Estimation of variance and mean by the Bayesian approach on Poisson EWMA (PEWMA) control chart

The Poisson Exponentially Weighted Moving Average (PEWMA) control chart is a widely used statistical tool for monitoring count data in quality control processes. Traditional estimation methods may not fully incorporate prior knowledge, potentially limiting the efficiency of process monitoring. This study introduces a Bayesian approach for estimating the mean and variance in PEWMA charts, leveraging prior distributions to enhance statistical process control. By integrating Bayesian inference, this approach aims to improve the accuracy and stability of parameter estimation, particularly in cases with small sample sizes. A Monte Carlo simulation was conducted to evaluate the sensitivity of the Bayesian PEWMA method under varying sample sizes, smoothing parameters, and prior distributions. The results indicate that Bayesian estimation enhances the stability of mean and variance estimates as sample sizes increase, leading to more reliable control limits. Additionally, higher smoothing parameters result in slightly lower initial mean estimates, yet the overall convergence pattern remains consistent. The study further analyzes control limit behavior and detection performance, demonstrating that Bayesian PEWMA effectively identifies process variations while maintaining stable control thresholds. These findings underscore the advantages of Bayesian inference in adaptive control charting, offering valuable insights into optimizing prior selection for robust variance estimation and improving process monitoring in dynamic environments.