Estimation of σ for Individuals Charts

Abstract

We study the problem of estimating σ from individuals' data in the presence of out-of-control and nonnormal conditions. We consider a large number of estimators for σ that have been proposed in the last 20 years. Using simulation, we compute the mean squared errors for 10 of these estimators, both before and after screening for outliers with median-absolute-deviation—based control limits. We also consider the effect of contaminated data on the performance of EWMA (exponentially weighted moving average) charts. We show, by simulation, that Boyles' dynamic linear model estimator is, after screening, often the best way of estimating σ. We also show that an approach that involves robustly testing for shifts prior to final estimation of σ can be approximated by an automatic method based on smoothing the data first and estimating σ from residuals; this method is often nearly as effective as that of Boyles and is sometimes better. We also find that there are some situations where noisy data are not amenable to accurate estimation of σ, even at a moderately large sample size. In such cases, engineering considerations and careful data analysis are required to obtain more accurate estimates.