Monitoring a sequence of Bernoulli random variables subject to gradual changes in the success rates where the success rates are unknown

Abstract

We look at a sequence of Bernoulli random variables where the success rates change from θ 1 to θ 2. We will assume that both the success rates before and after the change are unknown but increasing. We assume that this change does not happen abruptly but gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop late.