Average Run Length in TEWMA Control Charts: Analytical Solutions for AR(p) Processes and Real Data Applications

This study aims to examine the explicit solution for calculating the Average Run Length (ARL) on the triple exponentially weighted moving average (TEWMA) control chart applied to autoregressive model (AR(p)), where AR(p) is an autoregressive model of order p, representing a time series with dependencies on its p previous values. Additionally, the study evaluates the accuracy of both explicit and numerical integral equation (NIE) solutions for AR(p) using the TEWMA control chart, focusing on the absolute percentage relative error. The results indicate that the explicit and approximate solutions are in close agreement. Furthermore, the study investigates the performance of exponentially weighted moving average (EWMA) and TEWMA control charts in detecting changes in the process, using the relative mean index (RMI) as a measure. The findings demonstrate that the TEWMA control chart outperforms the EWMA control chart in detecting process changes, especially when the value of λ is sufficiently large. In addition, an analysis using historical data from the SET index between January 2024 and May 2024 and historical data of global annual plastic production, the results of both data sets also emphasize the superior performance of the TEWMA control chart.