Simultaneous sharp estimation of functions and their derivatives
type: Item
instance of: scholarly article
title: Simultaneous sharp estimation of functions and their derivatives
zbMATH Open document ID: 0930.62035
author: Sam Efromovich
published in: The Annals of Statistics
publication date: 1999-11-09T00:00:00Z
review text: There are a number of reasons for wishing to estimate both a function and its derivatives; for instance, first and second derivatives may be of intrinsic interest as measures of slope and curvature. So far the attention has focused on estimation where typically derivatives of a rate optimal estimate are rate optimal estimates of the corresponding derivatives. We consider the filtering statistical model where on the interval \([0,1]\) we observe random processes \(Y_n(t)\) having stochastic differential \[ dY_n(t)= f(t)dt+ n^{-1/2} dW(t), \quad 0\leq t\leq 1, \] and \(W(t)\) is a standard Brownian motion. We assume that \(f\) is one-periodic, including its \(\alpha\) derivatives, and belongs to the Sobolev class \(W^\alpha_2 Q=\{f: \int^1_0 [f^({(\alpha)} (t)]^2 dt\leq Q\), \(\alpha \geq 2\), \(0<Q< \infty\}\) where \(f^{(k)}\) denotes the \(k\) th derivative and \(f^{(0)} =f\). A data-driven estimate is given that, over the Sobolev space, is simultaneously asymptotically sharp minimax for estimating both the function and its derivatives under integrated squared error loss. It is also shown that linear estimates cannot be simultaneously asymptotically sharp minimax over a given Sobolev space.
Mathematics Subject Classification ID: 62G07
Mathematics Subject Classification ID: 62M05
Mathematics Subject Classification ID: 62M20
Mathematics Subject Classification ID: 62M09
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62J02
zbMATH DE Number: 1359517
zbMATH Keywords: adaptation
zbMATH Keywords: Sobolev functions
zbMATH Keywords: filtering
zbMATH Keywords: data-driven estimate
zbMATH Keywords: minimax
zbMATH Keywords: linear estimates
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1030563985
OpenAlex ID: W2027884397
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DOI: 10.1214/AOS/1030563985
description: scientific article; zbMATH DE number 1359517