This paper presents two approaches to derive an asymptotic distribution of the robust Adaptive Normalized Matched Filter (ANMF). More precisely, the ANMF has originally been derived under the assumption of Gaussian distributed noise where the variance is different between the observation under test and the set of secondary data. We propose in this work to relax the Gaussian hypothesis: we analyze the ANMF built with robust estimators, namely the M-estimators and the Tyler's estimator, under the Complex Elliptically Symmetric (CES) distributions framework. In this context, we derive two asymptotic distributions for this robust ANMF. Firstly, we combine the asymptotic properties of the robust estimators and the Gaussian-based distribution of the ANMF at finite distance. Secondly, we directly derive the asymptotic distribution of the robust ANMF. Then, Monte-Carlo simulations show the good approximation provided by the proposed methods. Moreover, for a non-asymptotic regime, the simulations provide very promising results.