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Exponentially weighted moving average (EWMA) type charts are very popular and efficient in monitoring various kind of statistics. Much researches have been done on the EWMA X ¯ chart with known process parameters. But, in practice, the process parameters used to set the control chart limits are often unknown and they need to be estimated from different Phase I samples. Moreover, because the shape of the run length distribution for the EWMA X ¯ chart changes with the mean shift, the median run length (MRL) can serve as a good alternative to evaluate the performance of the EWMA X ¯ chart. In this article, we will investigate the conditional properties of the EWMA X ¯ chart with unknown process parameters based on the MRL metric. In order to investigate the chart's properties, the average MRL (AMRL) and the standard deviation of MRL (SDMRL) will be used together when the process parameters are unknown. To prevent too many lower in-control MRL values, the adjusted control limits of the MRL based EWMA chart are obtained by using a bootstrap type approach and the results show that the adjusted control limits can give a good tradeoff between the in-control and out-of-control MRL performances. [ABSTRACT FROM AUTHOR]

As a useful tool in statistical process control (SPC), the exponential control chart is more and more popular for monitoring high‐quality processes. Considering both known and estimated parameter cases, the one‐sided exponential cumulative sum (CUSUM) charts are studied in this paper through a Markov chain approach. Because the shape of the run length (RL) distribution of the one‐sided exponential CUSUM charts is skewed and it also changes with the mean shift size and the number of Phase I samples used to estimate the process parameter, the median run length (MRL) is employed as a good alternative performance measure for the charts. The optimal design procedures based on MRL of the one‐sided exponential CUSUM charts with known and estimated parameters are discussed. By comparing the MRL performance of the chart with known parameters with the one of the chart with estimated parameters, we investigate the effect of estimated process parameters on the properties of the chart. Finally, an application is illustrated to show the implementation of the chart. [ABSTRACT FROM AUTHOR]

Run Rules X ˉ chart are often implemented for process monitoring. The process observations are usually assumed normally distributed, and the process parameters are assumed known. In fact, a very limited number of Phase I samples may only be available to the practitioners to estimate the process parameters, the control chart's properties will vary among different practitioners. Considering this variability, the performance of the Run Rules X ˉ chart with estimated parameters is investigated. To measure this variability, the standard deviation of average run length (SDARL) is used to evaluate the performance of control chart. The results show that the Run Rules X ˉ chart requires a much larger amount of Phase I data to sufficiently reduce the between-practitioners variability. Moreover, we also investigate the properties of Run Rules X ˉ chart with estimated parameters under the non-normal distributed data. Due to the limitation of the amount of Phase I data set in practice, a bootstrap method is applied to design the Run Rules X ˉ chart. Finally, an example is provided to illustrate the application of the Run Rules X ˉ chart with estimated parameters. [ABSTRACT FROM AUTHOR]

The exponentially weighted moving average (EWMA) chart is often designed assuming the process parameters are known. In practice, the parameters are rarely known and need to be estimated from Phase I samples. Different Phase I samples are used when practitioners construct their own control chart's limits, which leads to the "Phase I between-practitioners" variability in the in-control average run length (ARL) of control charts. The standard deviation of the ARL (SDARL) is a good alternative to quantify this variability in control charts. Based on the SDARL metric, the performance of the EWMA median chart with estimated parameters is investigated in this paper. Some recommendations are given based on the SDARL metric. The results show that the EWMA median chart requires a much larger amount of Phase I data in order to reduce the variation in the in-control ARL up to a reasonable level. Due to the limitation of the amount of the Phase I data, the suggested EWMA median chart is designed with the bootstrap method which provides a good balance between the in-control and out-of-control ARL values. [ABSTRACT FROM AUTHOR]

The Adaptive Exponentially Weighted Moving Average (AEWMA) chart is known to combine the Shewhart and the classical EWMA schemes in a smooth way. Its performance has been investigated under the assumptions that the data are free from outliers and the chart's parameters are known or can be accurately estimated from in-control historical samples. However, there are many situations where the process parameters (a) are estimated from a very limited number of samples (b) they can potentially contain unexpected outliers. Therefore, in this article, we develop an AEWMA median chart with known and estimated parameters to monitor the mean value of a normal process. Taking the 'Phase I between-practitioners variability' into account, both the Average of () and the Standard Deviation of the () are used to evaluate the conditional effect of the number of Phase I samples on the Phase II performance. Using a Markov Chain approach, it is shown that (a) the performance of the proposed AEWMA median chart is seriously affected when parameters are estimated compared with the known-parameter case, and (b) it requires a large amount of Phase I data to reduce the variation in the in-control distribution up to a reasonable level. Therefore, a bootstrap-based design approach is applied here and, the performance of the AEWMA median chart is compared with that of the existing Shewhart and EWMA median charts. The comparative results show that the AEWMA median chart represents a good alternative to achieve a reasonable balance for various shift sizes. [ABSTRACT FROM AUTHOR]

Abstract: Control charts are usually investigated under the assumption of known process parameters. In practice, however, the process parameters are rarely known and they have to be estimated from different Phase I data sets. The properties of control charts with estimated parameters are usually investigated with the unconditional average of the average run length. Control chart's performance is known to vary among practitioners because of the use of different Phase I data sets. Considering the between‐practitioners variability in control chart's performance, the standard deviation of the average run length is developed to reevaluate the properties of the synthetic X ¯ chart with estimated parameters. Because of the limited amount of Phase I data in practice, the bootstrap method is used as a good adjustment technique for the synthetic X ¯ chart's parameters. [ABSTRACT FROM AUTHOR]

A common assumption for most control charts is the fact that the process parameters are supposed to be known or accurately estimated from Phase I samples. But, in practice, this is not a realistic assumption and the process parameters are usually estimated from a very limited number of samples that, in addition, may contain some outliers. Recently, a median chart with estimated parameters has been proposed to overcome these issues and it has been investigated in terms of the unconditional Average Run Length (ARL). As this median chart with estimated parameters does not take the 'Phase I between-practitioners' variability into account, in this paper, we suggest to revisit it using the Standard Deviation of the ARL as a measure of performance. The results show that this Standard Deviation of the ARL-based median chart actually requires a much larger amount of Phase I data than previously recommended to sufficiently reduce the variation in the chart performance. Due to the practical limitation of the number of the Phase I data, the bootstrap method is recommended as a good alternative approach to define new dedicated control chart parameters. [ABSTRACT FROM AUTHOR]