The cumulative sum (CUSUM) technique is well-established in theory and practice of process control. For a variant of the CUSUM technique, the cumulative score chart, we investigate the effect of serial correlation on the in-control average run length (ARL). The Shewhart chart is a special case of the cumulative score chart. Using the fact that the cumulative score statistic is a correlated random walk with a reflecting and an absorbing barrier, we derive an approximate but closed-form expression for the ARL of a control variable that follows a first-order autoregressive process with normally distributed disturbances. We also give an expression for the asymptotic (large in-control ARL) case. Our method of approximation gives ARL values that are in good agreement with Monte Carlo estimates of the true values. For positive serial correlation the ARL decreases with increasing value of the correlation coefficient. For increasing negative serial correlation, the ARL may decrease or increase depending on the choice of the parameters of the chart; parameterizations can be found which are rather insensitive for negative serial correlation. We use our results to give recommendations on how to modify the control chart procedure in the presence of serial correlation.
Böhm, W., & Hackl, P. (1996). The effect of serial correlation on the in-control average run length of cumulative score charts. Journal of Statistical Planning and Inference, 54(1), 15–30. https://doi.org/10.1016/0378-3758(95)00153-0