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On the Equivalence of the Bernoulli and Geometric CUSUM Charts

The Bernoulli cumulative sum (CUSUM) chart has been shown to have good properties in detecting an increase in the nonconforming rate, p, based on the average number of observations to signal (ANOS). The geometric CUSUM chart has also been proposed, in particular for high-quality processes where p is very small. The geometric CUSUM chart has very good properties in a zero-state analysis because of a built-in headstart feature. In steady-state performance analyses, often a shift in the process has been assumed to occur immediately after a nonconforming item has been found. In this case, the performance of the geometric CUSUM chart is said to be better than that of the Bernoulli CUSUM chart. We explore the case where a process shift may occur at any time. We show that the steady-state properties of the Bernoulli CUSUM and geometric CUSUM charts are the same because the charts can be designed to be mathematically equivalent.