Nonparametric estimation of a function from noiseless observations at random points

被引：4

作者：

Bauer, Benedikt ^{[1
]}

Devroye, Luc ^{[2
]}

Kohler, Michael ^{[1
]}

Krzyak, Adam ^{[3
]}

Walk, Harro ^{[4
]}

机构：

[1] Tech Univ Darmstadt, Fachbereich Math, Schlossgartenstr 7, D-64289 Darmstadt, Germany

[2] McGill Univ, Sch Comp Sci, 3450 Rue Univ, Montreal, PQ H3A 2K6, Canada

[3] Concordia Univ, Dept Comp Sci & Software Engn, 1455 Bout Maisonneuve Ouest, Montreal, PQ H3G 1M8, Canada

[4] Univ Stuttgart, Fachbereich Math, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

来源：

JOURNAL OF MULTIVARIATE ANALYSIS | 2017年 / 160卷

基金：

加拿大自然科学与工程研究理事会;

关键词：

Multivariate scattered data approximation; Rate of convergence; Supremum norm error; OPTIMAL GLOBAL RATES; SCATTERED DATA INTERPOLATION; RADIAL BASIS FUNCTIONS; APPROXIMATION; CONVERGENCE;

D O I：

10.1016/j.jmva.2017.05.010

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

In this paper we study the problem of estimating a function from n noiseless observations of function values at randomly chosen points. These points are independent copies of a random variable whose density is bounded away from zero on the unit cube and vanishes outside. The function to be estimated is assumed to be (p, C)-smooth, i.e., (roughly speaking) it is p times continuously differentiable. Our main results are that the supremum norm error of a suitably defined spline estimate is bounded in probability by {ln(n)/n}(p/d) for arbitrary p and d and that this rate of convergence is optimal in minimax sense. (C) 2017 Elsevier Inc. All rights reserved.

引用

页码：93 / 104

页数：12

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