ZOTMPo-INAR(1) Process: Entropy and Modeling of Epidemiological Count Time Series Data

In this article, we propose a first-order integer-valued autoregressive (INAR(1)) model based on binomial thinning, in which the innovation sequence follows a zero-one-two-modified Poisson (ZOTMPo) distribution. The proposed model accommodates overdispersion and allows for inflation or deflation at low-count values, particularly at zero, one, and two, which are commonly observed in public health count time series. We derive the main probabilistic properties of the model and estimate the unknown parameters using the conditional maximum likelihood (CML) method. A Monte Carlo simulation study is conducted to evaluate the finite-sample performance of the estimators. Furthermore, we establish the information-theoretic properties of the model, specifically deriving the Shannon entropy and conditional entropy bounds to quantify the dynamical complexity and predictability of the stochastic process. The practical utility of the model is illustrated using two real-world datasets on dengue fever incidence and Escherichia coli (E. coli) enteritis. Model performance is assessed using standard information criteria and forecast accuracy measures, as well as the Euclidean distance between observed and fitted probabilities for zero, one, and two. Diagnostic checks, including analysis of residual autocorrelation, cumulative periodograms, and jump process behavior, provide further confirmation of the fitted model's adequacy. The results indicate that the proposed ZOTMPo-INAR(1) model provides an effective framework for modeling overdispersed count time series with a modified low-count structure.