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Yander Wiel

observations from a process (absent an abrupt change in level) are well described by an ARIMA model (Box and Jenkins, 1976), then minimum variance 1-step ahead forecasts of the process based on past data are easily formed. Furthermore, the forecast errors should be approximately iid. They can thus be monitored using standard control charts and false alarm rates can be gaged using tables and plots of ARLs found in the literaturę. Crowder (1987) tabulates ARLs for EWMAs. Goel and Wu (1971) give a nomogram for CUSUM ARLs. Hence, a control chart can easily be set up with a given false alarm ratę. But many important practical ąuestions about these charts remain unanswered: What does an alarm indicate? How ąuickly can a sudden shift in the level of the process trigger an alarm? Are the usual control charts sufficient for monitoring forecast errors? This paper helps answer these ąuestions by studying several methods of monitoring integrated moving average (IMA) processes.

Organization of this paper The next subsection reviews some relevant literaturę. Section 2 defines the class of first order IMAs and shows how a step shift in level affects the forecast errors. Section 3 describes 4 diflerent schemes for detecting step shifts in IMAs and compares their ARL performances. Special attention is given to designing CUSUM and EWMA charts for forecast errors from IMA processes. Section 4 derives the nuli distribution of the likelihood ratio test for detecting a level shift and gives an accurate approximation for upper taił probabilities. Section 5 gives a summary and concluding remarks.

Releyant literaturę Performance properties of monitoring schemes such as average run lengths (ARLs) have been widely tabulated for iid Gaussian seąuences and common schemes such as cumulative sum (CUSUM) charts and exponentially weighted moving average (EWMA) charts. See, for example, Lucas and Crosier (1982) and Lucas and Saccucci (1990). In addition, ARLs are easy to calculate for Shewhart individuals and X charts. Much less guidance, however, is available for choosing and designing monitoring schemes appropriate for autocorrelated data. Tracking signals have been used to monitor performance of forecasting systems for morę than 30 years. Brown's (1962) tracking signal is the cumulative sum of forecast errors divided by an EWMA of their absolute values. Trigg (1964) replaced the cumulative sum in Brown's numerator with an EWMA. Golder and Settle (1976) simulated ARLs of these tracking signals. Gardner (1983) gave morę extensive simulation results and introduced a tracking signal for detecting autocorrelation in the forecast errors—an indication that the forecasts can be improved.