Average run length of the long-memory autoregressive fractionally integrated moving average process of the exponential weighted moving average control chart

摘要

AbstractMeasurement of control chart efficiency by comparison of average run length (ARL) is widely implemented in quality control. The aim of this study is to evaluate the ARL, which is a solution of the integral equation obtained from the Exponential Weighted Moving Average (EWMA) statistic with a long-memory Autoregressive Fractionally Integrated Moving Average (ARFIMA) process. The derivation of the analytical ARL of the EWMA control chart and proof of the existence and uniqueness of the analytical ARL by Fixed Point theory are shown. Moreover, the numerical ARL carried out by the Composite Midpoint Rule technique of the EWMA control chart is demonstrated. A comparison between the analytical and numerical ARL is also illustrated. The findings indicated that analytical ARL of the EWMA control chart is more quickly computational than the numerical ARL. Therefore, the analytical ARL is an alternative method for measuring the efficiency of the EWMA control chart with the long-memory ARFIMA process.