  Robust estimators of location and dispersion are often used in the elliptical
model to obtain an uncontaminated and highly representative subsample by
trimming the data outside an ellipsoid based in the associated Mahalanobis
distance. Here we analyze some one (or $k$)-step Maximum Likelihood Estimators
computed on a subsample obtained with such a procedure. We introduce different
models which arise naturally from the ways in which the discarded data can be
treated, leading to truncated or censored likelihoods, as well as to a
likelihood based on an only outliers gross errors model. Results on existence,
uniqueness, robustness and asymptotic properties of the proposed estimators are
included. A remarkable fact is that the proposed estimators generally keep the
breakdown point of the initial (robust) estimators, but they could improve the
rate of convergence of the initial estimator because our estimators always
converge at rate $n^{1/2}$, independently of the rate of convergence of the
initial estimator.
