A note on an average run length calculation for the EWMA and other charts

In the last decade numerous papers have appeared in the statistical process monitoring literature where the in-control and out-of-control average run length (ARL) profiles of the exponentially weighted moving average chart, and some of its variants, have been approximated under the assumption that the signal event can be represented by a sequence of independent Bernoulli random variables. In this short note we show that these approximations, presented in the literature as being exact, can be very inaccurate when compared with the true ARL values. Moreover, the use of these poor ARL approximations has led to many flawed articles, which should be either corrected or retracted.