On the control chart with estimated parameters

Statistical monitoring of multivariate processes is becoming increasingly important in modern manufacturing environments. Typical equipment may have multiple key variables to be measured continuously. Hotelling’s T² chart was originally applied for monitoring the mean vector of multivariate quality measurements. In practical problems, estimated parameters are needed and their use will modify the properties of control charts. The Average Run Length (ARL), an indicator of the performance of the control charts, will be larger when the estimated parameters are used. As one contribution of the paper, we provide a rigorous proof of this phenomenon which has been reported in several empirical studies. Furthermore, in order to design an efficient T² chart with estimated parameters, it is necessary to have a method to calculate or approximate the ARL function. An existing approach in the literature is based on extensive Monte-Carlo simulations. In this paper, we propose a novel approach by providing an analytic approximation of the ARL function in the however limited case of univariate observations.