Further characterizations of design optimality and admissibility for partial parameter estimation in linear regression
type: Item
instance of: scholarly article
title: Further characterizations of design optimality and admissibility for partial parameter estimation in linear regression
zbMATH Open document ID: 0649.62070
author: Norbert Gaffke
published in: The Annals of Statistics
publication date: 1987-01-01T00:00:00Z
review text: The problem of finding optimal linear regression designs for the estimation of \(s<k\) linear functions Ka (where K is an \(s\times k\) matrix) of the k regression parameters \(a=(a_ 1,...,a_ k)^ t\) is considered. The optimality criteria \(\phi\) considered are real-valued functions on the set PD(s) of all positive definite \(s\times s\) matrices which are convex and decreasing. An (approximate) design \(\xi_ 0\) is \(\phi\)-optimal for this problem iff Ka is estimable under \(\xi_ 0\) and \(\xi_ 0\) minimizes \(\phi\) (J(\(\xi)\)) over the set of all \(\xi\) under which Ka is estimable, J(\(\xi)\) being the reduced information matrix of \(\xi\) for Ka. A new equivalence theorem for this minimization problem is presented and the results are strengthened for invariant designs. Design admissibility for the estimation of Ka is discussed and a necessary condition and a sufficient condition for admissibility are given with special focus on invariant designs. Examples are given to illustrate possible applications of the results.
Mathematics Subject Classification ID: 62K05
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 62C15
Mathematics Subject Classification ID: 49K35
zbMATH DE Number: 4060569
zbMATH Keywords: multiple quadratic regression
zbMATH Keywords: one-dimensional polynomial regression
zbMATH Keywords: linear estimation
zbMATH Keywords: Gauss-Markov estimator
zbMATH Keywords: approximate design
zbMATH Keywords: Chebyshev approximation
zbMATH Keywords: second order response surfaces
zbMATH Keywords: optimal linear regression designs
zbMATH Keywords: new equivalence theorem
zbMATH Keywords: minimization problem
zbMATH Keywords: invariant designs
zbMATH Keywords: Design admissibility
reviewed by: Hariharan K. Iyer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176350485
OpenAlex ID: W2029197614
DOI: 10.1214/AOS/1176350485
description: scientific article; zbMATH DE number 4060569