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Abstract:
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Control charting count data (binomial, Poisson, negative binomial etc.) became a popular topic during the last decade. Here we deal with the classic CUSUM chart and provide several new insights. First, by utilizing the Toeplitz-like structure of the CUSUM transition matrix we are able to calculate the average run length (ARL) for huge matrix dimensions. The latter appear, if the rational approximation of the optimal reference value k exhibits a large denominator. In this way, we are able to determine optimal CUSUM ARL envelops for benchmarking analyses. Note that for obtaining this ideal ARL curve we make use of an idea introduced by Yashchin (2019). He developed a simple recursion for the ARL while adding another state to the transition matrix. This makes it easy to calculate the randomization probability to meet exactly the in-control ARL condition, which is a prerequisite for a smooth ARL envelop. Second, we analyze combos of two CUSUM charts, both for detecting increases, to improve the detection performance for a range of mean shifts. Their detection power is measured by using the above ARL envelope. It turns out that the combo gets quite close to the ideal ARL curve.
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