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Recently, simultaneous monitoring of multivariate process mean and variability has received increasing attention in the literature of statistical process monitoring (SPM). However, the deleterious impact of parameter estimation on the capability of control charts designed for simultaneous monitoring of the mean vector and covariance matrix of multivariate processes has been clearly neglected. In this paper, we study the effect of estimation error on both in-control and out-of-control properties of the multivariate exponentially weighted moving average (EWMA)-based generalized likelihood ratio ( MELR ) chart. Simulation studies in terms of the average run length ( ARL ), the standard deviation of run length ( SDRL ), and the median run length ( MRL ) metrics are conducted to explore how the amount of Phase I reference samples affects the performance of the  MELR  chart. The results show that extra variability due to estimation error reduces the detecting capability of the  MELR  chart while increases its false alarm rate. Meanwhile, a real-life data is provided to illustrate poor Phase I estimation results in more false alarms than expected.

Multivariate ELR control chart with estimated mean vector and covariance matrix