2,180 research outputs found

    Ensemble Copula Coupling as a Multivariate Discrete Copula Approach

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    In probability and statistics, copulas play important roles theoretically as well as to address a wide range of problems in various application areas. In this paper, we introduce the concept of multivariate discrete copulas, discuss their equivalence to stochastic arrays, and provide a multivariate discrete version of Sklar's theorem. These results provide the theoretical frame for the ensemble copula coupling approach proposed by Schefzik et al. (2013) for the multivariate statistical postprocessing of weather forecasts made by ensemble systems.Comment: references correcte

    Robust risk aggregation with neural networks

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    We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g. the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: Firstly, we derive a dual representation of the considered problem and prove that strong duality holds. Secondly, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real world instance.Comment: Revised version. Accepted for publication in "Mathematical Finance

    Approximating multivariate distributions with vines

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    In a series of papers, Bedford and Cooke used vine (or pair-copulae) as a graphical tool for representing complex high dimensional distributions in terms of bivariate and conditional bivariate distributions or copulae. In this paper, we show that how vines can be used to approximate any given multivariate distribution to any required degree of approximation. This paper is more about the approximation rather than optimal estimation methods. To maintain uniform approximation in the class of copulae used to build the corresponding vine we use minimum information approaches. We generalised the results found by Bedford and Cooke that if a minimal information copula satis¯es each of the (local) constraints (on moments, rank correlation, etc.), then the resulting joint distribution will be also minimally informative given those constraints, to all regular vines. We then apply our results to modelling a dataset of Norwegian financial data that was previously analysed in Aas et al. (2009)
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