SUMMARYTo design highly efficient and robust detectors of a weak signal, an asymptotic approach to stable estimation exploiting redescending score functions is used. Two new indicators of robustness of detection, the detection error sensitivity and detection stability, are introduced. The optimal Neyman–Pearson rules maximizing detection efficiency under the guaranteed level of detection stability are written out. Under heavy‐tailed noise distributions, the proposed asymptotically stable detectors based on redescending score functions, namely, the minimum error sensitivity and the radical ones, outperform conventional linear bounded Huber's and redescending Hampel's detectors both on small and large samples. Copyright © 2013 John Wiley & Sons, Ltd.