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In this paper, we propose and assess a CFAR detector that can adjust its ldquodirectivityrdquo through a real scalar parameter. It relies on the usual assumption that a set of homogeneous training data is available and encompasses as special cases the well-known Kelly's GLRT and the recently introduced W-ABORT detector. More important, it can be tuned in order to control the level to which sidelobe signals are rejected. Such functionality is particularly important to contain the number of false alarms in presence of mismatched signals. We also consider a parametric detector which resorts to a diagonally loaded sample covariance matrix commonly adopted to take advantage of the presence of strong interferers. The performance assessment of such detector has shown that it can significantly outperform Kelly's GLRT in terms of prediction probabilities for matched signals and in terms of selectivity, but it is not strictly CFAR. We also propose to use the CFAR parametric detector as second stage of a two-stage tunable detector and show that such a two-stage detector can outperform already proposed tunable receivers in terms of selectivity. The analysis of the detectors is conducted assuming a homogeneous Gaussian environment; with reference to this scenario and to the CFAR detectors we derive analytical expressions for the probability of false alarm and the probability of detection for both matched and mismatched signals.