This work introduces a novel detector, bounded false-alarm rate (BFAR), for distinguishing true detections from noise in radar data, leading to improved accuracy in radar odometry estimation. Scanning frequency-modulated continuous wave (FMCW) radars can serve as valuable tools for localization and mapping under low visibility conditions. However, they tend to yield a higher level of noise in comparison to the more commonly employed lidars, thereby introducing additional challenges to the detection process. We propose a new radar target detector called BFAR which uses an affine transformation of the estimated noise level compared to the classical constant false-alarm rate (CFAR) detector. This transformation employs learned parameters that minimize the error in odometry estimation. Conceptually, BFAR can be viewed as an optimized blend of CFAR and fixed-level thresholding designed to minimize odometry estimation error. The strength of this approach lies in its simplicity. Only a single parameter needs to be learned from a training dataset when the affine transformation scale parameter is maintained. Compared to ad-hoc detectors, BFAR has the advantage of a specified upper-bound for the false-alarm probability, and better noise handling than CFAR. Repeatability tests show that BFAR yields highly repeatable detections with minimal redundancy. We have conducted simulations to compare the detection and false-alarm probabilities of BFAR with those of three baselines in non-homogeneous noise and varying target sizes. The results show that BFAR outperforms the other detectors. Moreover, We apply BFAR to the use case of radar odometry, and adapt a recent odometry pipeline, replacing its original conservative filtering with BFAR. In this way, we reduce the translation/rotation odometry errors/100 m from 1.3%/0.4◦ to 1.12%/0.38◦, and from 1.62%/0.57◦ to 1.21%/0.32◦, improving translation error by 14.2% and 25% on Oxford and Mulran public data sets, respectively.