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 detectingsmall and moderate shifts in data modeled by a cubic trend AR model when assessing theExtended EWMA chart in both analytical and process control to determine changes in processbehavior. Monitoring sensitivity is affected by control limit structure. The symmetrictwo-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 directiononly. Explicit run-length analysis is used to improve the shift-detection performance ofthe Extended ExponentiallyWeighted Moving Average (Extended EWMA) control chartfor an autoregressive model with cubic trend (Cubic trend AR) under asymmetric andsymmetric control-limit schemes. Average Run Length (ARL) of the Extended EWMAchart is given by an explicit formula for the Cubic trend AR model with exponential whitenoise. The analytical expression is compared with the trapezoidal, Simpson, and Boolequadrature numerical integral equation (NIE) approximations for accuracy verification.All three NIE techniques and their explicitly stated formulas yield excellently consistentresults with a percentage accuracy (%Acc) of approximately 100%. The explicit run-lengthformula 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 chartby employing run-length efficiency (ARL, MRL, SDRL), and overall efficiency (relativeindex, mean, median, standard deviation) was analyzed. Numerical results indicate thatthe Extended EWMA chart is faster and more consistent than the EWMA chart in detectingsmall 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 closingprices of Tesla, Inc. (TSLA). This setting detects both upward and downward priceswings. The empirical results indicate that the extended EWMA chart detects the shiftsfaster than the EWMA chart. The results indicate that the explicit analytical framework isan alternative for detecting small and moderate shifts under data with a cubic trend ARmodel for assessing the Extended EWMA chart in asymmetric one-sided and symmetrictwo-sided monitoring.