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ARL-unbiased CUSUM schemes to monitor binomial counts

Morais, M. C. ; Knoth, S.; Cruz, C. J.; Weiß, C. H.

Proceedings of the XIII International Workshop on Intelligent Statistical Quality Control, (2019), 123-145

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 pro- cess 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 swiftly detect both increases and decreases in the mean of independent and identically distributed and first-order autoregressive (AR(1)) binomial counts. These schemes take longer (in average) to trigger a false alarm than to detect any shifts in the process mean 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.
We also discuss the extension of this ARL-unbiased design for other independent and autocorrelated types of counts.