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.