ARL-unbiased CUSUM schemes to monitor binomial counts

Frontiers in Statistical Quality Control (Vol. 13), (2021), 77-98
https://link.springer.com/chapter/10.1007%2F978-3-030-67856-2_6

Counted output, such as the number of defective items per sample, is often assumed to have a marginal binomial distribution. The integer and asymmetrical nature of this distribution and the value of its target mean hinders the quality control practitioner from dealing with a chart for the process mean with a pre-stipulated in-control average run length (ARL) and the ability to swiftly detect not only increases but also decreases in the process mean. In this paper we propose ARL-unbiased cumulative sum (CUSUM) schemes to rapidly detect both increases and decreases in the mean of independent and identically distributed as well as first-order autoregressive (AR(1)) binomial counts. Any shift is detected more quickly than a false alarm is generated by these schemes and their in-control ARL coincide with the pre-specified in-control ARL. We use the R statistical software to provide compelling illustrations of all these CUSUM schemes.