Cumulative Sum Control Chart Applied to Monitor Shifts in the Mean of a Long-memory ARFIMAX(p,d*,q,r) Process with Exponential White Noise

The aim of this study is to derive the average run length (ARL) for detecting a changes in the process mean of a long-memory autoregressive fractionally integrated moving-average model with exogenous variables (ARFIMAX(p, d*, q, r)) process with exponential white noise on a cumulative sum (CUSUM) control chart. ARLs are derived using explicit formulas and the numerical integral equation (NIE) method, which is the solution for the integral equation. Moreover, proof of the existence and uniqueness of the proposed ARL based on Banach's fixed-point theorem are presented. The performances of the two ARLs were evaluated in terms of accuracy and computational time for monitoring shifts in the process mean for an ARFIMAX (p, d*, q, r) process on a CUSUM control chart. The results reveal that although their accuracies were similar, the explicit formula method consumed less computational time than the NIE method and so is recommended as a good alternative for this scenario.