This study addresses the impact of estimating in-control parameters on the performance of the Poisson cumulative sum (CUSUM) chart in detecting an increase in the mean when the counts are small. To reduce false signals induced by estimation uncertainty, recommendations prescribe impractically large sample sizes or propose methods that widen control limits. However, these methods are ineffective for small Poisson counts. We describe three approximate methods from the literature and introduce a novel exact approach that requires no heuristics or approximations, ensuring guaranteed in-control performance. The theoretical foundation behind this approach is generalizable to other distributions, such as other one-parameter distributions of the exponential family. Simulations confirm that existing methods fail to achieve the desired in-control performance, unlike the exact approach. We offer recommendations on how domain knowledge can often be incorporated into the exact approach to improve out-of-control performance. Finally, we validate the practical effectiveness of our proposed method through retrospective monitoring of chikungunya infections among returning international travelers reported to the GeoSentinel network. In our case study, the knowledge-based exact method successfully detects nine major outbreaks at the cost of only one false alarm. Existing methods show a lower positive predictive value for predicting major outbreaks.
Heidema et al. (Wed,) studied this question.