Optimal Bounded-Influence Procedure for Robust Sequential Change-Point Detection

In this paper, we study the problem of robust sequential change-point detection under Huber's gross error model. We incorporate ideas of the influence function from the classical offline robust statistics literature and propose a new definition called false alarm influence function to quantify the robustness of general sequential change detection procedures. Then, we propose a general family of robust detection procedures and derive their false alarm influence function. The result shows they all have bounded influence function which implies any single arbitrary outlier will only have limit impact on their detection performance. Furthermore, we construct the optimal robust bounded-influence procedure in that general family that has the smallest detection delay subject to the false alarm rate constraint and the false alarm influence function constraint. It turns out the optimal procedure is based on the truncated likelihood ratio statistic and has a simple form. Finally, we demonstrate the robustness and the detection efficiency of the optimal robust bounded-influence procedure through extensive simulations.