Effects of Parameter Estimation on Control Chart Properties: A Literature Review

Control charts are powerful tools used to monitor the quality of processes. In practice, control chart limits are often calculated using parameter estimates from an in-control Phase I reference sample. In Phase II of the monitoring scheme, statistics based on new samples are compared with the estimated control limits to monitor for departures from the in-control state. Many studies that evaluate control chart performance in Phase II rely on the assumption that the in-control parameters are known. Although the additional variability introduced into the monitoring scheme through parameter estimation is known to affect the chart performance, many studies do not consider the effect of estimation on the performance of the chart. This paper contains a review of the literature that explicitly considers the effect of parameter estimation on control chart properties. Some recommendations are made and future research ideas in this area are provided.

[1]  Marion R. Reynolds,et al.  EWMA CONTROL CHARTS FOR MONITORING THE MEAN OF AUTOCORRELATED PROCESSES , 1999 .

[2]  Daniel W. Apley.  Time Series Control Charts in the Presence of Model Uncertainty , 2002 .

[3]  Charles W. Champ,et al.  Designing Phase I ―X Charts with Small Sample Sizes , 2004 .

[4]  Tzong-Ru Tsai,et al.  On estimating control limits of X ̄ chart when the number of subgroups is small , 2005 .

[5]  Roger M. Sauter,et al.  Introduction to Statistical Quality Control (2nd ed.) , 1992 .

[6]  Charles W. Champ,et al.  Design Strategies for Individuals and Moving Range Control Charts , 1994 .

[7]  Willem Albers,et al.  New Corrections for Old Control Charts , 2003 .

[8]  Wolfgang Schmid,et al.  CUSUM control schemes for Gaussian processes , 1997 .

[9]  Douglas C. Montgomery,et al.  Using Control Charts to Monitor Process and Product Quality Profiles , 2004 .

[10]  W. Woodall,et al.  Adapting control charts for the preliminary analysis of multivariate observations , 1998 .

[11]  Wilbert C.M. Kallenberg,et al.  Estimation in Shewhart control charts: effects and corrections , 2004 .

[12]  Subhabrata Chakraborti.  Run length, average run length and false alarm rate of shewhart x-bar chart: exact derivations by conditioning , 2000 .

[13]  Dan Trietsch,et al.  Statistical Quality Control: A Loss Minimization Approach , 1999 .

[14]  C. H. Sim.  Combined X-bar and CRL Charts for the Gamma Process , 2003, Comput. Stat..

[15]  K. Govindaraju,et al.  On Statistical Design of the S 2 Control Chart , 2005 .

[16]  Michael S. Hamada.  Bayesian tolerance interval control limits for attributes , 2002 .

[17]  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 .

[18]  Sven Knoth,et al.  Control Charts for Time Series , 1997 .

[19]  Frederick S. Hillier,et al.  Mean and Variance Control Chart Limits Based on a Small Number of Subgroups , 1970 .

[20]  E. Castillo.  Computer Programs -- Evaluation of Run Length Distribution for X Charts with Unknown Variance , 1996 .

[21]  Antonio Fernando Branco Costa,et al.  X̄ charts with variable sample size , 1994 .

[22]  Steven E. Rigdon,et al.  COMPARING TWO ESTIMATES OF THE VARIANCE TO DETERMINE THE STABILITY OF A PROCESS , 1992 .

[23]  Charles W. Champ,et al.  Exact results for shewhart control charts with supplementary runs rules , 1987 .

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

[25]  Douglas C. Montgomery,et al.  A review of multivariate control charts , 1995 .

[26]  W. John Braun.  Run length distributions for estimated attributes charts , 1999 .

[27]  Marion R. Reynolds,et al.  Cusum Charts for Monitoring an Autocorrelated Process , 2001 .

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

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

[30]  Willem Albers,et al.  Empirical Non-Parametric Control Charts: Estimation Effects and Corrections , 2004 .

[31]  Douglas C. Montgomery,et al.  Research Issues and Ideas in Statistical Process Control , 1999 .

[32]  Kwok-Leung Tsui,et al.  Run-Length Performance of Regression Control Charts with Estimated Parameters , 2004 .

[33]  Charles W. Champ,et al.  An Analysis of Shewhart Charts with Runs Rules When No Standards Are Given , 1993 .

[34]  Douglas C. Montgomery,et al.  Short-run statistical process control: Q-chart enhancements and alternative methods , 1996 .

[35]  D. Hawkins.  Self‐Starting Cusum Charts for Location and Scale , 1987 .

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

[37]  Charles W. Champ.  Designing an ARL Unbiased R Chart , 2001 .

[38]  William H. Woodall,et al.  A Comparison of Multivariate Control Charts for Individual Observations , 1996 .

