Accurate Average Run Length Analysis for Detecting Changes in a Long-Memory Fractionally Integrated MAX Process Running on EWMA Control Chart

Numerical evaluation of the average run length (ARL) when detecting changes in the mean of an autocorrelated process running on an exponentially weighted moving average (EWMA) control chart has received considerable attention. However, accurate computation of the ARL of changes in the mean of a long-memory model with an exogenous (X) variable, which often occurs in practice, is challenging. Herein, we provide an accurate determination of the ARL for long-memory models such as the fractionally integrated MAX processes (FIMAX) with exponential white noise running on an EWMA control chart by using an analytical formula based on an integral equation. From a computational perspective, the analytical formula approach is accomplished by solving the solution for the integral equation obtained via the Fredholm integral equation of the second kind. Moreover, the existence and uniqueness of the solution for the analytical formula were confirmed via Banach’s fixed-point theorem. Its efficacy was compared with that of the ARL derived by using the well-known numerical integral equation (NIE) technique under the same circumstances in terms of the ARL percentage accuracy and computational processing time. The percentage accuracy was 100%, which indicates excellent agreement between the two methods, and the analytical formula also required much less computational processing time. An example to illustrate the effectiveness of the proposed approach with a process involving real data running on an EWMA control chart is also provided herein. The explicit formula method offers an accurate determination of the ARL and a new approach for validating its computation, especially for long-memory scenarios running on EWMA control charts.