Spline smoothing in regression models and asymptotic efficiency in \(L_ 2\)
type: Item
instance of: scholarly article
title: Spline smoothing in regression models and asymptotic efficiency in \(L_ 2\)
zbMATH Open document ID: 0596.62052
author: Michael Nussbaum
published in: The Annals of Statistics
publication date: 1985-01-01T00:00:00Z
review text: Consider the regression model \(y_ j=f(x_ j)+\xi_ j\), \(1\leq j\leq n\), where the \(x_ j's\) are equidistant in [0,1], the \(\xi_ j's\) are independent random variables with distribution N(0,1) and the function f is to be estimated. Let \(\| \cdot \|_ 2\) denote the usual norm on \(L^ 2([0,1])\), \(\| \cdot \|\) the semi-norm defined by \(\| f\|^ 2=n^{-1}\sum^{n}_{j=1}f^ 2(x_ j)\). Let, for \(m\in {\mathbb{N}}\), \(W^ m_ 2\) denote the usual Sobolev space on [0,1] and, for \(P>0\), \(W^ m_ 2(P)\) be defined by \(W^ m_ 2(P)=\{f\in W^ m_ 2| \| D^ mf\|^ 2\leq P\}\). Let \({\mathcal F}_ n\) be the class of measurable mappings \(\hat f:\) \({\mathbb{R}}^ n\times [0,1]\to {\mathbb{R}}.\) Using minimax filtering result in Gaussian white noise obtained by \textit{M. S. Pinsker} [Probl. Peredachi Inf. 16, No.2, 52-68 (1980; Zbl 0452.94003)] the author proves the following asymptotic minimax theorem: \[ \lim_{n\to +\infty}\inf_{\hat f\in {\mathcal F}_ n}\sup_{f\in W^ m_ 2(P)}n^{2m/\quad (2m+1)}E_ f(\|| \hat f-f\||^ 2)= \] \[ [P(2m+1)]^{1/(2m+1)}(mn^{-1}(m+1)^{-1})^{2m/(2m+1)}. \] The notation \(\|| \cdot \||\) stands for either \(\| \cdot \|_ 2\) or \(\| \cdot \|\). As part of the proof some eigenvalue estimates in spline theory are achieved.
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 65D10
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 3960784
zbMATH Keywords: spline smoothing
zbMATH Keywords: nonparametric regression estimation
zbMATH Keywords: integrated
zbMATH Keywords: mean square error
zbMATH Keywords: Fisher's bound for asymptotic variances
zbMATH Keywords: minimax
zbMATH Keywords: risk
zbMATH Keywords: Hilbert space
zbMATH Keywords: Sobolev space
zbMATH Keywords: minimax filtering
zbMATH Keywords: asymptotic minimax theorem
zbMATH Keywords: eigenvalue estimates
reviewed by: Alain Berlinet
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349651
OpenAlex ID: W2007800374
DOI: 10.1214/AOS/1176349651
Recommended article: Q4940648
Recommended article: Q3986301
Recommended article: A risk bound in Sobolev class regression
Recommended article: Asymptotic Minimax Risk for the White Noise Model on the Sphere
Recommended article: Consistent least squares nonparametric regression
Recommended article: A minimax approach to errors-in-variables linear models
Recommended article: Minimax nonparametric estimation on maxisets
Recommended article: Minimax estimation in a deconvolution problem
Recommended article: Frame-constrained total variation regularization for white noise regression
description: scientific article; zbMATH DE number 3960784