We investigate several important properties of an Adaptive Energy Detector (AED), which is originally proposed according to the Generalized Likelihood Ratio Test (GLRT), under the assumption that the signal steering vector is completely unknown. We show that the AED coincides with the Rao and Wald tests. We also give the exact statistical distribution, according to which one can easily derive the Probabilities of Detection (PD) and False Alarm (PFA), and we use the AED to design a parametrically tunable signal-mismatch detector. Compared to existing tunable detectors, the novel tunable detector is more flexible in governing the rejection of the mismatched signal; moreover, for the matched signal, the tunable detector can provide a higher PD than existing detectors. Two scalars, called the tunable parameters, control this functionality.