Robust Estimators of Location and Their Second-Order Asymptotic Relations

Let X 1, …, X n be independent random variables, identically distributed according to the distribution function F(x — θ), where θ is the parameter to be estimated. F is generally unspecified; we only assume that F has a symmetric density f. Three broad classes of robust estimators of θ, these of M-estimators, L-estimators, and R-estimators, are first briefly described. Denoting these estimators M n, L n, and R n, respectively, we give sufficient conditions under which these estimators are asymptotically equivalent in probability, i.e., $$\sqrt n (M_n - L_n )\xrightarrow{p}0$$ , etc., as n → ∞. These relations are supplemented by the rates of convergence in most cases.