Doctor of Philosophy / Department of Statistics / James J. Higgins / Outliers in the data impair traditional estimators of location, variance, and regression parameters so researchers tend to look for robust estimators, i.e., estimators that aren’t sensitive to outliers. These robust estimators can tolerate a certain proportion of outliers. Besides robustness, efficiency is another desirable property. Researchers try to find estimators that are efficient under standard conditions and use them when outliers exist in the data. In this study the robustness and efficiency of a class of estimators that we call [subscript n]C[subscript k ]estimators are investigated. Special cases of this method exist in the literature including U and generalized L-statistics. This estimation technique is based on taking all subsamples of size k from a sample of size n, finding the estimator of interest for each subsample, and specifying one of them, typically the median, or a linear combination of them as the estimator of the parameter of interest. A simulation study is conducted to evaluate these estimators under different distributions with small sample sizes. Estimators of location, scale, linear regression and multiple regression parameters are studied and compared to other estimators existing in the literature. The concept of data depth is used to propose a new type of estimator for the regression parameters in multiple regression.