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March, 1993 Lower Bounds for Contamination Bias: Globally Minimax Versus Locally Linear Estimation
Xuming He, Douglas G. Simpson
Ann. Statist. 21(1): 314-337 (March, 1993). DOI: 10.1214/aos/1176349028

Abstract

We study how robust estimators can be in parametric families, obtaining a lower bound on the contamination bias of an estimator that holds for a wide class of parametric families. This lower bound includes as a special case the bound used to establish that the median is bias minimax among location equivariant estimators, and it is tight or nearly tight in a variety of other settings such as scale estimation, discrete exponential families and multiple linear regression. The minimum variation distance estimator has contamination bias within a dimension-free factor of this bound. A second lower bound applies to locally linear estimates and implies that such estimates cannot be bias minimax among all Fisher-consistent estimates in higher dimensions. In linear regression this class of estimates includes the familiar M-estimates, GM-estimates and S-estimates. In discrete exponential families, yet another lower bound implies that the "proportion of zeros" estimates has minimax bias if the median of the distribution is zero, a common situation in some fields. This bound also implies that the information-standardized sensitivity of every Fisher consistent estimate of the Poisson mean and of the Binomial proportion is unbounded.

Citation

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Xuming He. Douglas G. Simpson. "Lower Bounds for Contamination Bias: Globally Minimax Versus Locally Linear Estimation." Ann. Statist. 21 (1) 314 - 337, March, 1993. https://doi.org/10.1214/aos/1176349028

Information

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

zbMATH: 0770.62023
MathSciNet: MR1212179
Digital Object Identifier: 10.1214/aos/1176349028

Subjects:
Primary: 62F35
Secondary: 62F10, 62G35

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.21 • No. 1 • March, 1993
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