CUSUM Chart for Correlated Control Variables

The cumulative sum (CUSUM) technique is well-established in theory and practice of process control. A comprehensive exposition of the method is given, e.g., by Wetherill and Brown (1991). A question that is seldom treated in the literature is that on the effect of serial correlation of the control variable. Johnson and Bagshaw (1974) investigate the effect of correlation on the run length distribution when the control variable follows a first order autoregressive or moving average process. They also give an approximate expression for the average run length of the CUSUM- technique for correlated control variables. In this paper we derive an exact expression for the average run length of a discretized CUSUM-technique, i.e., a technique that uses a scoring system for the observations of the control variable. The scoring system is that suggested by Munford (1980). Our results are derived for a control variable that is assumed to follow a first order autoregressive process and with normally distributed disturbances. After deriving in Section 2 the expression for the average run length we discuss its dependence on the process parameter and give a numerical illustration. In Section 3 we discuss corrections for the CUSUM-technique in order to keep the nominal risk for an out-of-control decision and compare our results with those given by Johnson and Bagshaw (1974). (author's abstract)