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This paper deals with the problem of detecting a signal known up to a scaling factor in the presence of Gaussian disturbance with unknown covariance matrix. A new detector based on the Rao test criterion is introduced and its invariance properties and constant false alarm rate (CFAR) behavior are studied. At the analysis stage, the performance of the new receiver is assessed, also in comparison with some classical adaptive radar detectors, both in the matched as well as in the mismatched signal case. Remarkably, the Rao test may achieve a matched detection performance which is commensurate with that of the generalized likelihood ratio test (GLRT)-based detectors if a sufficient number of training data is available. Moreover, it also exhibits better rejection capabilities of mismatched signals than the counterparts. In the last part of the work, a two-stage detector whose second stage coincides with the Rao test is devised. It represents a suitable means to restore the detection performance of the plain Rao test in the presence of a small number of training data. Finally, the performance of the aforementioned two-stage processor is analyzed in closed form and the CFAR behavior is proved.