Object detection plays a critical role in autonomous driving, but current state-of-the-art object detectors will inevitably fail in many driving scenes, which is unacceptable for safety-critical automated vehicles. Given the complexity of the real traffic scenarios, it is impractical to guarantee zero detection failure; thus, online failure prediction is of crucial importance to mitigate the risk of traffic accidents. Of all the failure cases, False Negative (FN) objects are most likely to cause catastrophic consequences, but little attention has been paid to the online FN prediction. In this paper, we propose a general introspection framework that can make online prediction of FN objects for black-box object detectors. In contrast to existing methods which rely on empirical assumptions or handcrafted features, we facilitate the FN feature extraction by an introspective FN predictor we designed in this framework. For this purpose, we extend the original concept of introspection to object-wise FN predictions, and propose a multi-branch cooperation mechanism to address the distinct foreground-background imbalance problem of FN objects. The effectiveness of the proposed framework is verified through extensive experiments and analysis, and the results show that our method successfully predicts the FN objects with 81.95% precision for 88.10% recall on the challenging KITTI Benchmark, and effectively improves object detection performance by taking FN predictions into consideration.