Probability distribution of CUSUM charting statistics

Abstract In parallel to the conventional study of the distribution of the first time to signal, this paper investigates the probability distribution of the article Cumulative Sum (CUSUM) charting statistics based on a recurrence relationship. The probability distribution of the CUSUM statistic can not only provide statistical significance of observations against the null hypothesis of being in control but also facilitate the analysis of the CUSUM chart in the steady-state scenario. Both the conditional case (CUSUM chart without restarting) and the cyclical case (CUSUM chart with restarting) are considered. It is shown that the distribution of CUSUM charts both with and without restarting approaches a stationary distribution, independent of their initial values. We also show that the null steady-state distribution of the unbounded CUSUM chart previously reported in the literature is a special case of this article as the control limit approaches infinity.