Count data often exhibit overdispersion driven by heavy tails or excess zeros, making standard models (e.g., Poisson, negative binomial) insufficient for handling outlying observations. We propose a novel contaminated discrete Weibull (cDW) framework that augments a baseline discrete Weibull (DW) distribution with a heavier-tail subcomponent. This mixture retains a single shifted-median parameter for a unified regression link while selectively assigning extreme outcomes to the heavier-tail subdistribution. The cDW distribution accommodates strictly positive data by setting the truncation limit c=1 as well as full-range counts with c=0. We develop a Bayesian regression formulation and describe posterior inference using Markov chain Monte Carlo sampling. In an application to hospital length-of-stay data (with c=1, meaning the minimum possible stay is 1), the cDW model more effectively captures extreme stays and preserves the median-based link. Simulation-based residual checks, leave-one-out cross-validation, and a Kullback-Leibler outlier assessment confirm that the cDW model provides a more robust fit than the single-component DW model, reducing the influence of outliers and improving predictive accuracy. A simulation study further demonstrates the cDW model's robustness in the presence of heavy contamination. We also discuss how a hurdle scheme can accommodate datasets with many zeros while preventing the spurious inflation of zeros in situations without genuine zero inflation.

翻译：计数数据常因重尾或过多零值而呈现过度离散性，使得标准模型（如泊松模型、负二项模型）难以有效处理离群观测。本文提出一种新颖的污染离散威布尔（cDW）框架，该框架通过引入重尾子成分对基线离散威布尔（DW）分布进行扩展。该混合模型保留单一的中位数偏移参数以实现统一的回归连接，同时将极端观测结果选择性地分配至重尾子分布。cDW分布通过设置截断限c=1适应严格正值数据，并通过c=0适应全范围计数数据。我们建立了贝叶斯回归框架，并描述了基于马尔可夫链蒙特卡洛抽样的后验推断方法。在住院时长数据（c=1，表示最短住院时长为1天）的应用中，cDW模型能更有效地捕捉极端住院时长并保持基于中位数的连接关系。基于模拟的残差检验、留一交叉验证以及Kullback-Leibler离群点评估均证实，cDW模型相比单成分DW模型具有更强的稳健性，能降低离群点影响并提升预测精度。模拟研究进一步展示了cDW模型在存在严重污染时的稳健表现。我们还讨论了如何通过跨栏方案处理含大量零值的数据集，同时避免在无真实零膨胀情况下产生虚假的零值膨胀效应。