Phase II monitoring of multivariate simple linear profiles with estimated parameters

In some applications of statistical process monitoring, a quality characteristic can be characterized by linear regression relationships between several response variables and one explanatory variable, which is referred to as a 'multivariate simple linear profile.' It is usually assumed that the process parameters are known in Phase II. However, in most applications, this assumption is violated; the parameters are unknown and should be estimated based on historical data sets in Phase I. This study aims to compare the effect of parameter estimation on the performance of three Phase II approaches for monitoring multivariate simple linear profiles, designated as MEWMA, MEWMA_3 and {MEWMA}/χ2. Three metrics are used to accomplish this objective: AARL, SDARL and CVARL. The superior method may be different in terms of the AARL and SDARL metrics. Using the CVARL metric helps practitioners make reliable decisions. The comparisons are carried out under both in-control and out-of-control conditions for all competing approaches. The corrected limits are also obtained by a Monte Carlo simulation in order to decrease the required number of Phase I samples for parameter estimation. The results reveal that parameter estimation strongly affects the in-control and out-of-control performance of monitoring approaches, and a large number of Phase I samples are needed to achieve a parameter estimation that is close to the known parameters. The simulation results show that the MEWMA and {MEWMA}/χ2methods perform better than the MEWMA_3 method in terms of the CVARL metric. However, the superior approach is different in terms of AARL and SDARL.