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 trends in the scaling have only a minor impact on stability. We further show that optimizing scoring rules (equivalently, minimizing the associated loss form) yields consistent tail-index estimators and that the classical Hill estimator arises as a special case. The performance of the proposed method 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.

翻译：本文提出一种基于评分规则的方法，用于对弗雷歇域中的预测分布进行排序，该方法能够区分不同的尾部指数。该方法建立在归一化次序统计量之上，利用适当的评分规则在分布框架下比较尾部极限分布，与保险理赔严重性尾部直接相关。在理论方面，通过残差估计理论，我们得到了基于归一化上阶统计量的经验尾部评分的一致性及渐近正态性。模拟结果表明，基于评分规则的方法能够在有限样本中区分不同的尾部行为，且缩放趋势对稳定性影响较小。我们进一步证明，优化评分规则（等价于最小化其对应的损失函数）可得到一致的尾部指数估计量，而经典的希尔估计量是其特例。所提方法的性能在多个尾部指数下与希尔估计量进行了比较研究。最后，我们分析了一组汽车理赔严重性数据集，演示了在精算场景中如何利用评分规则基于尾部预测对预测模型进行排序。