We investigate the asymptotic behavior of a weighted sample mean and covariance,where the weights are determined by the Mahalanobis distances with respect to initial robust estimators.We derive an explicit expansion for the weighted estimators. From this expansion it can be seen that reweighting does not improve the rate of convergence of the initial estimators.We also show that if one uses smooth $S$-estimators to determine the weights, the weighted estimators are asymptotically normal. Finally, we will compare the efficiency and local robustness of the reweighted $S$-estimators with two other improvements of $S$-estimators: $\tau$-estimators and constrained $M$-estimators.
"Asymptotics of reweighted estimators of multivariate location and scatter." Ann. Statist. 27 (5) 1638 - 1665, October 1999. https://doi.org/10.1214/aos/1017939145