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August 1999 Breakdown points and variation exponents of robust $M$-estimators in linear models
Ivan Mizera, Christine H. Müller
Ann. Statist. 27(4): 1164-1177 (August 1999). DOI: 10.1214/aos/1017938920

Abstract

The breakdown point behavior of $M$-estimators in linear models with fixed designs, arising from planned experiments or qualitative factors, is characterized. Particularly, this behavior at fixed designs is quite different from that at designs which can be corrupted by outliers, the situation prevailing in the literature. For fixed designs, the breakdown points of robust $M$-estimators (those with bounded derivative of the score function), depend on the design and the variation exponent (index) of the score function. This general result implies that the highest breakdown point within all regression equivariant estimators can be attained also by certain $M$-estimators: those with slowly varying score function, like the Cauchy or slash maximum likelihood estimator. The $M$-estimators with variation exponent greater than 0, like the $L_1$ or Huber estimator, exhibit a consider-ably worse breakdown point behavior.

Citation

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Ivan Mizera. Christine H. Müller. "Breakdown points and variation exponents of robust $M$-estimators in linear models." Ann. Statist. 27 (4) 1164 - 1177, August 1999. https://doi.org/10.1214/aos/1017938920

Information

Published: August 1999
First available in Project Euclid: 4 April 2002

zbMATH: 0959.62029
MathSciNet: MR1740104
Digital Object Identifier: 10.1214/aos/1017938920

Subjects:
Primary: 62F10 , 62F35
Secondary: 62J05 , 62J10 , 62K99

Keywords: $L_1$ estimator , Breakdown point , linear model , M-estimator , planned experiments , regular variation

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 4 • August 1999
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