NONPARAMETRIC ESTIMATES OF REGRESSION QUANTILES AND THEIR LOCAL BAHADUR REPRESENTATION

• 成果类型: Article
• 署名作者: CHAUDHURI, P
• 署名单位: University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176348119
• 发表日期: 1991
• 页码: 760-777
• 关键词: robust regression; kernel

摘要

Let (X,Y) be a random vector such that X is d-dimensional, Y is real valued and Y = theta-(X) + epsilon, where X and epsilon-are independent and the alpha-th quantile of epsilon-is 0 (alpha-is fixed such that 0 < alpha < 1). Assume that theta-is a smooth function with order of smoothness p > 0, and set r = (p - m)/(2p + d), where m is a nonnegative integer smaller than p. Let T(theta) denote a derivative of theta-of order m. It is proved that there exists a pointwise estimate T triple-over-dot-n of T(theta), based on a set of i.i.d. observations (X1,Y1),...,(X(n),Y(n)), that achieves the optimal nonparametric rate of convergence n-r under appropriate regularity conditions. Further, a local Bahadur type representation is shown to hold for the estimate T triple-over-dot-N and this is used to obtain some useful asymptotic results.