Run-length performance of AE-I and AE-II mean charts under phase-I parameter estimation

Adaptive exponentially weighted moving average charts are effective tools for detecting small-to-moderate shifts in a process mean. In practical applications, however, the incontrol mean and standard deviation are usually unknown and must be estimated from Phase I data before Phase II monitoring begins. This estimation uncertainty can affect both false-alarm behaviour and out-of-control detection performance. Although the adaptive exponentially weighted moving average-I chart has been studied under parameter estimation, the corresponding run-length behaviour of the parameter-free adaptive exponentially weighted moving average-II chart has not been systematically examined. This study evaluates the adaptive exponentially weighted moving average-II mean chart when process parameters are estimated from Phase I samples and compares it with the adaptive exponentially weighted moving average-I chart under a common in-control average run length calibration. Monte Carlo simulations with 100,000 replications are conducted to obtain zero-state and steady-state run-length profiles for different Phase I sample sizes and mean-shift magnitudes. Overall performance is summarized using integrated relative average run length and relative mean index measures. In addition to the normal model, a heavy-tailed t-distribution setting is considered as a robustness check. We also use a real industrial piston-ring dataset to illustrate practical implementation. The results show that parameter-estimation effects are strongest when the number of Phase I subgroups is small and diminish as m increases. Under calibrated in-control performance, the adaptive exponentially weighted moving average-II chart generally detects small-to-moderate shifts faster, whereas the adaptive exponentially weighted moving average-I chart remains competitive for moderate and mixed-range shifts.