Change-Detected ARIMA (1,1,1) Time Series Using the Approximated and Exact ARL of the MEWMA Scheme

This study proposes an explicit analytical formulation for calculating the Average Run Length (ARL) of a modified Exponentially Weighted Moving Average (MEWMA) control chart applied to an Autoregressive Integrated Moving Average of order (1,1,1) time series model with exponential white noise. The developed explicit formula provides a closed-form solution that enables efficient computation and theoretical insight into the control chart’s performance. The derived ARL values are validated against those obtained using the Numerical Integral Equation (NIE) method, ensuring accuracy and reliability. The comparative analysis shows that the explicit formula yields results that closely match those from the NIE method, with negligible absolute percentage differences while significantly reducing computational time. Simulation studies across various parameter settings and real-world applications, including climate and commodity price datasets, further confirm the consistency and practical utility of the proposed approach.