Zero inflation is a common nuisance while monitoring disease progression over time. This article proposes a new observation-driven model for zero-inflated and over-dispersed count time series. The counts given from the past history of the process and available information on covariates are assumed to be distributed as a mixture of a Poisson distribution and a distribution degenerated at zero, with a time-dependent mixing probability, 𝜋𝑡${\pi }_t$${\pi }_t$. Since, count data usually suffers from overdispersion, a Gamma distribution is used to model the excess variation, resulting in a zero-inflated negative binomial regression model with mean parameter 𝜆𝑡${\lambda }_t$${\lambda }_t$. Linear predictors with autoregressive and moving average (ARMA) type terms, covariates, seasonality and trend are fitted to 𝜆𝑡${\lambda }_t$${\lambda }_t$ and 𝜋𝑡${\pi }_t$${\pi }_t$ through canonical link generalized linear models. Estimation is done using maximum likelihood aided by iterative algorithms, such as Newton-Raphson (NR) and Expectation and Maximization. Theoretical results on the consistency and asymptotic normality of the estimators are given. The proposed model is illustrated using in-depth simulation studies and two disease datasets.

中文翻译：

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零膨胀计数时间序列的自回归和移动平均模型

在监测疾病随时间的进展时，零通货膨胀是一种常见的麻烦事。本文针对零膨胀和过度分散的计数时间序列提出了一种新的观察驱动模型。从过程的过去历史中给出的计数和协变量的可用信息被假设为分布为泊松分布和在零处退化的分布的混合，具有与时间相关的混合概率，𝜋𝑡${\pi }_{吨}$${\pi }_t$. 由于计数数据通常存在过度分散，因此使用 Gamma 分布对过度变化进行建模，从而产生具有均值参数的零膨胀负二项式回归模型𝜆𝑡${\lambda }_{吨}$${\lambda }_t$. 具有自回归和移动平均 (ARMA) 类型项、协变量、季节性和趋势的线性预测器适合𝜆𝑡${\lambda }_{吨}$${\lambda }_t$和𝜋𝑡${\pi }_{吨}$${\pi }_t$通过规范链接广义线性模型。估计是使用迭代算法辅助的最大似然完成的，例如 Newton-Raphson (NR) 和期望与最大化。给出了估计量的一致性和渐近正态性的理论结果。使用深入的模拟研究和两个疾病数据集来说明所提出的模型。