Modeling Over-dispersed Count Time Series via the Poisson X-Exponential INAR(1) Process: Theory, Estimation, and Applications

Abstract

This paper proposes a novel integer-valued autoregressive process of or-der one, the PXED-INAR(1), based on the recently introduced Poisson X-Exponential Distribution (PXED). This new model provides a flexible frame-work for analyzing discrete time series data that exhibit overdispersion and skewness, features commonly observed in fields such as epidemiology, finance, and insurance.

We derive key statistical properties of the PXED-INAR(1) process, in-cluding stationarity, moments, autocorrelation, and conditional distributions. Parameter estimation is performed using three methods: Conditional Least Squares (CLS), Conditional Maximum Likelihood Estimation (CMLE), and the Method of Moments (MM). A comprehensive Monte Carlo simulation study compares the estimators’ bias and root mean squared error under various scenarios.

The practical relevance of the model is illustrated using real-world count data. The PXED-INAR(1) process is benchmarked against classical and con-temporary models, including Poisson-INAR(1), Poisson-Lindley-INAR(1), and NXLD-INAR(1). Results show that the PXED-INAR(1) model consistently achieves better fit and predictive accuracy based on likelihood-based criteria and forecast errors.