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The coherent radar detection against a background of compound-Gaussian clutter with inverse gamma texture is studied and three detectors: One-step generalised likelihood ratio test (1S-GLRT), maximum a posteriori GLRT (MAP-GLRT) and two-step GLRT (2S-GLRT) are proposed. The detectors have the same structure with their test statistics and modified thresholds, respectively, related to the scale and the shape parameters of the texture, which can also be formulated in a matched filter (MF) form. Subsequently, the performance assessments are given by their probability of detection and probability of false alarm. The authors find that the probability of false alarm is dependent on the shape parameter, meaning the detectors have no CFAR property. When the shape parameter and the number of the integrated radar pulses satisfy certain condition, it has no relation with the shape parameter and then the detectors have CFAR property. Finally, simulation results show that: (i) 1S-GLRT and MAP-GLRT have the same performance for fixed probability of false alarm and 2S-GLRT bears slightly bad performance; (ii) the performance of 1S-GLRT is much closer to the adaptive coherence estimator (ACE) and is better than that of the Kelly GLRT and (iii) the 1S-GLRT is robust when parameter estimation errors exist.