Statistical Design of a One-sided CUSUM Control Chart to Detect a Mean Shift in a FIMAX Model with Underlying Exponential White Noise
Keywords:
Average Run Length (ARL), integral equation, fractionally integrated moving average process with exogenous variableAbstract
Control charts comprise an important statistical technique used to monitor the quality of a process, of which the cumulative sum (CUSUM) chart is effective at detecting small-to-moderate shifts in the parameter of interest of a process such as a fractionally integrated moving average with exogenous variables (FIMAX) model with underlying exponential white noise. When a specific size of the mean shift is assumed, the CUSUM control chart can be optimally designed in terms of the average run length (ARL). Herein, the ARL is derived by using analytical integral equations as explicit formulas with proven existence and uniqueness based on Banach's fixed-point theorem. This approach was compared with the ARL derived by using the numerical integral equation (NIE) method for out-of-control situations. In simulations studies, the precision of the proposed ARL method based on explicit formulas achieved the same accuracy as the NIE method in terms of percentage accuracy for a wide variety of out-of-control situations. Meanwhile, the computational time required for the explicit formulas was far shorter than for the NIE method. Furthermore, we illustrate the practicability of the explicit formulas method by using real data following a FIMAX model with exponential white noise running on a CUSUM control chart.
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