Breakdown Points and Variation Exponents of Robust M-Estimators in Linear Models

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 prevaili...

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Bibliographic Details
Published in:The Annals of statistics 1999-08, Vol.27 (4), p.1164-1177
Main Authors: Mizera, Ivan, Muller, Christine H.
Format: Article
Language:English
Subjects:
62F10
62F35
62J05
62J10
62K99
Breakdown point
Estimators
Exact sciences and technology
Experiment design
Experimental design
L_1 estimator
Least squares
Linear and Generalized Linear Models
Linear inference, regression
linear model
Linear models
Linear regression
M-estimator
Mathematics
Maximum likelihood estimation
Maximum likelihood estimators
Objective functions
Parametric inference
planned experiments
Point estimators
Probability and statistics
regular variation
Sciences and techniques of general use
Statistics
T distribution
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Summary: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 L1or Huber estimator, exhibit a considerably worse breakdown point behavior.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1017938920