Explicit Run Length Evaluation Under Asymmetric One-Sided and Symmetric Two-Sided Control-Limit Schemes for a Cubic Trend AR Model and Its Application

Control charts are necessary for statistical purposes and serve as an alternative for detecting small and moderate shifts in data modeled by a cubic trend AR model when assessing the Extended EWMA chart in both analytical and process control to determine changes in process behavior. Monitoring sensitivity is affected by control limit structure. The symmetric two-sided control limits detect shifts in both upward and downward directions equally, while the asymmetric one-sided control limits are used to detect shifts in a specific direction only. Explicit run-length analysis is used to improve the shift-detection performance of the Extended Exponentially Weighted Moving Average (Extended EWMA) control chart for an autoregressive model with cubic trend (Cubic trend AR) under asymmetric and symmetric control-limit schemes. Average Run Length (ARL) of the Extended EWMA chart is given by an explicit formula for the Cubic trend AR model with exponential white noise. The analytical expression is compared with the trapezoidal, Simpson, and Boole quadrature numerical integral equation (NIE) approximations for accuracy verification. All three NIE techniques and their explicitly stated formulas yield excellently consistent results with a percentage accuracy (%Acc) of approximately 100%. The explicit run-length formula does not require repeated numerical integration and thus saves processing effort. The run-length ability of the extended EWMA chart was compared with the EWMA chart by employing run-length efficiency (ARL, MRL, SDRL), and overall efficiency (relative index, mean, median, standard deviation) was analyzed. Numerical results indicate that the Extended EWMA chart is faster and more consistent than the EWMA chart in detecting small and moderate shifts in the process, whereas large shifts are only slightly different, all of this under both the asymmetric one-sided and symmetric two-sided control limits. The usefulness of the proposed chart is demonstrated by applying it to the monthly closing prices of Tesla, Inc. (TSLA). This setting detects both upward and downward price swings. The empirical results indicate that the extended EWMA chart detects the shifts faster than the EWMA chart. The results indicate that the explicit analytical framework is an alternative for detecting small and moderate shifts under data with a cubic trend AR model for assessing the Extended EWMA chart in asymmetric one-sided and symmetric two-sided monitoring.