A new forecasting method based on the concept of the profile predictive likelihood function is proposed for discrete‐valued processes. In particular, generalized autoregressive moving average (GARMA) models for Poisson distributed data are explored in detail. Highest density regions are used to construct forecasting regions. The proposed forecast estimates and regions are coherent. Large‐sample results are derived for the forecasting distribution. Numerical studies using simulations and two real data sets are used to establish the performance of the proposed forecasting method. Robustness of the proposed method to possible misspecifications in the model is also studied.

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

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