Good robust estimators can be tuned to combine a high breakdown point and a specified asymptotic efficiency at a central model. This happens in regression with MM- and tau-estimators among others. However, the finite-sample efficiency of these estimators can be much lower than the asymptotic one. To overcome this drawback, an approach is proposed for parametric models, which is based on a distance between parameters. Given a robust estimator, the proposed one is obtained by maximizing the likelihood under the constraint that the distance is less than a given threshold. For the linear model with normal errors and using the MM estimator and the distance induced by the Kullback-Leibler divergence, simulations show that the proposed estimator attains a finite-sample efficiency close to one, while its maximum mean squared error under pointwise outlier contamination is smaller than that of the MM estimator. The same approach also shows good results in the estimation of multivariate location and scatter.