Improving the ARL profile and the accuracy of its calculation for Poisson EWMA charts

The Poisson exponentially weighted moving average (PEWMA) chart was proposed in 1998 by Borror, Champ, and Rigdon to monitor the mean of counts of nonconformities. This chart regrettably fails to have an in-control average run length (ARL) larger than any out-of-control ARL, ie, the PEWMA chart is ARL-biased. Moreover, due to the discrete character of its control statistic the PEWMA chart and the resulting subtleties of its ARL calculation, it is difficult to set the control limits in such way that the in-control ARL takes a desired value, say ARL0. In this paper, we establish an improved Markov chain technique to approximate the ARL of EWMA for count output; propose two ARL-unbiased counterparts of the PEWMA chart; and use the R statistical software to provide illustrations of these charts with a decidedly superior ARL profile and an in-control ARL equal to ARL0. We also compare the ARL performance of the proposed charts with the one of a few competing control charts for the mean of independent and identically distributed (i.i.d.) Poisson counts.