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Statistical Process Monitoring with Integrated Moying Average Noise 257 to detect smali shifts. The comparisons also show that the Likelihood Ratio scheme usually does not compete well against CUSUMs and EWMAs. For this reason we have not extensively tabulated its ARL properties as we did for the other schemes.

ARL comparisons among schemes Having looked at each of 4 classes of monitoring schemes individually, what can be said about relative performance among the classes?

Figurę 7 shows optimal ARL curves for IMA processes with parameters ranging from X = 0 in the upper left panel to X = 1 in the lower right one. Each curve is a function of the shift size p and shows the minimum ARL that can be achieved within the given class of schemes subject to the constraint ARL₀ = 500. Generally, no single scheme within a class can attain the minimum ARL for several shift sizes so the optimal scheme changes with p. For example, within the CUSUM curves, k changes with p to achieve the minimum ARL. Thus, Figurę 5 should be used to judge how well a CUSUM chart optimized for one value of p will perform for others.

The conclusions drawn from Figurę 7 are remarkably simple. Namely, for a given sized shift, a CUSUM can be designed to perform at least as well as, and often better than, any of the other schemes. For the ARL criterion CUSUMs always perform best and Shewhart individuals charts worst. The ordering between the EWMA and the likelihood ratio scheme, however, depends on X.

Shewhart individuals charts sometimes perform miserably and never do better than the others. In the case of random walks (X = 1), all schemes have eąually poor performances. In fact, for random walks, the optimal member of each class reduces to a Shewhart individuals chart. A finał broad observation from Figurę 7 has already been madę but bears repeating: it is substantially morę difficult to detect level shifts in IMAs as X increases.

Hypothesis Testing

Concepts such as run length distributions are important for ‘bn-line” monitoring applications because control charts are meant to be continually updated as a process generates new data. Occasionally, however, it is useful to ask ‘bff-line” ąuestions about past data. For example, one might like to know how much evidence there is that a step shift in the level of a process occurred in a given set of data. Hypothesis testing can be useful in this context. This