When a Cusum Stops, What Confidence Is There that the Alarm Is Not False?

In the framework of the Cusum procedure, the evolution of a false alarm has a well-understood stochastic behavior. So, if observations preceding an alarm were to exhibit a behavior that is significantly different, there would be reason to reject the hypothesis that the alarm is false. We develop a test of this difference. The method is applied to detecting a change in a Covid-19 context involving a possible increase of a mean and in a context involving a possible increase in the probability of a rare event. Supplementary materials for this article are available online.