Autoregressive and moving average models for zero-inflated count time series

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, ${\pi }_t$. 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 ${\lambda }_t$. Linear predictors with autoregressive and moving average (ARMA) type terms, covariates, seasonality and trend are fitted to ${\lambda }_t$ and ${\pi }_t$ 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.