Robust Sequential Change Detection: The Approach Based on Breakdown Points and Influence Functions

Sequential change-point detection has many important applications in industrial quality control, signal detection, and clinical trials. However, many classical procedures may fail when the observed data are contaminated by outliers, even if the percentage of outliers is very small. In this paper, we focus on the problem of robust sequential change-point detection in the presence of a small proportion of random outliers. We first study the statistical detection properties of a general family of detection procedures under Huber’s gross error model. Moreover, we incorporate ideas of the breakdown point and the influence function from the classical offline robust statistics literature and propose their new definitions to quantify the robustness of general sequential change-point detection procedures. Then, we derive the breakdown points and influence functions of our proposed family of detection procedures, which provide a quantitative analysis of the robustness of these procedures. Moreover, we find the optimal robust bounded-influence procedure in that general family that has the smallest detection delay subject to the constraints on the false alarm rate influence function. It turns out the optimal procedure is based on truncation of the scaled 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 and compute numerical approximations of breakdown points and influence functions of some procedures to have a quantitative understanding of the robustness of different procedures.