Asymptotically Optimal Quickest Change Detection under a Nuisance Change

This paper considers the problem of quickest detection of a change in distribution where the signal may undergo both nuisance and critical changes. Our goal is to detect the critical change without raising a false alarm over the nuisance change. We formulate the quickest change detection (QCD) problem in the presence of a nuisance change following Lorden's formulation. We propose a window-limited sequential change detection procedure based on the generalized likelihood ratio test statistic for the problem of QCD in which both nuisance and critical changes may occur. We derive a recursive update scheme for our proposed test statistic and show that our test is asymptotically optimal under mild technical conditions. We compare our proposed stopping rules with a naive 2-stage stopping time, which attempts to detect the changes using separate CuSum stopping procedures for the nuisance and critical changes. Simulations suggest that our proposed stopping time outperforms the naive 2-stage procedures.