Maximal determinant over a certain class of matrices and its application to D-optimality of designs
type: Item
instance of: scholarly article
title: Maximal determinant over a certain class of matrices and its application to D-optimality of designs
zbMATH Open document ID: 1239.15005
author: Katarzyna Filipiak
author: Augustyn Markiewicz
author: Rafał Róẓański
published in: Linear Algebra and its Applications
publication date: 2012-03-08T00:00:00Z
review text: In the first part of the paper the problem of maximizing of determinants is considered. Firstly the determinants of \(t\times t\) matrices of type \(\alpha I_t-(P+P^T)\) are considered, where \(I_t\) is an identity matrix, \(P_t\) is a permutation matrix and \(\alpha>2\) is a scalar. The authors find the permutation matrix \(P\) that maximizes this determinant. Secondly the determinants of \(t\times t\) matrices of type \(\alpha I_t+P+P^T\) are considered, where \(P_t\) is a permutation matrix with zero diagonal and \(\alpha>2,5\) is a scalar. The authors also find the permutation matrix with zero diagonal that maximizes this determinant. In the second part of the paper these results are used to characterize \(D\)-optimal complete block designs under an interference model. The criterion is based on the eigenvalues of the information matrices of designs. Previously in the literature only the problem of determining \(D\)-optimal designs under a model with block effects as the only nuisance parameters over the class of designs were considered [\textit{J. A. John} and \textit{T. J. Mitchell}, ``Optimal incomplete block designs'', J. R. Stat. Soc., Ser. B 39, 39--43 (1977; Zbl 0354.05015); \textit{N. Gaffke}, ``D-optimal block designs with at most six varieties'', J. Stat. Plann. Inference 6, 183--200 (1982; Zbl 0481.62058); \textit{K. Balasubramanian} and \textit{A. Dey}, ``D-optimal designs with minimal and nearly minimal number of units'', J. Stat. Plann. Inference 52, No. 2, 255--262 (1996; Zbl 0846.62055)].
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 62K05
Mathematics Subject Classification ID: 62K10
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 6013292
zbMATH Keywords: eigenvalues
zbMATH Keywords: determinants
zbMATH Keywords: D-optimality of designs
zbMATH Keywords: inference model
zbMATH Keywords: information matrix
zbMATH Keywords: maximal determinant
zbMATH Keywords: block designs
reviewed by: Dmitry Artamonov
MaRDI profile type: Publication
full work available at URL: j.laa.2011.04.036
OpenAlex ID: W2080502902
cites work: D-optimal designs with minimal and nearly minimal number of units
cites work: Optimality of neighbour balanced designs
cites work: Optimality of neighbor balanced designs under mixed effects model.
cites work: Optimality of type I orthogonal arrays for general interference model with correlated observa\-tions
cites work: Optimality and efficiency of circular neighbor balanced designs for correlated observations
cites work: Optimal designs for a mixed interference model
cites work: Connectedness of complete block designs under an interference model
cites work: On the E-optimality of complete designs under an interference model
cites work: D-optimal block designs with at most six varieties
cites work: Q4124844
cites work: Determinants of block tridiagonal matrices
cites work: Inequalities: theory of majorization and its applications
cites work: Q4849481
DOI: 10.1016/J.LAA.2011.04.036
description: scientific article; zbMATH DE number 6013292