Design and Analysis of Control Charts for Standard Deviation with Estimated Parameters

Abstract

Design and analysis of control charts for standard deviation with estimated parametersSchoonhoven, M.; Riaz, M.; Does, R.J.M.M. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. This paper concerns the design and analysis of the standard deviation control chart with estimated limits. We consider an extensive range of statistics to estimate the in-control standard deviation (Phase I) and design the control chart for real-time process monitoring (Phase II) by determining the factors for the control limits. The Phase II performance of the design schemes is assessed when the Phase I data are uncontaminated and normally distributed as well as when the Phase I data are contaminated. We propose a robust estimation method based on the mean absolute deviation from the median supplemented with a simple screening method. It turns out that this approach is e cient under normality and performs substantially better than the traditional estimators and several robust proposals when contaminations are present.

A Robust Control Chart

“…An unbiased estimator of σ is given by , where d ∗ ( c , n , k ) is the normalizing constant. In our study, we use the corrected normalizing constants given in Schoonhoven et al …”

A Robust Standard Deviation Control Chart

“…The advantage of this estimator is that it is less sensitive to outliers than R (cf. Schoonhoven et al 2011). The resulting estimator is denoted by MD s .…”

“…Additionally, we look at an adaptive trimmer based on the mean deviation from the median, a statistic more resistant to diffuse outliers (cf. Schoonhoven, Riaz, and Does 2011). For diffuse outliers, we think that a control chart for individual observations would detect outliers more quickly.…”

Quality Quandaries: How to Set Up a Robust Shewhart Control Chart for Dispersion?

“…What remains is the determination of the factors U II and L II in order to obtain the desirable in-control performance. Schoonhoven et al (2011) presented a formula for U II and L II of the Phase II standard deviation control chart based on the pooled mean of the sample standard deviations,S S. They tested this formula for charts derived from a broad range of Phase I estimators and concluded that the formula is suitable when the variance of the estimator is close to the variance ofS S. Subsequently, Schoonhoven and Does (2013) showed that the variance of the estimator given in [5] is close toS S. Hence, the formula presented by Schoonhoven et al (2011) can also be applied in the procedure discussed here. This is a nice result because the formula is a plug-in so no simulations are required to obtain the constants.…”