Determining the ARL for a Shift in the Mean of a Long-Memory ARFIMA(1, d, 1)(1, D, 1)s Process with Exponential White Noise Running on a CUSUM Control Chart

Abstract

The CUSUM control chart is a well-known statistical process monitoring tool that is sensitive to small-to-moderate shifts in a process parameter. Herein, we provide a solution for the average run length (ARL) for a shift in the mean of a long-memory  process with exponential white noise running on a CUSUM control chart based on the analytical integral equation (AIE) approach. Its existence and uniqueness were proven by applying Banach's fixed-point theorem. The numerical integral equation (NIE) method was used to verify the accuracy of the proposed AIE approach by comparing the percentage accuracies for various out-of-control ARL situations. We also recorded their computational times. The results of using the AIE and NIE approaches are in close agreement (percentage accuracy > 95%) but the time required to compute the ARL using the former was significantly shorter. A major advantage of using the AIE approach was in finding the optimal values for reference value (k) and control limit (b) very quickly when the process is in-control. Moreover, their optimal values for the minimum out-of-control ARL for the mean of a long-memory   process with exponential white noise running on a CUSUM control chart are also derived. An illustrative example with real data is given to illustrate the practicability of the proposed method. The practical scenario involves analysing natural gas futures prices using the AIE method on the CUSUM control chart in order to track changes in stock prices, which is the point in making profitable decisions for investors.