Detecting a downward shift in a proportion using a geometric CUSUM chart

In monitoring an ordered stream of discrete items from a repetitive process, the geometric CUSUM chart may be used to detect sudden shifts from an acceptable level for a process-proportion (p) such as fraction nonconforming. Much of the investigative effort for this CUSUM scheme has been concentrated on the detection of upward shifts, and a recent paper has provided guidance to quality engineers in choosing the parameters (k, h) of such a scheme. In this article, the corresponding task of aiding the choice of parameters for detecting a downward shift is addressed. It is shown, using extensive numerical investigations, that the use of a value for the parameter k based on the Sequential Probability Ratio is not optimal when one is using steady-state evaluation of the detection performance of the CUSUM scheme. Tables are presented listing recommended values of parameters for detection of five sizes of downward shift, for each of 27 in-control levels for p in the range 0.20 to 0.001. Interpolation and extrapolation to find parameter values for other in-control levels of p are also considered, and a range of examples presented. There is an equivalence between a geometric CUSUM scheme and a Bernoulli CUSUM scheme, so that the results of this investigation may also be used in choosing parameter values for a Bernoulli CUSUM chart.