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We look at a sequence of Bernoulli random variables where the success rates change from θ1 to θ2. 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.
Marlo Brown (Tue,) studied this question.