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 (<i>p</i>) 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 (<i>k</i>, <i>h</i>) 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 <i>k</i> 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 <i>p</i> in the range 0.20 to 0.001. Interpolation and extrapolation to find parameter values for other in-control levels of <i>p</i> 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.

在对重复性生产过程产生的离散有序数据流进行监控时，几何累积和（geometric CUSUM）控制图可用于检测过程比例参数<i>p</i>（如不合格品率（fraction nonconforming））的可接受水平发生的突发偏移。此前针对该CUSUM方案的研究大多聚焦于向上偏移的检测，近期已有文献为质量工程师提供了该方案参数(<i>k</i>, <i>h</i>)的选择指南。本文则针对检测向下偏移时的参数选择问题展开研究。通过大量数值实验可知，当采用稳态方法评估CUSUM方案的检测性能时，基于序贯概率比（Sequential Probability Ratio）确定的参数<i>k</i>取值并非最优。本文给出了参数推荐取值表，覆盖0.20至0.001区间内27个受控过程比例参数<i>p</i>水平，可分别用于检测5种不同幅度的向下偏移。本文还探讨了通过插值与外推法获取其他受控<i>p</i>水平对应参数取值的方法，并给出了多个应用示例。几何CUSUM方案与伯努利（Bernoulli）CUSUM方案具有等价性，因此本研究的结果同样可用于伯努利CUSUM控制图的参数选择。