A Semi-Bayesian Method for Shewhart Individual Control Charts

Abstract Shewhart control limits for individual observations are traditionally based on the average of the moving ranges. The performance of this control chart behaves quite well if the underlying distribution is normal and the sample size is greater than 250. Under non-normality it is recommended to use control charts based on non-parametric statistics. The drawback of these individual control charts is that at least 1,000 observations are needed to obtain appropriate results. In this paper we propose an alternative individual control chart which behaves quite well under non-normality for moderate sample sizes in the range of 250 through 1,000 observations. To apply this control chart one starts with an initial guess for the density function of the characteristic under study. Based on this initial guess and the observed data a density function can be derived by means of an approximation with Bernstein polynomials. The in-control and out-of-control performance of the proposed control chart and the traditional control charts are studied by simulation. If the initial guess is appropriate, then for non-normal data and moderate sample sizes in the order of 250 through 1,000 observations, the new method performs better than the individual control charts based on the average of the moving ranges or based on non-parametric statistics. So for these sample sizes we have tried to close the gap.

[1]  Marion R. Reynolds,et al.  Robustness to non-normality and autocorrelation of individuals control charts , 2000 .

[2]  Prakash Vaman Joshi.  Bayesian approach to quality control , 1976 .

[3]  Ronald J. M. M. Does,et al.  Statistical process control in industry , 1998 .

[4]  Connie M. Borror,et al.  Robustness of the EWMA Control Chart to Non-Normality , 1999 .

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

[6]  George Tagaras,et al.  Comparing the Effectiveness of Various Bayesian X Control Charts , 2002, Oper. Res..

[7]  H. Shore.  Process Control for Non-Normal Populations Based on an Inverse Normalizing Transformation , 2001 .

[8]  David Olwell.  Topics in Statistical Process Control. , 1996 .

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

[10]  F. Famoye.  Continuous Univariate Distributions, Volume 1 , 1994 .

[11]  Casper J. Albers,et al.  Estimating a density by adapting an initial guess , 2003, Comput. Stat. Data Anal..

[12]  Ronald J. M. M. Does,et al.  Shewhart-Type Control Charts for Individual Observations , 1993 .

[13]  M. A. Girshick,et al.  A BAYES APPROACH TO A QUALITY CONTROL MODEL , 1952 .

[14]  W. Shewhart.  The Economic Control of Quality of Manufactured Product. , 1932 .

[15]  Philippe Girard,et al.  Bayesian Analysis of Autocorrelated Ordered Categorical Data for Industrial Quality Monitoring , 2001, Technometrics.

[16]  George W. Sturm,et al.  An empirical bayes strategy for analysing manufacturing data in real time , 1991 .

[17]  Casper Johannes Albers,et al.  Distributional inference : the limits of reason , 2003 .

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

[19]  G. B. Wetherill,et al.  Quality Control and Industrial Statistics , 1975 .

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

[21]  Ronald J. M. M. Does,et al.  A Comparison of Shewhart Individuals Control Charts Based on Normal, Non‐parametric, and Extreme‐value Theory , 2003 .

[22]  Jyh-Jen Horng Shiau,et al.  Statistical process monitoring using an empirical Bayes multivariate process control chart , 2001 .

[23]  Charles P Quesenberry.  SPC methods for quality improvement , 1997 .

[24]  Youn Min Chou,et al.  Transforming Non-Normal Data to Normality in Statistical Process Control , 1998 .

[25]  Peter R. Nelson,et al.  The Effect of Non-Normality on the Control Limits of X-Bar Charts , 1976 .

[26]  Ralph B. D'Agostino,et al.  Goodness-of-Fit-Techniques , 2020 .

[27]  Steven A. Yourstone,et al.  Non‐Normality and the Design of Control Charts for Averages* , 1992 .

[28]  Philip W. Woodward,et al.  An application of Bayesian methods in SPC , 1993 .

[29]  Fred Spiring,et al.  Introduction to Statistical Quality Control , 2007, Technometrics.

[30]  Robert N. Rodriguez.  Recent Issues in Statistical Process Control: SAS® Solutions Using Statistical Modeling Procedures , 1996 .

[31]  José Muñoz Perez,et al.  Estimating the quantile function by Bernstein polynomials , 1987 .

[32]  W. Schaafsma,et al.  A semi-Bayesian method for nonparametric density estimation , 1999 .

[33]  C. B. Roes,et al.  Shewhart-type Charts in Statistical Process Control , 1995 .

[34]  Ulrich Menzefricke,et al.  ON THE EVALUATION OF CONTROL CHART LIMITS BASED ON PREDICTIVE DISTRIBUTIONS , 2002 .

[35]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .