The Robustness of the Multivariate EWMA Control Chart

Control charts are graphical tools widely used to monitor manufacturing processes to quickly detect any change in a process that may result in a change in product quality. The statistic plotted on a control chart is based on samples of n ≥ 1 observations (rational subgroups) that may be taken at regular sampling intervals. However, there are numerous practical applications using individual observations (n = 1), as in many chemical and process industries or when the rate of production is slow. See, for example, Montgomery (1997, pp. 221-222) and Ryan (2000, p. 133) for many applications using individual observations. The standard approach to control charts has been univariate, with the most frequently investigated charts being those designed to monitor the mean µ of a single normally distributed process variable X. However, with rapidly evolving data-acquisition technology, it is now common practice to simultaneously monitor several, usually correlated, quality variables. Using separate univariate charts does not account for correlation between variables , so a multivariate control chart is more suitable. Two multivariate control charts that have received attention in the statistical process control (SPC) literature are the Shewhart-type χ 2 chart, originating in the work of Hotelling (1947), and the multivariate exponentially weighted moving average (MEWMA) chart, proposed much more recently by Lowry et al. (1992). Both of these charts were originally developed for the problem of monitoring the mean vector, say µ µ µ µ, of a continuous multivariate process variable (vector), say x. The χ 2 chart can be relatively easily applied and is helpful in identifying large shifts in µ µ µ µ, but is ineffective at detecting small and moderate-sized shifts. Unlike the χ 2 chart, the MEWMA chart accumulates information from past observations, making it more sensitive in detecting small, sustained shifts. The survey good discussions on multivariate control charts and extensive lists of references. When monitoring the mean vector, control chart design is usually predicated on the approximate multivari-ate normality (multinormality) of process data. Non-normality generally is not a major concern with " large " subgroups, because the central limit theorem applies and the sample mean vector x is approximately multi-normal, for any practical distribution of the individual observations. However, with small samples—quite common in SPC practices—from a non-normal population , x may be far from multinormal. The problem is most severe with the smallest sized subgroups, individual observations. The univariate non-normality problem was recently …