RUN-LENGTH DISTRIBUTIONS OF SPECIAL-CAUSE CONTROL CHARTS FOR CORRELATED PROCESSES

We derive run-length distributions of the special-cause control chart proposed by Alwan and Roberts for correlated observations, given that the assignable cause to be detected is a shift in the process mean. Bo th recursive and closed-form solutions are derived for the run-length distribution, average run length (ARL), and standard deviation of the run length (SRL) for any AR(p) process, and approximate solutions are derived for the more general ARMA(p,q) processes. The expressions deri ved do not depend on the type of shift in the process mean. Numerical results are illustrated for the ARL and SRL of the ARMA(1,1) model, gi ven that the shift in the mean is a step shift. These results show tha t the ARL and SRL of the special-cause control chart are relatively sm aller when the process is negatively rather than positively autocorrel ated. Regardless of the sign of the autocorrelation, the shape of the probability mass function of the run length reveals that the probabili ty of detecting shifts very early is substantially higher for the spec ial-cause chart than for more traditional control charts. Early detect ion makes the cause of the signal easier to identify, resulting in a m ore rapid rate of continuous quality improvement. There are some cases , however, when traditional charts, which are simpler to implement, sh ould be considered even when the process is autocorrelated.