Investigations on the discrimination ability of multivariate scoring rules

Probabilistic forecasts in form of predictive distributions over future quantities have become more and more important in many different fields, including meteorology, hydrology, epidemiology and economics.
Along with the growing prevalence of probabilistic models the need for tools to evaluate the appropriateness of models and forecasts emerges. Various measures have been developed to address this topic.
In their seminal paper, Gneiting and Raftery (2007) study so called proper scoring rules as summary measures to evaluate probabilistic forecasts by assigning a single numerical score based on the predictive distribution and the event that materializes. Such a proper scoring rule encourages the forecaster to make careful assessments.
For univariate quantities there is a vast selection of proper scoring rules and their properties are understood quite well. However, for multivariate quantities there is only a small number of proper scoring rules available and there does not yet exist much research on their properties. The most prominent example of a strictly proper multivariate scoring rules is the energy score.
A scoring rule should not only be strictly proper, but it should also assign significantly different score values to probabilistic forecasts of models that are significantly wrong. This property is refered to as discrimination ability of a scoring rule.
To assess the discrimination ability of a scoring rule we use the Diebold-Mariano test, which is a crucial element in the evaluation of scoring rules. In this thesis we mainly focus on the discrimination ability of scoring rules for multivariate distributions with a special emphasis on the correct modelling of the dependence structure. The energy score has been criticized for its poor ability to distinguish between forecasting distributions with different dependence structure, whereas it detects very well errors in location and scale.
The discrimination ability of the energy score depends on the choice of the parameter β, which is very often fixed to 1. However, β can be chosen to be any value in the interval (0, 2). Thus, the main topic of this thesis is to study the discrimination ability of the energy score for various choices of the parameter β and to compare it with other known scoring rules like the Dawid-Sebastiani score and the variogram score. An extensive simulation study shows that the discrimination ability of the energy score typically improves with smaller parameter β. Therefore, a new multivariate strictly proper scoring rule is introduced that arises as a scaling limit of the energy score as β tends to zero and its properties are studied.