  We discuss a new weighted likelihood method for parametric estimation. The
method is motivated by the need for generating a simple estimation strategy
which provides a robust solution that is simultaneously fully efficient when
the model is correctly specified. This is achieved by appropriately weighting
the score function at each observation in the maximum likelihood score
equation. The weight function determines the compatibility of each observation
with the model in relation to the remaining observations and applies a
downweighting only if it is necessary, rather than automatically downweighting
a proportion of the observations all the time. This allows the estimators to
retain full asymptotic efficiency at the model. We establish all the
theoretical properties of the proposed estimators and substantiate the theory
developed through simulation and real data examples. Our approach provides an
alternative to the weighted likelihood method of Markatou et al. (1997, 1998).
