Multiparameter bandwidth processes and adaptive surface smoothing
type: Item
instance of: scholarly article
title: Multiparameter bandwidth processes and adaptive surface smoothing
zbMATH Open document ID: 0795.62039
author: Kathryn A. Prewitt
author: Hans-Georg Müller
published in: Journal of Multivariate Analysis
publication date: 1993-12-06T00:00:00Z
review text: For the problem \(y_{i,n}= g(x_{i,n})+ \varepsilon_{i,n}\), \(i=1,2,\dots,n\) and \(x_{i,n}\in \mathbb{R}^ m\), and for the nonparametric estimation of the function \(g\) using the kernel method, the authors provide a sequence of window processes that converge towards a Gaussian process, having mean and covariance known. The limit process gives a criterion to choose the optimal window. Their procedure is as follows: a classical analysis of bias and variance shows that the general optimal window is \(b_ i= t_ i n^{- (1/2k+m)}\), \(i=1,\dots,m\), where \(k\) is the smoothness degree of \(g\). Hence it is only necessary to determine the \(t_ i\). Then, they prove: \(\gamma_ n (\widehat{g}^{(\nu)} (x,t)- D^{(\nu)} g(x))\) converges towards \(\eta ({\mathbf t})\), \({\mathbf t}= (t_ 1,\dots,t_ m)\) and \(\eta({\mathbf t})\) a Gaussian process, where \(\widehat{g}^{(\nu)}\) is the estimator of \(D^{(\nu)}g\), \(0<a_ 1\leq t_ i\leq a_ 2<\infty\), and taking as window the \(b_ i\). In particular, when \({\mathbf t}\) is fixed this theorem gives the asymptotic mean quadratic error (AMSE). Theoretical optimal estimators can be obtained minimizing the AMSE and introducing then this value in the estimators.
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 62G20
zbMATH DE Number: 445135
zbMATH Keywords: curve estimation
zbMATH Keywords: data-adaptative smoothing
zbMATH Keywords: density estimation
zbMATH Keywords: functional limit theorem
zbMATH Keywords: limiting Gaussian process
zbMATH Keywords: window processes
zbMATH Keywords: optimal window
zbMATH Keywords: bias
zbMATH Keywords: variance
zbMATH Keywords: asymptotic mean quadratic error
zbMATH Keywords: optimal estimators
reviewed by: José R. León
MaRDI profile type: Publication
full work available at URL: jmva.1993.1067
OpenAlex ID: W2061674424
DOI: 10.1006/JMVA.1993.1067
description: scientific article; zbMATH DE number 445135