On iterated logarithm laws for linear arrays and nonparametric regression estimators
type: Item
instance of: scholarly article
title: On iterated logarithm laws for linear arrays and nonparametric regression estimators
zbMATH Open document ID: 0733.60047
published in: The Annals of Probability
publication date: 1991-01-01T00:00:00Z
review text: Let \[ \hat g(x)=\sum^{n}_{i=1}Y_ iL\{(x-x_ i)/h_ n\}/\sum^{n}_{i=1}L\{(x-x_ i)/h_ n\} \] be the kernel estimator for the smooth function g in the nonparametric regression model \(Y_ i=g(x_ i)+e_ i\), \(i\leq n\), where each \(x_ i\) is a fixed d- dimensional vector, \(Y_ i\) is a random variable, \(e_ 1,e_ 2,...,e_ n\) are i.i.d. with E \(e_ 1=0\), E \(e^ 2_ 1=1\), L is a kernel function and \(h_ n\) is the bandwidth. Under modest hypotheses, a law of the iterated logarithm for \(\hat g(x)-E\hat g(x)\) is proved, thus shedding light on the rate of convergence of \(\hat g(x)-E\hat g(x)\). This extends work in the case \(d=1\) due to \textit{W. Härdle} [Ann. Stat. 12, 624-635 (1984; Zbl 0591.62030)] and \textit{U. Stadtmüller} [Ann. Probab. 12, 35-44 (1984; Zbl 0537.60023)].
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60G05
zbMATH DE Number: 4213126
zbMATH Keywords: triangular array
zbMATH Keywords: kernel estimator
zbMATH Keywords: nonparametric regression model
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: rate of convergence
reviewed by: R. James Tomkins
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990449
OpenAlex ID: W2068499146
DOI: 10.1214/AOP/1176990449
author: Peter Hall
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description: scientific article; zbMATH DE number 4213126