On the Relationship Between Two CUSUM Charts

The cumulative sum (CUSUM) chart is a fundamental tool in statistical process control (SPC) and sequential change‐point detection. Two closely related formulations appear in the literature: the standard CUSUM introduced by Page in 1954 and the CUSUM studied by Moustakides in 1986, the latter of which is shown to be exactly optimal under Lorden's minimax criterion. Although these two versions differ in their recursive definitions, their close relationship is not always explicitly articulated in the SPC literature. In this technical note, we clarify the relationship between these two formulations. On the one hand, we show that the standard CUSUM statistic is the positive part of the Moustakides CUSUM statistic. Thus, when the decision threshold is positive, the two corresponding control charts yield identical stopping times. Consequently, the classical SPC formulation of the standard CUSUM inherits the exact optimality established for the Moustakides CUSUM under reasonable practical settings. On the other hand, when the decision threshold is non‐positive, a case that is primarily of theoretical interest, these two CUSUM charts are indeed different. We provide illustrative examples for Gaussian data in this case, showing that the values of the two CUSUM charts can be significantly different, whereas the difference in their values is always bounded by 1. We hope that this technical note provides a clear theoretical link between these two formulations and helps clarify their relationship and optimality properties.