Local invariance principles and their application to density estimation
type: Item
instance of: scholarly article
title: Local invariance principles and their application to density estimation
zbMATH Open document ID: 0794.60019
DOI: 10.1007/BF01311347
author: Emmanuel Rio
publication date: 1994-08-15T00:00:00Z
review text: Let \(x_ 1,\dots,x_ n\) be independent random variables with uniform distribution over \([0,1]^ d\), and \(X^{(n)}\) be the centered and normalized empirical process associated to \(x_ 1, \dots,x_ n\). Given a Vapnik-Chervonenkis class \({\mathcal S}\) of bounded functions from \([0,1]^ d\) into \(\mathbb{R}\) of bounded variation, we apply the one- dimensional dyadic scheme of Komlós, Major and Tusnády to get the best possible rate in Dudley's uniform central limit theorem for the empirical process \(\{X^{(n)} (h):h \in {\mathcal S}\}\). When \({\mathcal S}\) fulfills some extra condition, we prove there exists some sequence \(B_ n\) of Brownian bridges indexed by \({\mathcal S}\) such that \[ \sup_{h \in {\mathcal S}} | X^{(n)} (h)-B_ n(h) |=O \bigl( n^{-1/2} \log n \vee n^{-1/(2d)} \sqrt {K({\mathcal S}) \log n} \bigr) \text{ a.s.}, \] where \(K({\mathcal S})\) denotes the maximal variation of the elements of \({\mathcal S}\). This result is then applied to maximal deviations distributions for kernel density estimators under minimal assumptions on the sequence of bandwith parameters. We also derive some results concerning strong approximations for empirical processes indexed by classes of sets with uniformly small perimeter. For example, it follows from \textit{J. Beck}'s paper [Z. Wahrscheinlichkeitstheorie Verw. Geb. 70, 289-306 (1985; Zbl 0554.60037)] that the above result is optimal, up to a possible factor \(\sqrt {\log n}\), when \({\mathcal S}\) is the class of Euclidean balls with radius less than \(r\).
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 569021
zbMATH Keywords: Vapnik-Chervonenkis class
zbMATH Keywords: central limit theorem
zbMATH Keywords: Brownian bridges
zbMATH Keywords: strong approximations
zbMATH Keywords: empirical processes
reviewed by: Eduardo García-Río
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569021