Worst-Case Misidentification Control in Sequential Change Diagnosis Using the Min-CuSum

Abstract

The problem of sequential change diagnosis is considered, where a sequence of independent random elements is accessed sequentially, there is an abrupt change in its distribution at some unknown time, and there are two main operational goals: to quickly detect the change, and to accurately identify upon stopping the post-change distribution among a finite set of alternatives. The focus is on the min-CuSum algorithm, which raises an alarm as soon as a CuSum statistic that corresponds to one of the post-change alternatives exceeds a certain threshold. We obtain, under certain assumptions, non-asymptotic upper bounds on its conditional probability of misidentification given that a false alarm did not occur. When, in particular, the data are generated over independent channels and the change can occur in only one of them, its worst-case—with respect to the change point—conditional probability of misidentification given that there was not a false alarm is shown to decay exponentially fast in the threshold. As a corollary, in this setup, the min-CuSum is shown to asymptotically minimize Lorden’s detection delay criterion, simultaneously for every post-change scenario, within the class of schemes that satisfy prescribed bounds on both the false alarm rate and the worst-case conditional probability of misidentification, in a regime where the latter does not go to zero faster than the former. Finally, these theoretical results are also illustrated in simulation studies.