In this note, we revisit the $g$ and $h$ control charts that are commonly used for monitoring the number of conforming cases between the two consecutive appearances of nonconformities. It is known that the process parameter of these charts is usually unknown and estimated by using the maximum likelihood estimator and the minimum variance unbiased estimator. However, the minimum variance unbiased estimator in the control charts has been inappropriately used in the quality engineering literature. This observation motivates us to provide the correct minimum variance unbiased estimator and investigate theoretical and empirical biases of these estimators under consideration. Given that these charts are developed based on the underlying assumption that samples from the process should be balanced, which is often not satisfied in many practical applications, we propose a method for constructing these charts with unbalanced samples.

翻译：在本说明中,我们重新审视了通常用于监测连续两次不符合同现象之间相符情况数量的G美元和h美元对照图,已知这些图表的流程参数通常未知,并且使用最大可能性估计器和最小差异无偏差估计器来估计这些图表的流程参数,然而,在质量工程文献中,控制图中的最低差异无偏差估计器被不当使用。这一观察促使我们提供正确的最低差异不偏差估计器,并调查这些估算器的理论和经验偏差。鉴于这些图表的编制所依据的基本假设是:该流程的样本应当平衡,而在许多实际应用中往往不能满足这一假设,因此我们建议了一种方法,用不平衡的样本构建这些图表。