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Yander Wiel

exponentially weighted moving averages (EWMAs), Shewhart individuaJs charts, and a scheme based on likelihood ratio statistics. Comparing ARL properties of these schemes shows that CUSUMs can be designed that are at least as good as, and sometimes better than, any of the others. Some graphical aids are provided for designing CUSUMs and EWMAs in this context.

The likelihood ratio statistic is also studied in the context of hypothesis testing. A recursive formula is developed to compute its exact nuli distribution. A much simpler approximation for upper taił areas of the distribution is also given. The approximation is based on an extended Bonferroni inequality and is adeąuate to approximate P-values.

Introduction: Correlated Data and Statistical Monitoring

Too often a stream of manufacturing data that wanders about as in the top panel of Figurę 1 is subjected to a monitoring scheme that expects observations to behave like independent and identically distributed (iid) random variables. The lower panel of Figurę 1 shows an exponentially weighted moving average (EWMA) control chart of the data where the EWMA gives a weight of 0.12 to the most recent observation. The 3-sigma control limits are calculated by estimating the process variance using successive differences. Of course, this local measure of variance is smaller than the total variance because it does not include the variability due to the meandering level. Thus, the control chart shows a lack of Statistical control.” For very good reasons, all of the standard control charts for continuous measurements use local measures of variability like moving ranges or estimates based on sequential subsamples. Hence, they will show that the process is not iid.

Engineers wanting to monitor this kind of data are usually well aware that the process wanders and they do not want a monitoring scheme that will continually tell them what they already know. A common way to Tix” the problem is to widen the control limits until the control chart rarely signals. This clearly reduces the number of uninformative alarms. But does it make the chart morę useful? The answer depends upon the application as the following examples indicate.

Example 1—widening control limits makes sense The top panel of Figurę 1 shows 500 measurements of the power of laser light entering successive pieces of optical flber at a laboratory inspection station. Every second measurement from about 1 week of production is shown. The eąuipment measures the fraction of energy launched into the fiber that is dissipated before reaching the exit end. If the power entering the fiber is too Iow the measurement will be too