Development of average run length formulas for EWMA control chart under the AR(1) with quadratic trend model for detecting and monitoring process variability

This paper proposes an explicit formula for calculating the average run length (ARL) of an exponentially weighted moving average (EWMA) control chart for monitoring process mean shifts.The study focuses on the AR(1) model with quadratic trends, because this model combines two essential components in time series analysis including the autocorrelation and long-term trend analysis.It is useful for producing accurate and insightful data predictions.Additionally, it is used in various contexts such as economic forecasting, financial analysis, and environmental studies.Furthermore, the study presents a technique for estimating the ARL using the numerical integral equation (NIE) method.This enables a comparison between the results of the explicit formula and the numerical integral equation method.The two ARL solutions obtained from the explicit formula and numerical integral equation method are very similar identical with an absolute percentage difference of less than 0.001.Thereby, the explicit formula accurately corresponds to the NIE method.Additionally, the explicit formulas are more computationally efficient as they require fewer computations compared to the NIE approach