This paper provides an original asymptotic analysis of robust adaptive detectors performance in the context of non-Gaussian observations. We focus on a single-steering case in homogeneous environment and analyze the properties of different adaptive detectors such as Adaptive (Normalized) Matched Filter (AMF/ANMF), Kelly’s GLRT, and Rao test when an estimator of the covariance matrix is plugged-in. When the noise distribution turns to be non-Gaussian, the detectors relying on the Sample Covariance Matrix (SCM) can perform poorly and an interesting alternative is the use of M-estimators. In this context, we show that, from Complex Elliptically Symmetric (CES) samples, the distribution of a detection statistic built with M-estimators can be accurately approximated by the one of the same statistic built from the SCM of an equivalent Gaussian setting. The loss due to this approximation is theoretically derived and shown to be negligible in most cases. This explicit equivalent statistic is especially interesting since it allows to tune robust plug-in detectors with well established results from the Gaussian detection framework. Furthermore, this approach provides new insights on the robust estimation tools behavior. Finally, some simulations illustrate the interest of the proposed results.