Suboptimal properties of page's CUSUM and shiryayev-roberts procedures in change-point problems with dependent observations

We construct a simple counterexample to the conjectures of Pollak (1985) and Yakir, Krieger and Pollak (1999), which state that Page's CUSUM procedure and the Shiryayev-Roberts procedure are asymptotically minimax optimal for dependent observations. Moreover, our example shows that the close relationship between open-ended tests and change-point detection procedures no longer holds for dependent observations. As a consequence, the standard approach which constructs change-point detection procedures based on asymptotically optimal openended tests does not in general provide asymptotically optimal procedures for dependent observations.