Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice

• 成果类型: Article
• 署名作者: Spokoiny, VG
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1998
• 页码: 1356-1378
• 关键词: Nonparametric regression; residual variance; points; jump

摘要

We propose a method of adaptive estimation of a regression function which is near optimal in the classical sense of the mean integrated error. At the same time, the estimator is shown to be very sensitive to discontinuities or change-points of the underlying function f or its derivatives. For instance, in the case of a jump of a regression function, beyond the intervals of length (in order) n(-1) log n around change-points the quality of estimation is essentially the same as if locations of jumps were known. The method is fully adaptive and no assumptions are imposed on the design, number and size of jumps. The results are formulated in a nonasymptotic way and can therefore be applied for an arbitrary sample size.