On estimation of the \(L_r\) norm of a regression function
type: Item
instance of: scholarly article
title: On estimation of the \(L_r\) norm of a regression function
zbMATH Open document ID: 0921.62103
author: V. G. Spokoiny
author: O. V. Lepski
author: Arkadi Nemirovski
publication date: 1999-03-30T00:00:00Z
review text: Let \(X(t)\), \(t\in[0,1]\), be a process satisfying the stochastic differential equation \[ dX(t)=f(t)dt+n^{-1/2}dW(t), \] where \(f(t)\) is an unknown function, and \(W(t)\), \(t\in[0,1]\), is the standard Wiener process. The authors' concern is to estimate the functional \[ \| f\|_r=\left[ \int^1_0\bigl| f(t)\bigr|^r dt\right]^{1/r}\quad \text{with given }r\geq 1. \] The asymptotic behavior of \[ {\mathcal B}(\widehat f_n)=\sup_{f\in\Sigma (\beta,L)}E\bigl| \widehat f_n-\| f\|_r \bigr|\text{ and }{\mathcal B}^*(n)= \inf_{\widehat f_n}\sup_{f\in\Sigma (\beta,L)}E\bigr| \widehat f_n-\| f\|_r\bigr | \] is studied. In particular, the main result shows that the optimal rate of convergence of the estimator of the functional \(\| f\|_r\) is better than the rate of convergence of nonparametric estimates of \(f(t)\).
Mathematics Subject Classification ID: 62M05
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62M09
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 1271431
MaRDI profile type: Publication
full work available at URL: s004409970006
OpenAlex ID: W2078729747
DOI: 10.1007/S004409970006
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1271431