Scoring Rules with Normalized Upper Order Statistics for Tail Inference

2026-03-25

AI Summary

This study addresses the challenge of discriminating and ranking predictive models based on their ability to capture tail behavior in heavy-tailed settings, such as insurance claim severity. The authors propose a novel tail scoring rule grounded in normalized upper order statistics, which, for the first time, integrates this statistic with proper scoring rules for tail inference. Theoretical analysis establishes the consistency and asymptotic normality of the proposed score and demonstrates that the classical Hill estimator arises as a special case. Simulations confirm the method’s effectiveness in distinguishing between distinct tail behaviors, while an application to real-world auto insurance claims data successfully enables comparative assessment and ranking of competing models according to their tail predictive performance.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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 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.

Problem

• tail inference
• scoring rules
• tail index
• predictive distributions
• extreme value theory

Innovation

• scoring rules
• tail inference
• normalized upper order statistics
• Fréchet domain
• Hill estimator