On optimality properties of simple block designs in the approximate design theory
type: Item
instance of: scholarly article
title: On optimality properties of simple block designs in the approximate design theory
zbMATH Open document ID: 0531.62070
DOI: 10.1016/0378-3758(83)90038-1
author: Friedrich Pukelsheim
published in: Journal of Statistical Planning and Inference
publication date: 1983-01-01T00:00:00Z
review text: Consider an experiment with v treatments and b blocks. An exact block design of size n is a set of integers \(n_{ij}\in \{0,1,...,n\}\) with \(i=1,2,...,v\), and \(j=1,2,3,...,b\) such that their sum is n in which the experimenter observes \(n_{ij}\) times with the i-th treatment in the j- th block. This paper deals with approximate block designs which are sets of weights \(w_{ij}\in [0,1]\) with sum equal to 1, meaning thereby that \(w_{ij}\) fraction of all observations is to be taken on the ith treatment in the j-th block. In this paper optimality properties of approximate block designs are discussed under variations of (a) the class of competing designs, (b) the optimality criterion, (c) the parametric function to be investigated, and (d) the statistical model (interaction effects, additive effects). Optimality means here universal optimality, uniform optimality, and \(j_ p\)-optimality. The optimal designs turn out to be the product of their treatment and block marginals, and uniform designs whose support is specified well in advance. A geometric account of uniform optimality is also given. The classical balanced incomplete block designs (BIBD) are shown to be simultaneously \(j_ p\)-optimal for a maximal system of identifiable parameters.
Mathematics Subject Classification ID: 62K05
Mathematics Subject Classification ID: 62K10
zbMATH DE Number: 3843018
zbMATH Keywords: approximate block designs
zbMATH Keywords: balanced incomplete block designs
zbMATH Keywords: universal optimality
zbMATH Keywords: simultaneous j(p) optimality
zbMATH Keywords: E-optimality
zbMATH Keywords: product designs
zbMATH Keywords: uniform designs
zbMATH Keywords: D-optimality
MaRDI profile type: Publication
full work available at URL: 0378-3758(83)90038-1
OpenAlex ID: W2051186659
cites work: Optimality of certain asymmetrical experimental designs
cites work: Some classes of optimality criteria and optimal designs for complete two- way layouts
cites work: Optimum designs in complete two-way layouts
cites work: Optimum continuous block designs
cites work: Minimax Designs in Two Dimensional Regression
cites work: Balanced Designs with Unequal Numbers of Replicates
cites work: On the Nonrandomized Optimality and Randomized Nonoptimality of Symmetrical Designs
cites work: General equivalence theory for optimum designs (approximate theory)
cites work: Q4072671
cites work: Q4167423
cites work: Optimale Versuchspläne bei zweifach klassifizierten Beobachtungsmodellen. (Optimal experimental plans for two-fold classified observation models)
cites work: Inequalities: theory of majorization and its applications
cites work: On linear regression designs which maximize information
cites work: Q5627526
cites work: Quadratic Subspaces and Completeness
cites work: On a class of variance balanced block designs
description: scientific article; zbMATH DE number 3843018