Random environment Z-valued AR(1) process with poisson difference marginal distributions

This paper introduces a new r states random environment Z-valued autoregressive process of order one with Poisson difference (PD) marginal distributions, abbreviated as RrPDZAR(1) process. The results indicate that this process is distributed as a difference of two independent random environment INAR(1) processes with Poisson marginal distribution. The RrPDZAR(1) process exhibits local ergodic behavior within specific intervals. Additionally, some properties including the conditional expectation and variance, distribution and correlation, are derived. Yule-Walker, four-stage weighted least squares and conditional maximum likelihood estimates are considered and their performances are evaluated in numerical simulations under different cases. Finally, the proposed model is applied to motor vehicle theft data and football match scores, and compared with other models.