Guaranteed conditional design of the median chart with estimated parameters

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.

[1]  P. Maravelakis,et al.  A CUSUM control chart for monitoring the variance when parameters are estimated , 2011 .

[2]  Bing Xing Wang,et al.  The Design of the ARL‐Unbiased S2 Chart When the In‐Control Variance Is Estimated , 2015, Qual. Reliab. Eng. Int..

[3]  Ying Zhang,et al.  RUN RULES $\bar{X}$ CHARTS WHEN PROCESS PARAMETERS ARE UNKNOWN , 2010 .

[4]  William H. Woodall,et al.  The Difficulty in Designing Shewhart X̄ and X Control Charts with Estimated Parameters , 2015 .

[5]  William H. Woodall,et al.  Guaranteed conditional performance of the S2 control chart with estimated parameters , 2015 .

[6]  Philippe Castagliola,et al.  Computational Statistics and Data Analysis an Ewma Chart for Monitoring the Process Standard Deviation When Parameters Are Estimated , 2022 .

[7]  Stefan H Steiner,et al.  Assessing the effect of estimation error on risk-adjusted CUSUM chart performance. , 2012, International journal for quality in health care : journal of the International Society for Quality in Health Care.

[8]  Philippe Castagliola,et al.  Some Recent Developments on the Effects of Parameter Estimation on Control Charts , 2014, Qual. Reliab. Eng. Int..

[9]  Michael B. C. Khoo,et al.  A Control Chart Based on Sample Median for the Detection of a Permanent Shift in the Process Mean , 2005 .

[10]  Gemai Chen,et al.  THE MEAN AND STANDARD DEVIATION OF THE RUN LENGTH DISTRIBUTION OF X̄ CHARTS WHEN CONTROL LIMITS ARE ESTIMATED Gemai Chen , 2003 .

[11]  William H. Woodall,et al.  A Reevaluation of the Adaptive Exponentially Weighted Moving Average Control Chart When Parameters are Estimated , 2015, Qual. Reliab. Eng. Int..

[12]  Philippe Castagliola,et al.  The median chart with estimated parameters , 2013 .

[13]  Charles W. Champ,et al.  The Performance of Exponentially Weighted Moving Average Charts With Estimated Parameters , 2001, Technometrics.

[14]  Giovanni Celano,et al.  THE EXACT RUN LENGTH DISTRIBUTION AND DESIGN OF THE S2 CHART WHEN THE IN-CONTROL VARIANCE IS ESTIMATED , 2009 .

[15]  Philippe Castagliola,et al.  The variable sampling interval X̄ chart with estimated parameters , 2012, Qual. Reliab. Eng. Int..

[16]  Min Zhang,et al.  Exponential CUSUM Charts with Estimated Control Limits , 2014, Qual. Reliab. Eng. Int..

[17]  Michael B. C. Khoo,et al.  Optimal designs of the variable sample size and sampling interval X¯ chart when process parameters are estimated , 2015 .

[18]  Joseph J. Pignatiello,et al.  On Estimating X̄ Control Chart Limits , 2001 .

[19]  Marion R. Reynolds,et al.  Shewhart x-charts with estimated process variance , 1981 .

[20]  Shey-Huei Sheu,et al.  The Generally Weighted Moving Average Control Chart for Monitoring the Process Median , 2006 .

[21]  L. Allison Jones,et al.  The Statistical Design of EWMA Control Charts with Estimated Parameters , 2002 .

[22]  S. Psarakis,et al.  EFFECT OF ESTIMATION OF THE PROCESS PARAMETERS ON THE CONTROL LIMITS OF THE UNIVARIATE CONTROL CHARTS FOR PROCESS DISPERSION , 2002 .

[23]  Fadel M. Megahed,et al.  Geometric Charts with Estimated Control Limits , 2013, Qual. Reliab. Eng. Int..

[24]  Charles W. Champ,et al.  Effects of Parameter Estimation on Control Chart Properties: A Literature Review , 2006 .

[25]  William H. Woodall,et al.  Another Look at the EWMA Control Chart with Estimated Parameters , 2015 .

[26]  Muhammad Riaz,et al.  Design schemes for the X control chart , 2009, Qual. Reliab. Eng. Int..

[27]  Gemai Chen,et al.  The run length distributions of the R, s and s2 control charts when is estimated , 1998 .

[28]  Philippe Castagliola,et al.  The EWMA median chart with estimated parameters , 2016 .

[29]  Philippe Castagliola,et al.  The synthetic [Xbar] chart with estimated parameters , 2011 .

[30]  Muhammad Riaz,et al.  Design and Analysis of Control Charts for Standard Deviation with Estimated Parameters , 2011 .

[31]  Ying Zhang,et al.  The Variable Sample Size X¯ Chart with Estimated Parameters , 2012, Qual. Reliab. Eng. Int..

[32]  Eugenio K. Epprecht,et al.  Effect of the Amount of Phase I Data on the Phase II Performance of S2 and S Control Charts , 2015 .

[33]  Anne R. Driscoll,et al.  The c-Chart with Bootstrap Adjusted Control Limits to Improve Conditional Performance , 2016, Qual. Reliab. Eng. Int..

[34]  Axel Gandy,et al.  Guaranteed Conditional Performance of Control Charts via Bootstrap Methods , 2011, 1111.4180.

[35]  Charles W. Champ,et al.  The Run Length Distribution of the CUSUM with Estimated Parameters , 2004 .

[36]  Mahmoud A. Mahmoud,et al.  The Performance of the Multivariate Adaptive Exponentially Weighted Moving Average Control Chart with Estimated Parameters , 2016, Qual. Reliab. Eng. Int..