On the Design of Control Charts with Guaranteed Conditional Performance under Estimated Parameters

When designing control charts the in‐control parameters are unknown, so the control limits have to be estimated using a Phase I reference sample. To evaluate the in‐control performance of control charts in the monitoring phase (Phase II), two performance indicators are most commonly used: the average run length (ARL) or the false alarm rate (FAR). However, these quantities will vary across practitioners due to the use of different reference samples in Phase I. This variation is small only for very large amounts of Phase I data, even when the actual distribution of the data is known. In practice, we do not know the distribution of the data, and it has to be estimated, along with its parameters. This means that we have to deal with model error when parametric models are used and stochastic error because we have to estimate the parameters. With these issues in mind, choices have to be made in order to control the performance of control charts. In this paper, we discuss some results with respect to the in‐control guaranteed conditional performance of control charts with estimated parameters for parametric and nonparametric methods. We focus on Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) control charts for monitoring the mean when parameters are estimated.