A novel lifetime distribution is presented, derived from compounding the Poisson and two-parameter Chris-Jerry distributions. Various of its statistical characteristics are determined, encompassing aspects such as the moments, probability generating function, and hazard rate function. Statistical inference regarding the model parameters is explored through maximum likelihood estimation. A simulation study is carried out to evaluate the bias and mean square error of the estimated values. The importance of the proposed distribution is illustrated within the framework of an integer-valued first-order autoregressive process, which we shall call the INAR(1) process. Additionally, a simulation study is conducted employing the conditional maximum likelihood and conditional least squares methods. The practical significance of the proposed model is validated through its application to real datasets. In the realm of statistical quality control, we devise a cumulative sum control chart for monitoring the process mean. To illustrate its applicability, we conduct both simulation and real-data analysis.
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