Approximating the ARL to Monitor Small Shifts in the Mean of an AR Fractionally Integrated with an exogenous variable Process Running on an EWMAControl Chart

Control charts are used to monitor processes and detect changes in a given control scheme. The Exponential Weighted Moving Average (EWMA) control chart is a well-recognized control chart used to detect small changes in parameters. The efficiency of the chart studied is usually achieved using ARL. Approximating ARL using the Gauss-Legendre quadrature method, also known as NIE,. This approach is used to evaluate the ARL of developments, such as explicit formulas because it provides a robust way to validate their validity and accuracy. Moreover, it evaluates the performance of control charts for time series under exponential white noise. Exponential white noise is obtained from a long-memory fractionally integrated AR with exogenous variables or the long-memory ARFIX process. Under the long-memory ARFIX model, the proposed technique compares the control chart's performance to an explicit formula using the criterion of percentage accuracy. The results of the comprehensive numerical study include investigations into a wide range of out-of-control processes and situations. Specifically, the results from the accuracy percentage in all cases are more than 95%, which means that the proposed technique is accurate and completely consistent with the well-defined explicit formula. Therefore, it is recommended that it be used in this situation. There are examples from real data that were found to be consistent with the research results.