LOWER BOUNDS FOR CONTAMINATION BIAS - GLOBALLY MINIMAX VERSUS LOCALLY LINEAR-ESTIMATION

• 成果类型: Article
• 署名作者: HE, XM; SIMPSON, DG
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176349028
• 发表日期: 1993
• 页码: 314-337
• 关键词: high breakdown-point; regression-models; Robustness; tests; scale

摘要

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'' estimate 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.