Une mesure de la déviation quadratique d'estimateurs non paramétriques. (A measure of quadratic deviation of nonparametric estimators)
type: Item
instance of: scholarly article
title: Une mesure de la déviation quadratique d'estimateurs non paramétriques. (A measure of quadratic deviation of nonparametric estimators)
zbMATH Open document ID: 0599.62043
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77269
review text: Let \(\theta\) be either the density f of \(U_ 1\) or the regression function \(r=E(V_ 1| U_ 1)\) or the product rf and let \({\hat \theta}{}_ n\) be a kernel-type or an orthogonal series estimator of \(\theta\) based on n identically distributed \({\mathbb{R}}^ d\)-valued random variables \((U_ i,V_ i)_{1\leq i\leq n}\). Nonrandom sequences \((a_ n)_{n\in {\mathbb{N}}^*}\) and \((b_ n)_{n\in {\mathbb{N}}^*}\) are determined such that the statistics \[ (a_ n\int | {\hat \theta}_ n-\theta | d\mu -b_ n)_{n\in {\mathbb{N}}^*}, \] converge to a Gaussian distribution \(N(0,\sigma^ 2)\) (\(\mu\) is positive, \(\sigma\)- finite and absolutely continuous w.r.t. the Lebesgue measure). Both independent and mixing case are dealt with. It must be noticed that \(a_ n\), \(b_ n\) and \(\sigma\) do not depend on the mixing function. The basic tools of the paper are Gaussian approximations in Hilbert spaces and the Karhunen-Loeve expansion.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 3967678
zbMATH Keywords: kernel-type estimator
zbMATH Keywords: density estimation
zbMATH Keywords: nonparametric regression
zbMATH Keywords: mean square error
zbMATH Keywords: asymptotic normality
zbMATH Keywords: orthogonal series estimator
zbMATH Keywords: Gaussian distribution
zbMATH Keywords: mixing
zbMATH Keywords: Gaussian approximations
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: Karhunen- Loeve expansion
author: Paul Doukhan
author: José R. León
author: Frédéric Portal
reviewed by: Alain Berlinet
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3967678