A note on the g and h control charts

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.