Speaker: Alfred Müller, Universität Siegen
There is an increasing interest in recent years in methods for assessing the quality of probabilistic forecasts by so called scoring rules. For forecasting general multivariate distributions, however, there are only a very few scoring rules that are considered in the literature. In their fundamental paper, Gneiting and Raftery (2007) considered the so called energy score as an example of a scoring rule that is strictly proper for arbitrary multivariate distributions. Pinson and Tastu (2013) started a debate on the discrimination ability of this scoring rule with respect to the dependence structure.In this paper we want to contribute to this discussion by deriving dependence uncertainty bounds for the energy score and the related multivariate Gini mean difference. This means that we derive bounds for the score under the assumption that we only know the marginals of the distributions, but do not know anything about the dependence structure, i.e. the copula. Using methods from stochastic orderings we will derive some analytical bounds that are sharp in some cases. In other cases we will derive interesting numerical bounds by using a variant of a swapping algorithm. It turns out that some of these bounds are attained for some non-standard copulas that are of interest in their own right.
The talk is based on joint work with Carole Bernard (Grenoble) and Marco Oesting (Siegen).