Strong approximation of quantile processes by iterated Kiefer processes

• 成果类型: Article
• 署名作者: Deheuvels, P
• 署名单位: Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1019160265
• 发表日期: 2000
• 页码: 909-945
• 关键词: Empirical Processes; partial sums; logarithm; THEOREMS; increments; LAWS

摘要

The notion of a kth iterated Kiefer process K(nu, t; k) for k is an element of N and nu, t is an element of R is introduced. We show that the uniform quantile process beta (n)(t) may be approximated on [0, 1] by n(-1/2) K(n, t; k), at an optimal uniform almost sure rate of O(n(-1/2+1/2k+1+0(1))) for each k is an element of N. Our arguments are based in part on a new functional limit law, of independent interest, for the increments of the empirical process; Applications include an extended version of the uniform Bahadur-Kiefer representation, together with strong limit theorems for nonparametric functional estimators.