The Chain-Ladder (CL) method remains the dominant macro-level technique for claims reserving in non-life insurance, yet its classical formulation lacks a coherent probabilistic foundation. Existing stochastic extensions-including the Mack model and the Over-Dispersed Poisson (ODP) framework-provide measures of uncertainty but rely on second-moment assumptions or quasi-likelihood variance structures without clear generative interpretations. This paper develops a Negative Binomial Chain-Ladder (NB-CL) model that embeds the CL method within a full likelihood-based framework. The key contribution is a micro-level derivation showing that the negative binomial distribution arises naturally from a Poisson-Gamma construction: claims arrive according to a Poisson process with Gamma-distributed accident-year heterogeneity, and aggregation yields negative binomial incremental counts. This derivation gives the dispersion parameter $κ$ a structural interpretation as accident-year heterogeneity, rather than an ad-hoc overdispersion adjustment. The NB-CL model generalises the Poisson Chain-Ladder model in the limit $κ\to \infty$, shares the point estimates of the ODP model while differing in its variance function (quadratic vs. linear), and unifies the Chain-Ladder family within a single probabilistic hierarchy. A parametric bootstrap procedure is developed to incorporate both process and parameter uncertainty. Simulation studies confirm near-nominal coverage under correct specification once the dispersion parameter is bias-corrected, and a controlled degradation under model misspecification. Empirical illustrations on claim count data (Australian motor bodily injury) and paid amounts (Taylor-Ashe) document both the structural reading of $κ$ and the working-approximation status of the model in the amounts case.

翻译：链梯法仍是非寿险索赔准备金估计中最主要的宏观技术，但其经典公式缺乏一致的概率基础。现有的随机扩展——包括Mack模型和过度分散Poisson（ODP）框架——虽能提供不确定性度量，却依赖于二阶矩假设或准似然方差结构，缺乏清晰的生成性解释。本文发展的负二项链梯模型将链梯法嵌入完全似然框架。核心贡献在于微观层面的推导：负二项分布自然地源于Poisson-Gamma构造——索赔按照Poisson过程发生，其中事故年异质性服从Gamma分布，聚合后得到负二项增量计数。该推导赋予离散参数κ结构性的异质性解释，而非临时的过度分散调整。负二项链梯模型在极限κ→∞时推广了Poisson链梯模型，其点估计与ODP模型一致但方差函数不同（二次型vs线性），并将链梯族模型统一于单一概率分层框架中。本文开发了参数自助法以整合过程不确定性和参数不确定性。模拟研究证实：在正确设定下，经离散参数偏差校正后覆盖率接近名义水平；在模型误设下呈现可控退化。基于索赔次数数据（澳大利亚机动车人身伤害）和赔付金额数据（Taylor-Ashe）的实证分析，既展示了κ的结构性解读，也说明了该模型在赔付金额场景下的工作近似状态。