On the design of control charts with guaranteed conditional performance under estimated parameters

Abstract

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.

Parameter Estimation Effect of the Homogeneously Weighted Moving Average Chart to Monitor the Mean of Autocorrelated Observations With Measurement Errors

“…As stated in the Introduction, the estimation of the process parameters (i.e. the mean ( 0 ) and standard deviation ( 0 )) significantly reduces the performance of any monitoring scheme, (see for instance, [8]- [10]). Thus, the scheme's capability to respond swiftly to changes in the statistical processes weakens; hence, the investigation of parameter estimation when the underlying process mean is under the combined effect of autocorrelation and measurement errors needs to be conducted.…”

“…However, in Case U, the monitoring procedure needs to be implemented in a twophase approach, i.e. Phase I and Phase II (see the following review publications by [8]- [10] for more details). The retrospective implemention of a monitoring scheme is done in Phase I in order to estimate the distribution parameters and determine the control limits using an IC reference sample.…”

The effect of measurement errors on the performance of the homogenously weighted moving average X¯ monitoring scheme with estimated parameters

“…Since it is well-known that the estimation of the process parameters significantly degrades the performance of any monitoring scheme (see [17][18][19][20]), the investigation of the effect of parameter estimation on the performance of the HWMA ̅ scheme is of great importance. In Case U, the process parameters are estimated in Phase I (using m reference samples each of size n) when the process is deemed to be IC.…”

“…Therefore, it is very important to investigate the performance of the existing and new monitoring schemes under the assumption of unknown process parameters. Other recent contributions to parameter estimation effect can be found in the review paper by [20].…”

Optimal design of one‐sided exponential cumulative sum charts with known and estimated parameters based on the median run length

“…Much researches on the exponential charts are based on the assumption that the process parameter is known. In general, the process parameter is often unknown, and it is usually estimated from different in‐control Phase I data set (see Goedhart et al 23 and Does et al 24 ). However, different in‐control Phase I data sets collected from practitioners cause the variability in the estimations of the process parameter, which leads to the variability among the performance of the exponential control charts designed with the estimated process parameter.…”