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The optimal control chart procedure. (English) Zbl 1249.93178
Summary: The Moving Average (MA) chart, the Exponentially Weighted Moving Average (EWMA) chart and the CUmulative SUM (CUSUM) chart are the most popular schemes for detecting shifts in a relevant process parameter. Any control chart system of span \(k\) is specified by a partition of the space \({\mathbb R}^k\) into three disjoint parts. We call this partition as the control chart frame of span \(k\). A shift in the process parameter is signalled at time \(t\) by having the vector of the last \(k\) sample characteristics fall out of the central part of this frame. The optimal frame of span \(k\) is selected in order to maximize the Average Run Length (ARL) if shift in the relevant process parameter is on an acceptable level and to minimize it on a rejectable level. We have proved in this article that the set of all frames of span \(k\) with an appropriate metric is a compact space and that the ARL for continuously distributed sample characteristics is continuous as a function of the frame. Consequently, there exists the optimal frame among systems of span \(k\). General attitude to control chart systems is the common platform for universal control charts with the particular point for each sample and variable control limits plotted one step ahead.
MSC:
93E20 Optimal stochastic control
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
62F15 Bayesian inference
62P30 Applications of statistics in engineering and industry; control charts
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