Consistent least squares nonparametric regression
type: Item
instance of: scholarly article
title: Consistent least squares nonparametric regression
zbMATH Open document ID: 0624.62040
DOI: 10.1016/0378-3758(87)90107-8
author: Paul D. Whitney
published in: Journal of Statistical Planning and Inference
publication date: 1987-01-01T00:00:00Z
review text: The regression model is studied. Let \(\{\epsilon_ i\}\) be an independent sequence of identically distributed random variables with mean equal to zero. Let f be a function `almost' in a known finite dimensional collection of functions, F. The data \(\{(z_ i,t_ i)\}\) are modeled as \(z_ i=f(t_ i)+\epsilon_ i.\) The least squares estimate examined is the minimizer over \(h\in S+F\) of \(\sum^{n}_{i=1}(z_ i-h(t_ i))^ 2\). S is a compact (in sup-norm) set of functions. Given a weak condition on the covariates \(\{t_ i\}\), the least squares estimate, \(f^*_ n\), converges in sup-norm to f almost surely. Under further conditions on S, derivatives of \(f^*_ n\) converge almost surely to the derivative of f.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62J02
Mathematics Subject Classification ID: 62J99
zbMATH DE Number: 4013765
zbMATH Keywords: nonparametric regression
zbMATH Keywords: Sobolev space
zbMATH Keywords: Hölder space
zbMATH Keywords: Banach spaces
zbMATH Keywords: almost sure convergence
zbMATH Keywords: strong consistency theorems
zbMATH Keywords: least squares estimate
zbMATH Keywords: sup-norm
zbMATH Keywords: derivatives
MaRDI profile type: Publication
full work available at URL: 0378-3758(87)90107-8
OpenAlex ID: W1971253988
cites work: Q4073917
cites work: Minimaxity of the method of regularization on stochastic processes
cites work: Q3909512
cites work: Linear estimation for approximately linear models
cites work: Smoothing noisy data with spline functions
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description: scientific article; zbMATH DE number 4013765