<p>This paper explores the challenges in the first-order integer-valued autoregressive process, particularly focusing on the incorporation of distribution assumptions within the marginals. The traditional dilemma of identifying the distribution of innovations is a critical factor influencing the transition probability. In response to these challenges, the paper introduces a first-order integer-valued autoregressive process with Poisson new XLindley distributed marginals. Beyond investigating statistical characteristics, explicit consideration of the distribution of the thinning operator and innovations enables a comprehensive assessment of estimation and forecasting methods. The primary objective is to compare the performance of the first-order integer-valued autoregressive process with Poisson new XLindley innovations, shedding light on the efficacy of this distributional assumption in enhancing modeling and predictive capabilities. The findings contribute valuable insights to the ongoing discourse on refining distribution assumptions within the first-order integer-valued autoregressive framework.</p>