A MINIMAX-BIAS PROPERTY OF THE LEAST ALPHA-QUANTILE ESTIMATES

A natural measure of the degree of robustness of an estimate T is the maximum asymptotic bias B(T)(epsilon) over an epsilon-contamination neighborhood. Martin, Yohai and Zamar have shown that the class of least alpha-quantile regression estimates is minimax bias in the class of M-estimates, that is, they minimize B(T)(epsilon), with a depending on epsilon. In this paper we generalize this result, proving that the least alpha-quantile estimates are minimax bias in a much broader class of estimates which we call residual admissible and which includes most of the known robust estimates defined as a function of the regression residuals (e.g., least median of squares, least trimmed of squares, S-estimates, tau-estimates, M-estimates, signed R-estimates, etc.). The minimax results obtained here, likewise the results obtained by Martin, Yohai and Zamar, require that the carriers have elliptical distribution under the central model.