  The aim of this paper is to present a new estimation procedure that can be
applied in many statistical frameworks including density and regression and
which leads to both robust and optimal (or nearly optimal) estimators. In
density estimation, they asymptotically coincide with the celebrated maximum
likelihood estimators at least when the statistical model is regular enough and
contains the true density to estimate. For very general models of densities,
including non-compact ones, these estimators are robust with respect to the
Hellinger distance and converge at optimal rate (up to a possible logarithmic
factor) in all cases we know. In the regression setting, our approach improves
upon the classical least squares from many aspects. In simple linear regression
for example, it provides an estimation of the coefficients that are both robust
to outliers and simultaneously rate-optimal (or nearly rate-optimal) for large
class of error distributions including Gaussian, Laplace, Cauchy and uniform
among others.
