An Explicit ARL Evaluation of the DHWMA Control Chart for MAX(q,r) Processes Under Symmetric and Asymmetric Control Limits: A Financial Application

Statistical process monitoring for autocorrelated processes with exogenous variables requires control chart methods that can account for both temporal dependence and the distributional characteristics of the process. This study develops an explicit analytical expression for the average run length (ARL) of the Double Homogeneously Weighted Moving Average (DHWMA) control chart for moving average processes with exogenous variables, denoted by MAX(q,r), with exponential white noise. The proposed framework considers both symmetric two-sided control limits and asymmetric one-sided control limits to examine process shifts under different monitoring directions. The asymmetric exponential innovation structure further provides a setting for evaluating the proposed methodology beyond the conventional symmetric distributional assumption. The accuracy of the proposed explicit formulation is evaluated against the Numerical Integral Equation (NIE) method for several MAX(q,r) processes. The results show that the explicit formulation produces ARL values in close agreement with those obtained from the NIE method while substantially reducing the computational time. The DHWMA control chart is subsequently compared with the Homogeneously Weighted Moving Average (HWMA) and cumulative sum (CUSUM) control charts in terms of ARL, standard deviation of run length (SDRL), and median run length (MRL). The simulation results show that the DHWMA chart generally provides lower out-of-control run-length measures than the competing methods. A real-data application using weekly financial data further supports the simulation findings, with the DHWMA chart showing faster detection of process shifts. Overall, the proposed explicit formulation provides an efficient approach for ARL evaluation, while the DHWMA chart demonstrates favorable monitoring performance for MAX(q,r) processes under both symmetric and asymmetric control limit settings.