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December, 1993 A Minimax-Bias Property of the Least $\alpha$-Quantile Estimates
Victor J. Yohai, Ruben H. Zamar
Ann. Statist. 21(4): 1824-1842 (December, 1993). DOI: 10.1214/aos/1176349400


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.


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Victor J. Yohai. Ruben H. Zamar. "A Minimax-Bias Property of the Least $\alpha$-Quantile Estimates." Ann. Statist. 21 (4) 1824 - 1842, December, 1993.


Published: December, 1993
First available in Project Euclid: 12 April 2007

zbMATH: 0797.62027
MathSciNet: MR1245771
Digital Object Identifier: 10.1214/aos/1176349400

Primary: 62F35

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.21 • No. 4 • December, 1993
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