, and model building.

Abstract

Motivated by the nice characterization of copulas A for which d(A, At) is maximal as established independently by Nelsen [11] and Klement & Mesiar [7], we study maximum asymmetry with respect to the conditioning-based metric D1 going back to Trutschnig [12]. Despite the fact that D1(A, At) is generally not straightforward to calculate, it is possible to provide both, a characterization and a handy representation of all copulas A maximizing D1(A, At). This representation is then used to prove the existence of copulas with full support maximizing D1(A, At). A comparison of D1- and d-asymmetry including some surprising examples rounds off the paper.

Abstract

In this paper we discuss a natural extension of infinite discrete partition-of-unity copulas which were recently introduced in the literature to continuous partition of copulas with possible applications in risk management and other fields. We present a general simple algorithm to generate such copulas on the basis of the empirical copula from high-dimensional data sets. In particular, our constructions also allow for an implementation of positive tail dependence which sometimes is a desirable property of copula modelling, in particular for internal models under Solvency II.

Abstract

In this paper, the scale mixtures of multivariate skew slash distributions is introduced. The probability density function with some additional properties are discussed. The first four order moments, skewness and kurtosis of this distribution are calculated. Furthermore, the first two moments of its quadratic forms are obtained. In particular, the linear transformation, stochastic representation and hierarchical representation are studied. In the end, the EM algorithm is proposed.

Abstract

For functions of several variables there exist many notions of monotonicity, three of them being characteristic for resp. distribution, survival and co-survival functions. In each case the “degree” of monotonicity is just the basic one of a whole scale.

Copulas are special distribution functions, and stable tail dependence functions are special co-survival functions. It will turn out that for both classes the basic degree of monotonicity is the only one possible, apart from the (trivial) independence functions. As a consequence a “nesting” of such functions depends on particular circumstances. For nested Archimedean copulas the rather restrictive conditions known so far are considerably weakened.

Abstract

We show that any distribution function on ℝd with nonnegative, nonzero and integrable marginal distributions can be characterized by a norm on ℝd+1, called F-norm. We characterize the set of F-norms and prove that pointwise convergence of a sequence of F-norms to an F-norm is equivalent to convergence of the pertaining distribution functions in the Wasserstein metric. On the statistical side, an F-norm can easily be estimated by an empirical F-norm, whose consistency and weak convergence we establish.

The concept of F-norms can be extended to arbitrary random vectors under suitable integrability conditions fulfilled by, for instance, normal distributions. The set of F-norms is endowed with a semigroup operation which, in this context, corresponds to ordinary convolution of the underlying distributions. Limiting results such as the central limit theorem can then be formulated in terms of pointwise convergence of products of F-norms.

We conclude by showing how, using the geometry of F-norms, we may characterize nonnegative integrable distributions in ℝd by simple compact sets in ℝd+1. We then relate convergence of those distributions in the Wasserstein metric to convergence of these characteristic sets with respect to Hausdorff distances.

Abstract

We revisit Sklar’s Theorem and give another proof, primarily based on the use of right quantile functions. To this end we slightly generalise the distributional transform approach of Rüschendorf and facilitate some new results including a rigorous characterisation of an almost surely existing “left-invertibility” of distribution functions.

Abstract

This paper proposes a Bayesian estimation algorithm to estimate Generalized Partition of Unity Copulas (GPUC), a class of nonparametric copulas recently introduced by [18]. The first approach is a random walk Metropolis-Hastings (RW-MH) algorithm, the second one is a random blocking random walk Metropolis-Hastings algorithm (RBRW-MH). Both approaches are Markov chain Monte Carlo methods and can cope with ˛at priors. We carry out simulation studies to determine and compare the efficiency of the algorithms. We present an empirical illustration where GPUCs are used to nonparametrically describe the dependence of exchange rate changes of the crypto-currencies Bitcoin and Ethereum.