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The present article examines the performance of the Shiryaev–Roberts (SR) procedure for Markov‐dependent count time series, using the Poisson INARCH(1) model as the representative data‐generating count process. For the purpose of easier performance evaluation, a comparative analysis with existing cumulative sum (CUSUM) results from the literature is provided. In particular, the zero‐state and the steady‐state behavior of the two control schemes is considered with regard to the average run length (ARL), the median run length, and the extra quadratic loss as performance measures. The comparison shows that SR performs at least as well as its more popular competitor in detecting changes in the process distribution. In terms of usability, however, the SR procedure has a practical advantage, which is illustrated by an application to a real data set. Moreover, a parametric bootstrap study based on a second data example investigates the effects of parameter estimation on the chart's true ARL to false alarm. In sum, the research reveals the SR chart to be the better tool for monitoring Markov‐dependent counts.

Monitoring stochastic processes with control charts is the main field of application in statistical process control. For a Poisson hidden Markov model (HMM) as the underlying process, we investigate a Shewhart individuals chart, an ordinary Cumulative Sum (CUSUM) chart, and two different types of log‐likelihood ratio (log‐LR) CUSUM charts. We evaluate and compare the charts' performance by their average run length, computed either by utilizing the Markov chain approach or by simulations. Our performance evaluation includes various out‐of‐control scenarios as well as different levels of dependence within the HMM. It turns out that the ordinary CUSUM chart shows the best overall performance, whereas the other charts' performance strongly depend on the particular out‐of‐control scenario and autocorrelation level, respectively. For illustration, we apply the HMM and the considered charts to a data set about weekly sales counts.

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