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control chart; frame of span $k$; average run length; probability distribution; compact metric space
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.
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