Estimation in Shewhart control charts: effects and corrections

Abstract

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,

Design schemes for the X̄ control chart

“…Taking ARL 0 as starting point is usually not the best option as ARL 0 is strongly determined by the occurrence of extreme long runs, which is often not relevant in practice, see in this respect Does and Schriever 12 . The last option we address is to determine the factor on the basis of the probability that the run length is at most a specified value x, see Nedumaran and Pignatiello 5 and Albers and Kallenberg 6 .…”

Correction factors for Shewhart and control charts to achieve desired unconditional ARL

“…This notation is similar to that of Albers and Kallenberg (2004), who derived corrections for the one-sided individuals control chart. From (11), we can write for any function g of P(E i |μ,σ ) …”

Properties and Use of the Shewhart Method and Its Followers

“…Kramer and Schmid (2000) study modifications and residuals for the Shewhart method to keep the false alarm rate under control in spite of the estimation of parameters. Albers and Kallenberg (2004) conclude that for the Shewhart method the sample size of the estimate has to be very large. Andersson et al (2005) make the same conclusion for other methods.…”