Enhancing the Ability of the EWMA Control Chart to Detect Changes in the Mean of a Time-Series Model

We improved the ability of the exponentially weighted moving average (EWMA) control chart to detect small shifts in the mean of a long-memory fractionally integrated autoregressive process with an exogenous variable under exponential white noise. We first designed the structure of the control chart and then evaluated its performance in terms of the average run length (ARL) via a simulation study. We first derived an analytical ARL using explicit formulas by solving integral equations and an approximated ARL derived by utilizing the numerical integral equation approach. Banach's fixed-point theorem proved that the analytical ARL exists and is unique. We then compared the out-of-control ARL values using both methods via a simulation study; the out-of-control ARL results for the analytical and approximated ARLs were similar. Moreover, the methods provided comparable accuracy in terms of the percentage difference in expected ARL and standard deviation of the run length. However, the explicit formula approach proved to be more advantageous in terms of faster computational speed and is thus recommended in this situation. An illustrative example using real data is also provided to demonstrate the practicability of the analytical ARL method.