  Estimators derived from a divergence criterion such as $\varphi-$divergences
are generally more robust than the maximum likelihood ones. We are interested
in particular in the so-called MD$\varphi$DE, an estimator built using a dual
representation of $\varphi$--divergences. We present in this paper an iterative
proximal point algorithm which permits to calculate such estimator. This
algorithm contains by its construction the well-known EM algorithm. Our work is
based on the paper of \citep{Tseng} on the likelihood function. We provide
several convergence properties of the sequence generated by the algorithm, and
improve the existing results by relaxing the identifiability condition on the
proximal term, a condition which is not verified for most mixture models and
hard to be verified for non mixture ones. Since convergence analysis uses
regularity conditions (continuity and differentiability) of the objective
function, which has a supremal form, we find it useful to present some
analytical approaches for studying such functions. Convergence of the EM
algorithm is discussed here again in a Gaussian and Weibull mixtures in the
spirit of our approach. Simulations are provided to confirm the validity of our
work and the robustness of the resulting estimators against outliers.
