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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