The influence of the estimation of parameters in Shewhart control charts is investigated. It is shown by simulation and asymptotics that (very) large sample sizes are needed to accurately determine control charts if estimators are plugged in. Correction terms are developed to get accurate control limits for common sample sizes in the in-control situation. Simulation and theory show that the new corrections work very well. The performance of the corrected control charts in the out-of-control situation is studied as well. It turns out that the correction terms do not disturb the behavior of the control charts in the out-of-control situation. On the contrary, for moderate sample sizes the corrected control charts remain powerful and therefore, the recommendation to take at least 300 observations can be reduced to 40 observations when corrected control charts are applied. Copyright Springer-Verlag 2004Statistical process control; Phase II control limits; second order unbiasedness; out-of-control, 62F12, 62P30,

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Standard control chart practice typically assumes normality and uses estimated parameters using data from an in-control process. However, because of the extreme quantiles involved, large relative errors will result for common performance characteristics such as the out-of-control signal probability or the average run length. Due to the estimation, such performance characteristics are stochastic and hence the relative errors involved can be analyzed in various ways. To access the effects of these various ways of estimation, we look at some exceedance probabilities. It is demonstrated how corrections can be derived to bring the estimated false alarm rates close to their nominal values. Exact results are given, followed by simple approximations. The latter reveal the way in which the corrections depend on the underlying parameters, thus allowing a sensible approach in practice. Some illustration is provided, as well as a brief analysis of the out-of-control behavior of the corrected charts.

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Rasch model, growth curves, regression, itembanking,

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