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Since many safety-critical systems, such as surgical robots and autonomous driving cars operate in unstable environments with sensor noise and incomplete data, it is desirable for object detectors to take the localization uncertainty into account. However, there are several limitations of the existing uncertainty estimation methods for anchor-based object detection. 1) They model the uncertainty of the heterogeneous object properties with different characteristics and scales, such as location (center point) and scale (width, height), which could be difficult to estimate. 2) They model box offsets as Gaussian distributions, which is not compatible with the ground truth bounding boxes that follow the Dirac delta distribution. 3) Since anchor-based methods are sensitive to anchor hyper-parameters, their localization uncertainty could also be highly sensitive to the choice of hyper-parameters. To tackle these limitations, we propose a new localization uncertainty estimation method called UAD for anchor-free object detection. Our method captures the uncertainty in four directions of box offsets (left, right, top, bottom) that are homogeneous, so that it can tell which direction is uncertain, and provide a quantitative value of uncertainty in [0, 1]. To enable such uncertainty estimation, we design a new uncertainty loss, negative power log-likelihood loss, to measure the localization uncertainty by weighting the likelihood loss by its IoU, which alleviates the model misspecification problem. Furthermore, we propose an uncertainty-aware focal loss for reflecting the estimated uncertainty to the classification score. Experimental results on COCO datasets demonstrate that our method significantly improves FCOS, by up to 1.8 points, without sacrificing computational efficiency.