Scoring rules with normalized upper order statistics for tail inference

Abstract This paper proposes a scoring-rule-based method for ranking predictive distributions in the Fréchet domain that is able to distinguish between different tail indices. The approach is built on normalized order statistics and exploits proper scoring rules to compare tail limit distributions in a distributional framework, with direct relevance for insurance claim-severity tails. On the theoretical side, consistency and asymptotic normality for empirical tail scores based on normalized upper order statistics are obtained through residual estimation theory. Simulation results demonstrate that the scoring-rule-based approach is capable of discriminating between different tail behaviors in finite samples and that systematic scale variation has only a minor impact on stability. We further show that optimizing scoring rules yields consistent tail-index estimators and that the classical Hill estimator arises as a special case. The performance of the tail-index estimator based on the Energy score is investigated and compared with the Hill estimator across a range of tail indices. Lastly, we analyze an automobile claim-severity data set to demonstrate how scoring rules can be used to rank predictive models based on tail predictions in actuarial settings.