The CUSUM control chart is a popular method used to monitor the performance of production processes. The performance of the CUSUM is generally evaluated with the assumption that the process parameters are known. In practice, the parameters are rarely known and are frequently replaced with estimates from an incontrol reference sample. We discuss the run length distribution of the CUSUM with estimated parameters and provide a method for approximating this distribution and moments. We evaluate the performance of the CUSUM with estimated parameters in a variety of practical situations.
Key Words: Average Run Length; Control Charts; Cumulative Sum.
By L. ALLISON JONES, Auburn University, Auburn, AL 36849-5247
CHARLES W. CHAMP, Georgia Southern University, Statesboro, GA 30460-8093
STEVEN E. RIGDON, Southern Illinois University, Edwardsville, IL 60206-1653
SINCE its introduction by Page (1954), the cumulative sum (CUSUM) control chart has become a popular tool for monitoring the performance of industrial processes. Over the past half century, the CUSUM and its properties have received much attention in the statistical literature. Of particular interest to us is the literature that discusses the run length distribution and the average run length (ARL) of the CUSUM chart for monitoring the process mean. The run length of a control chart is a discrete random variable that is defined as the number of plotted statistics before a signal is observed, and the ARL is the expected value of this random variable. Many authors have studied various aspects of the run length distribution of the CUSUM. An admittedly inexhaustive list includes Kemp (1961), Brook and Evans (1972), Woodall (1983), Gold (1989), and Luce~no and Puig-Pey (2000). Fellner (1990) provided an algorithm for computing the ARL, and Gan (1993) gave an algorithm for computing the run length distribution of the CUSUM. Several authors, including Goel and Wu (1971), Woodall (1986), and Gan (1991), have discussed procedures for designing CUSUM control charts. Hawkins and Olwell (1998) gave a very thorough discussion of many aspects of the CUSUM control chart.
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