BIAS-ROBUST ESTIMATES OF REGRESSION-BASED ON PROJECTIONS

• 成果类型: Article
• 署名作者: MARONNA, RA; YOHAI, VJ
• 署名单位: Universidad de San Andres Argentina; University of Buenos Aires; Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176349160
• 发表日期: 1993
• 页码: 965-990
• 关键词: high breakdown-point; multivariate scatter; scale

摘要

A new class of bias-robust estimates of multiple regression is introduced. If y and x are two real random variables, let T(y, x) be a univariate robust estimate of regression of y on x through the origin. The regression estimate T(y, x) of a random variable y on a random vector x = (x1,...,x(p))' is defined as the vector t is-an-element-of R(p) which minimizes sup\\lambda\\=1\T(y - t'x,lambda'x)\s(lambda'x), where s is a robust estimate of scale. These estimates, which are called projection estimates, are regression, affine and scale equivariant. When the univariate regression estimate is T(y, x) = median(y/x), the resulting projection estimate is highly bias-robust. In fact, we find an upper bound for its maximum bias in a contamination neighborhood, which is approximately twice the minimum possible value of this maximum bias for any regression and affine equivariant estimate. The maximum bias of this estimate in a contamination neighborhood compares favorably with those of Rousseeuw's least median squares estimate and of the most bias-robust GM-estimate. A modification of this projection estimate, whose maximum bias for a multivariate normal with mass-point contamination is very close to the minimax bound, is also given. Projection estimates are shown to have a rate of consistency of n1/2. A computational version of these estimates, based on subsampling, is given. A simulation study shows that its small sample properties compare very favorably to those of other robust regression estimates.