Explicit Closed-Form Expression for Run-Length Evaluation of the Double-Modified EWMA Control Chart Under ARX and ARFIX Models: Application to Major Crude Oil Benchmarks

Control charts are used in statistical process control (SPC) to keep track of processes and identify changes in the way that they work. The control limits around the center line are uniform, which means they react the same way to changes going in either direction. In contrast, linear charts use imbalance to make it easier to identify individual data points. Therefore, using imbalance in the creation of control charts helps keep track of and maintain consistency with data that has a big impact on results when it goes beyond predetermined limits. In this study, we look at both one-sided and two-sided control charts by getting an explicit closed-form formula for the double-modified EWMA control chart’s average run length (ARL). The study is mostly about developing better ways to spot things using autoregressive fractionally integrated models and external variables (ARX and ARFIX) in the presence of exponential white noise. The ARL is used to test how well the proposed chart works in both modeling systems. The NIE method is used to prove that the explicit closed-form ARL formula works. The closed-form ARL expression is shown to be valid under the given ARX and ARFIX model assumptions, exponential white noise errors, stationarity conditions, and fixed one-sided or two-sided control limits. The results show that %RPC has a value below 10⁻⁶, and the computation times for the ARX and ARFIX models remain below 1.6 s and 3 s, respectively, after that point. To show how much better it is, the suggestion is compared to Type-EWMA control charts, such as classical and modified EWMA charts, in terms of run-length efficiency using ARL and SDRL, as well as overall efficiency by the relative index and with mean and standard deviation. The simulation study checks how well the proposed chart works in both symmetric two-sided and asymmetric one-sided frameworks. For the crude oil application, the one-sided upper control chart is used to detect abrupt upward price shifts, which may indicate precautionary demand shocks, market uncertainty, and risk spillovers to financial markets. According to the findings, the suggested chart is able to identify shifts at a faster rate than both traditional EWMA charts and modified EWMA charts, which demonstrates that it is beneficial in a real setting.