Cautious parameter learning for a joint mean and variance monitoring CUSUM scheme with guaranteed in-control performance

Joint monitoring of the mean and variance of a normal process is crucial in quality engineering for assessing homoscedasticity, determining process capability, and recognizing that changes in scale often accompany changes in location. A CUSUM scheme for monitoring both mean and variance must ensure overall performance, even with estimated parameters. Approaches guaranteeing minimum in-control performance are standard, but they lose power with small Phase I samples. Learning approaches allow parameter re-estimation with new data, while cautious schemes define update rules to minimize sample contamination from out-of-control observations. This article presents a CUSUM scheme incorporating cautious parameter learning and guaranteed in-control performance for joint monitoring of mean and variance with normal, independent observations. Our novel approach compares parameter estimation procedures based on conditional expected delay and false alarm probability across a series of change points for a profile assessment, rather than a point-based assessment. Computational results show that the cautious learning procedure using a conservative external updating rule is preferred even when the initial sample size is small, as it provides the best balance between conditional expected delay and false alarm probability from a practical perspective.