Phase‐II Monitoring the Ratio of Population Means of a Bivariate Normal Distribution Based on Subgroup Samples

ABSTRACT In various industrial contexts, there is a need for online monitoring of quality characteristics represented as the ratio of two normal random variables. This includes scenarios such as maintaining the correct proportion between ingredients or elements in a product, assessing product performance before and after specific operations like chemical reactions, and monitoring chemical or physical properties computed as ratios. This paper introduces two new Phase II Shewhart monitoring schemes for monitoring the ratio of population means of bivariate normal distributions when parameters are known and unknown, respectively. Notably, the size of each sample unit can vary across different subgroups, enhancing the adaptability of the chart implementation in diverse manufacturing settings. We use the average run length (ARL) for evaluating the Phase II performance of the proposed monitoring schemes. Simulation results are provided to demonstrate the statistical performance of the investigated monitoring schemes under the known and unknown parameter cases. Two real‐data examples are presented for illustration.