AI 中文总结
研究集体风险模型中可信度排序问题，指出标准模型规范可能违反单调性，通过给出反例说明小附加索赔可能降保费，进而形式化适用于该模型的可信度排序及充分条件，并进行数值研究和实证说明。
AI 中文摘要
集体风险模型是保险费率制定中通过结合索赔频率和索赔严重程度来对总损失进行建模的基本框架。可靠的经验评级系统的一个关键结构要求是单调排序属性：过去经验较差的投保人应得到对未来损失的随机更大预测，从而保费更高。经典随机效应模型下单变量结果的这种可信度型单调性在保险文献中有充分记载。然而，将此原则扩展到集体风险模型并非易事，因为相关历史本质上是多变量的，涉及索赔次数和个体索赔金额。因此，尽管其具有实际重要性，但尚未系统地为总损失的预测分布开发相应的可信度型排序。本文给出一个例子表明标准集体风险模型规范可能违反单调性：添加一个额外但足够小的索赔可能会降低保费，从而产生战略报告的不良激励并可能破坏经验评级的完整性。鉴于保险中的这种病态情况，我们形式化了一种适用于集体风险模型的可信度排序，并得出易于处理的充分条件，在这些条件下总损失的预测分布在过去经验中是单调的，排除了这种病态违反情况。我们的理论结果还伴有数值研究和使用真实保险数据的实证说明。
英文摘要
The collective risk model is a fundamental framework in insurance ratemaking for modeling aggregate losses by combining claim frequency and claim severity components. A key structural requirement for a reliable experience rating system is a monotone ordering property: policyholders with worse past experience should receive a stochastically larger prediction for future losses, and hence a higher premium. Such credibility-type monotonicity is well documented in the insurance literature for univariate outcomes under classical random-effect models. However, extending this principle to the collective risk model is nontrivial because the relevant history is inherently multivariate, involving both claim counts and individual claim amounts. Hence, despite its practical importance, a corresponding credibility-type ordering has not been systematically developed for predictive distributions of aggregate loss. In this paper, we provide an example showing that standard collective risk model specifications can violate monotonicity: adding an additional but sufficiently small claim may decrease the premium, thereby creating perverse incentives for strategic reporting and potentially undermining the integrity of experience rating. Motivated by this pathology in view of insurance, we formalize a credibility order tailored to collective risk models and derive tractable sufficient conditions under which the predictive distribution of aggregate loss is monotone in past experience, ruling out such pathological violations. Numerical studies and an empirical illustration using real insurance data accompany our theoretical results.
