## The Annals of Statistics

- Ann. Statist.
- Volume 21, Number 1 (1993), 314-337.

### Lower Bounds for Contamination Bias: Globally Minimax Versus Locally Linear Estimation

Xuming He and Douglas G. Simpson

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 21, Number 1 (1993), 314-337.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349028

**Digital Object Identifier**

doi:10.1214/aos/1176349028

**Mathematical Reviews number (MathSciNet)**

MR1212179

**Zentralblatt MATH identifier**

0770.62023

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F35: Robustness and adaptive procedures

Secondary: 62F10: Point estimation 62G35: Robustness

**Keywords**

Bias minimax contamination model discrete exponential family invariance linear regression M-estimate minimum distance estimate scale model sensitivity

#### Citation

He, Xuming; Simpson, Douglas G. Lower Bounds for Contamination Bias: Globally Minimax Versus Locally Linear Estimation. Ann. Statist. 21 (1993), no. 1, 314--337. doi:10.1214/aos/1176349028. https://projecteuclid.org/euclid.aos/1176349028