  Robust estimators for Generalized Linear Models (GLMs) are not easy to
develop because of the nature of the distributions involved. Recently, there
has been an increasing interest in this topic, especially in the presence of a
possibly large number of explanatory variables. Transformed M-estimators (MT)
are a natural way to extend the methodology of M-estimators to the class of
GLMs and to obtain robust methods. We introduce a penalized version of
MT-estimators in order to deal with high-dimensional data. We prove, under
appropriate assumptions, consistency and asymptotic normality of this new class
of estimators. The theory is developed for redescending $\rho$-functions and
Elastic Net penalization. An iterative re-weighted least squares algorithm is
given, together with a procedure to initialize it. The latter is of particular
importance, since the estimating equations might have multiple roots. We
illustrate the performance of this new method for the Poisson family under
several type of contaminations in a Monte Carlo experiment and in an example
based on a real dataset.
