Alternative to Detecting Changes in the Mean of an Autoregressive Fractionally Integrated Process with Exponential White Noise Running on the Modified EWMA Control Chart

The modified exponentially weighted moving-average (modified EWMA) control chart is an improvement on the traditional EWMA control chart. Herein, we provide more details about the modified EWMA control chart using various values of an additional design parameter for detecting small-to-moderate shifts in the process mean of an autoregressive fractionally integrated (ARFI(p, d)) process with exponential white noise running thereon. The statistical performances of the two charts were evaluated in terms of the average run length (ARL) obtained by solving integral equations (IEs). This provides an exact formula with proven existence and uniqueness verified by applying Banach's fixed-point theorem. The accuracy of the proposed formula for the ARL was compared with the ARL derived by using the numerical IE technique for the out-of-control state. Although their accuracies were identical for various out-of-control situations and long-term memory processes, the exact formula method required less than 0.01 s to compute the ARL whereas the numerical IE method took 3-4 s. The strengths of using the exact formula are that it is simple to calculate and the computational time is significantly reduced. Comparing their standard deviations of the run length and median run lengths yielded the same results. Finally, practical examples with real-life data corresponding to ARFI(p, d) processes with exponential white noise are provided to demonstrate the applicability of the proposed exact formula.