[39]  Richard A. Johnson,et al.  The Influence of Reference Values and Estimated Variance on the Arl of Cusum Tests , 1975 .

[40]  Charles P. Quesenberry,et al.  DPV Q charts for start-up processes and short or long runs , 1991 .

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

[42]  Joseph J. Pignatiello,et al.  Monitoring Process Dispersion without Subgrouping , 2000 .

[43]  Charles W. Champ,et al.  Properties of the T2 Control Chart When Parameters Are Estimated , 2005, Technometrics.

[44]  Joe H. Sullivan,et al.  A Self-Starting Control Chart for Multivariate Individual Observations , 2002, Technometrics.

[45]  Daniel W. Apley,et al.  Design of Exponentially Weighted Moving Average Control Charts for Autocorrelated Processes With Model Uncertainty , 2003, Technometrics.

[46]  Tzong-Ru Tsai,et al.  An alternative control chart approach based on small number of subgroups , 2004 .

[47]  Lawrence G. Tatum.  Robust estimation of the process standard deviation for control charts , 1997 .

[48]  Wolfgang Schmid,et al.  The influence of parameter estimation on the ARL of Shewhart type charts for time series , 2000 .

[49]  Joseph J. Pignatiello,et al.  On constructing T2 control charts for on-line process monitoring , 1999 .

[50]  Gunabushanam Nedumaran,et al.  p-CHART CONTROL LIMITS BASED ON A SMALL NUMBER OF SUBGROUPS , 1998 .

[51]  D. Montgomery,et al.  A Combined Adaptive Sample Size and Sampling Interval X Control Scheme , 1994 .

[52]  Wing-Keung Wong,et al.  R-charts for the exponential, Laplace and logistic processes , 2003 .

[53]  Zhaojun Wang,et al.  THE MEDIAN ABSOLUTE DEVIATIONS AND THEIR APPLICATIONS TO SHEWHART CONTROL CHARTS , 2002 .

[54]  N. José Alberto Vargas,et al.  Robust Estimation in Multivariate Control Charts for Individual Observations , 2003 .

[55]  David M. Rocke.  Xq and Rq charts: Robust control charts , 1992 .

[56]  Benjamin M. Adams,et al.  Robustness of Forecast-Based Monitoring Schemes , 1998 .

[57]  Douglas M. Hawkins,et al.  The Changepoint Model for Statistical Process Control , 2003 .

[58]  Elisabeth J. Umble,et al.  Cumulative Sum Charts and Charting for Quality Improvement , 2001, Technometrics.

[59]  Charles P. Quesenberry,et al.  The Effect of Sample Size on Estimated Limits for and X Control Charts , 1993 .

[60]  David M. Rocke.  Robust control charts , 1989 .

[61]  Benjamin M. Adams,et al.  Robust Monitoring of Contaminated Data , 2005 .

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

[63]  Antonio Fernando Branco Costa,et al.  X̄ Chart with Variable Sample Size and Sampling Intervals , 1997 .

[64]  Enrique Del Castillo.  Run length distributions and economic design of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-x , 1996 .

[65]  Frederick S. Hillier,et al.  X-Bar- and R-Chart Control Limits Based on A Small Number of Subgroups , 1969 .

[66]  Thomas P. Ryan,et al.  The estimation of sigma for an X chart: MR/d2 or S/c4? , 1990 .

[67]  J. A. Nachlas,et al.  X charts with variable sampling intervals , 1988 .

[68]  Charles W. Champ,et al.  Adjusting the S-Chart for Detecting Both Increases and Decreases in the Standard Deviation , 1994 .

[69]  Kwok-Leung Tsui,et al.  Effects of estimation errors on cause-selecting charts , 2005 .

[70]  C. H. Sim.  Inverse Gaussian Control Charts for Monitoring Process Variability , 2003 .

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

[72]  Charles W. Champ,et al.  Analysis of the Shewhart S-Chart with Runs Rules When No Standards Are Given , 1995 .

[73]  Gemai Chen,et al.  A NOTE ON EWMA CHARTS FOR MONITORING MEAN CHANGES IN NORMAL PROCESSES , 2002 .

[74]  H. Levene,et al.  The Effectiveness of Quality Control Charts , 1950 .

[75]  Frederick S. Hillier,et al.  Small Sample Probability Limits for the Range Chart , 1967 .

[76]  Sheldon M. Ross,et al.  An Improved Estimator of σ in Quality Control , 1995, Probability in the Engineering and Informational Sciences.

[77]  R. F.,et al.  Statistical Method from the Viewpoint of Quality Control , 1940, Nature.

[78]  Wilbert C.M. Kallenberg,et al.  Are estimated control charts in control? , 2001 .