This paper presents a novel approach to stochastic mortality modelling by using the Conway--Maxwell--Poisson (CMP) distribution to model death counts. Unlike standard Poisson or negative binomial distributions, the CMP is a more adaptable choice because it can account for different levels of variability in the data, a feature known as dispersion. Specifically, it can handle data that are underdispersed (less variable than expected), equidispersed (as variable as expected), and overdispersed (more variable than expected). We develop a Bayesian formulation that treats the dispersion level as an unknown parameter, using a Gamma prior to enable a robust and coherent integration of the parameter, process, and distributional uncertainty. The model is calibrated using Markov chain Monte Carlo (MCMC) methods, with model performance evaluated using standard statistical criteria such as residual analysis and scoring rules. An empirical study using England and Wales male mortality data shows that our CMP-based models provide a better fit for both existing data and future predictions compared to traditional Poisson and negative binomial models, particularly when the data exhibit overdispersion. Finally, we conduct a sensitivity analysis with respect to prior specification to assess robustness.

翻译：本文提出了一种随机死亡率建模的新方法，即采用 Conway-Maxwell-Poisson (CMP) 分布对死亡计数进行建模。与标准的泊松分布或负二项分布不同，CMP 分布具有更强的适应性，因为它能够刻画数据中不同水平的变异性，这一特性称为离散度。具体而言，该方法能够处理欠离散（变异性低于预期）、等离散（变异性符合预期）以及过离散（变异性高于预期）的数据。我们建立了一个贝叶斯框架，将离散度水平视为未知参数，并采用 Gamma 先验分布，以实现参数不确定性、过程不确定性与分布不确定性的稳健且一致的整合。模型通过马尔可夫链蒙特卡洛 (MCMC) 方法进行校准，并利用残差分析和评分规则等标准统计准则评估模型性能。一项基于英格兰和威尔士男性死亡率数据的实证研究表明，与传统的泊松模型和负二项模型相比，我们基于 CMP 的模型对现有数据和未来预测均提供了更好的拟合效果，尤其是在数据呈现过离散特征时。最后，我们针对先验分布的设定进行了敏感性分析，以评估模型的稳健性。