This paper presents a new approach to robust estimation of a location parameter, focusing on asymptotic properties of estimators for contaminated normal distributions. The author introduces estimators that are asymptotically most robust among all translation invariant estimators. These estimators are intermediaries between the sample mean and sample median, and are defined by minimizing a function of the errors, $\rho(x_i - T)$, where $\rho$ is a non-constant function. The most robust estimator corresponds to a specific form of $\rho$, which is related to Winsorizing. The paper also discusses the asymptotic normality of these estimators and their performance under different models of contamination. It shows that the most robust estimator minimizes the asymptotic variance under a set of contaminating distributions. The paper further explores the minimax properties of these estimators, showing that they achieve optimal performance in certain cases. The results are applied to the contaminated normal distribution, where the most robust estimator is shown to minimize the maximal asymptotic variance. The paper concludes with a discussion of the implications of these results for robust estimation in the presence of contamination.This paper presents a new approach to robust estimation of a location parameter, focusing on asymptotic properties of estimators for contaminated normal distributions. The author introduces estimators that are asymptotically most robust among all translation invariant estimators. These estimators are intermediaries between the sample mean and sample median, and are defined by minimizing a function of the errors, $\rho(x_i - T)$, where $\rho$ is a non-constant function. The most robust estimator corresponds to a specific form of $\rho$, which is related to Winsorizing. The paper also discusses the asymptotic normality of these estimators and their performance under different models of contamination. It shows that the most robust estimator minimizes the asymptotic variance under a set of contaminating distributions. The paper further explores the minimax properties of these estimators, showing that they achieve optimal performance in certain cases. The results are applied to the contaminated normal distribution, where the most robust estimator is shown to minimize the maximal asymptotic variance. The paper concludes with a discussion of the implications of these results for robust estimation in the presence of contamination.