Bayesian optimal designs for linear regression models (Q1184231)

Given a linear regression model \(y=f'(x)\theta+\varepsilon\) and a suitable vector \(c\) (or, more general, a suitable matrix \(A\)), the design problem for estimating \(c'\theta\) (or \(A'\theta\)) is considered, leading to \(c\) (or \(\psi\))-optimality criterion. Furthermore, prior information on \(\theta\) is given, condensed in a mean value \(\mu\) and a precision matrix \(R\). Using geometrical arguments and duality theory, the authors prove a Bayesian version of Elfving's theorem for both criteria. Special attention, however, is given to \(c\)-optimality, especially to optimal one-point designs and to conditions, under which classical \(c\)-optimal designs are Bayesian \(c\)-optimal. Main references are: besides [\textit{G. Elfving}, Ann. Math. Stat. 23, 255--262 (1952; Zbl 0047.13403)], [\textit{J. Pilz}, