Robust designs and optimality of least squares for regression problems
type: Item
instance of: scholarly article
title: Robust designs and optimality of least squares for regression problems
zbMATH Open document ID: 0542.62064
DOI: 10.1016/0378-3758(84)90048-X
author: N. E. Zubov
published in: Journal of Statistical Planning and Inference
publication date: 1984-01-01T00:00:00Z
review text: Considered is the linear regression \(y_ i=a_ 0+a^ Tx_ i+\psi(x_ i)+\epsilon_ i\) where the errors \(\epsilon_ i\) are uncorrelated with mean zero and variance \(\sigma^ 2\), \(\psi\) is an unknown ''contamination function'' bounded by a known convex function \(\phi\), \(x_ i\in X\subset {\mathbb{R}}^ k\) and \(a_ 0\), \(a^ T\) are the parameters to be estimated. The mean square error matrix and \(\Phi_ p\)-design-optimality criteria are combined to find for the minimax approach w.r.t. the design the (linear) estimator and to \(\psi\) a best pair of estimator and design. Sufficient conditions are given that in case \(k=1\) the least squares estimator and a two point design are a best pair for all \(\Phi_ p\)- criteria and - in case k arbitrary - that the least squares estimator and a uniform design measure on a sphere whose radius depends on \(\sigma^ 2\) and \(\phi\) are a best pair for \(\Phi_ 1\) and \(\Phi_{\infty}\).
Mathematics Subject Classification ID: 62K05
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62F35
zbMATH DE Number: 3862270
zbMATH Keywords: unbiased linear estimators
zbMATH Keywords: optimal estimator
zbMATH Keywords: robust design
zbMATH Keywords: phi(p) optimality
zbMATH Keywords: contamination function
zbMATH Keywords: mean square error matrix
zbMATH Keywords: minimax approach
zbMATH Keywords: best pair of estimator and design
zbMATH Keywords: least squares estimator
reviewed by: O. Krafft
MaRDI profile type: Publication
cites work: A Basis for the Selection of a Response Surface Design
cites work: Q5588210
cites work: Q4080580
cites work: Q4042534
cites work: General equivalence theory for optimum designs (approximate theory)
cites work: Q3856001
cites work: Optimal robust designs: Linear regression in \(R^ k\).
cites work: Linear estimation for approximately linear models
cites work: Q3851467
Wikidata QID: Q126977471
full work available at URL: 0378-3758(84)90048-x
OpenAlex ID: W1992920951
description: scientific article; zbMATH DE number 3862270