A Minimax-Bias Property of the Least $\alpha$-Quantile Estimates

Abstract

A natural measure of the degree of robustness of an estimate $\mathbf{T}$ is the maximum asymptotic bias $B_\mathbf{T}(\varepsilon)$ over an $\varepsilon$-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 $B_\mathbf{T}(\varepsilon)$, with $\alpha$ depending on $\varepsilon$. 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.