THE ORDER OF THE REMAINDER IN DERIVATIVES OF COMPOSITION AND INVERSE OPERATORS FOR P-VARIATION NORMS

• 成果类型: Article
• 署名作者: DUDLEY, RM
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176325354
• 发表日期: 1994
• 页码: 1-20
• 关键词: differentiability; estimators

摘要

Many statisticians have adopted compact differentiability since Reeds showed in 1976 that it holds (while Frechet differentiability fails) in the supremum (sup) norm on the real line for the inverse operator and for the composition operator (F,G) bar arrow pointing right F . G with respect to F. However, these operators are Frechet differentiable with respect to p-variation norms, which for p > 2 share the good probabilistic properties of the sup norm, uniformly over all distributions on the line. The remainders in these differentiations are of order \\.\\gamma for gamma > 1. In a range of cases p-variation norms give the largest possible values of gamma on spaces containing empirical distribution functions, for both the inverse and composition operators. Compact differentiability in the sup norm cannot provide such remainder bounds since, over some compact sets, differentiability holds arbitrarily slowly.