### Abstract:

Common estimators of the mean g(θ) = Jx dFθ(x) in skewed distribution models may be sensitive to contamination by a few large observations. It is then desirable to consider robust estimators g(θ) .The approach of Hampel (1968), who defines an estimator θ for the parameter vector θ to be optimal B-robust if it is asymptotically efficient subject to a given upper bound on the norm of its influence function, is used to construct optimal robust estimators for g(θ) . An estimator g(θ) is defined to be functional invariant when it preserves the robustness and optimality properties of a robust estimator θ . The invariance of the optimal B-robust estimators are used to construct optimal B-robust estimator for the mean of multi-parameter distributions. An algorithm for computing the optimal B-robust score function for any distribution is developed. An optimal B-robust L-estimator for the location-scale family is also constructed. Asymptotic relative efficiencies of the optimal B-robust estimators for the mean of the lognormal and Weibull distributions are computed and compared with those for several other robust and nonrobust estimators. Type II censoring is considered as a method to achieve B-robustness. The optimal proportion of trimming is defined as that proportion which produces the smallest asymptotic MSE in the class of censored data estimators subject to some upper bound on the influence function. Several common estimators for censored data, including the maximum likelihood, a modified maximum likelihood [Tiku et al., 1986] and an L-estimator [Chernoff, et al. ,1967], are shown to have larger MSE than the optimal B-robust estimator with the same upper bound on the influence function. The optimal proportions of trimming are computed for the MLE and L-estimator of the mean of lognormal and Weibull distributions. A simulation study of nine estimators for the mean of a lognormal distribution shows that the optimal B-robust estimator has the smallest MSE for the sample size and contamination cases considered. All B-robust estimators considered are found to be better than the nonrobust ones with regard to both MSE and bias.