A general bahadur representation of M-estimators and its application to linear regression with nonstochastic designs

• 成果类型: Article
• 署名作者: He, XM; Shao, QM
• 署名单位: University of Oregon
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1032181172
• 发表日期: 1996
• 页码: 2608-2630
• 关键词: iterated logarithm; models; LAW; asymptotics

摘要

We obtain strong Bahadur representations for a general class of M-estimators that satisfies Sigma(i) psi(x(i), theta) = o(delta(n)), where the x(i)'s are independent but not necessarily identically distributed random variables. The results apply readily to M-estimators of regression with nonstochastic designs. More specifically, we consider the minimum L(p) distance estimators, bounded influence GM-estimators and regression quantiles. Under appropriate design conditions, the error rates obtained for the first-order approximations are sharp in these cases. We also provide weaker and more easily verifiable conditions that suffice for an error rate that is suboptimal but strong enough for deriving the asymptotic distribution of Ill-estimators in a wide variety of problems.