A Combined Shewhart-CUSUM Chart with Switching Limit

The common Shewhart-CUSUM chart deploys an additional Shewhart limit to expand a single CUSUM chart by triggering quick alarms for large changes in the parameter of interest. Here, we utilize this supplementary limit to initiate the CUSUM accumulation. That is, we switch between an accumulation and a silent phase. The new switching limit’s value resides between the reference value of the CUSUM chart and the usual Shewhart limit. Thus, for the case that the CUSUM statistic is equal to zero, a further observation has to be more substantial than this new limit to engage the summing process. We demonstrate the setup and analyze the new combination for independent Poisson distributed data and a more involved time series model with Poisson marginals, the Poisson INAR(1). Moreover, we also test the novel chart’s robustness against hypothetical misspecification such as undetected overdispersion or autocorrelation. It turns out that this kind of combination features patterns between a pure CUSUM and a stand-alone Shewhart chart and, hence, constitutes a solid alternative to both single charts as well as to the ordinary Shewhart-CUSUM. Finally, in the context of possible extensions, a real data set from semiconductor industry with apparently overdispersed counts is considered and the application to Gaussian variables data is briefly discussed.