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, πt. 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. Linear predictors with autoregressive and moving average (ARMA) type terms, covariates, seasonality and trend are fitted to λt and πt 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.