Worst-Case Misidentification Control in Sequential Change Diagnosis using the min-CuSum

TL;DR

This paper analyzes a sequential change diagnosis method using min-CuSum, demonstrating its exponential decay in misidentification probability and asymptotic optimality in detection delay under worst-case conditions.

Contribution

It introduces a worst-case misidentification control framework for sequential change diagnosis using min-CuSum, proving exponential decay and asymptotic optimality.

Findings

Misidentification probability decays exponentially with threshold.

Algorithm asymptotically minimizes Lorden's detection delay.

Theoretical results are supported by simulation studies.

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, upon stopping, to accurately identify the post-change distribution among a finite set of alternatives. The algorithm that raises an alarm as soon as the CuSum statistic that corresponds to one of the post-change alternatives exceeds a certain threshold is studied. When 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, this algorithm is shown to asymptotically minimize Lorden's detection delay criterion, simultaneously for every possible post-change distribution, within the class of schemes that satisfy prescribed bounds on the false alarm rate and the worst-case conditional probability of misidentification, as the former goes to zero sufficiently faster than the latter. Finally, these theoretical results are also illustrated in simulation studies.