This paper presents two different approaches to derive the asymptotic distributions of the robust Adaptive Normalized Matched Filter (ANMF) under both H0 and H1 hypotheses. More precisely, the ANMF has originally been derived under the assumption of partially homogenous Gaussian noise, i.e. 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 analyse two asymptotic performance characterization of this robust ANMF. The first approach consists in exploiting the asymptotic distribution of the different covariance matrix estimators while the second approach is to directly exploit the asymptotic distribution of the ANMF distribution built with these estimates.