The Poisson-Lindley Distribution: Some Characteristics, with Its Application to SPC

Statistical process control (SPC) is a significant method to monitor processes and ensure quality. Control charts are the most important tools in SPC. As production processes and production parts become more complex, there is a need to design control charts using more complex distributions. One of the most important control charts to monitor the number of nonconformities in production processes is the C-chart, which uses the Poisson distribution as a quality characteristic distribution. However, to fit the Poisson distribution to the count data, equality of mean and variance should be satisfied. In some cases, such as biological and medical sciences, count data exhibit overdispersion, which means that the variance of data is greater than the mean. In such cases, we can use the Poisson-Lindley distribution instead of the Poisson distribution to model the count data. In this paper, we first discuss some important characteristics of the Poisson-Lindley distribution. Then, we present parametric and bootstrap control charts when the observations follow the Poisson-Lindley distribution and analyze their performance. Finally, we provide a simulated example and a real-world dataset to demonstrate the implementation of control charts. The results show the good performance of the proposed control charts.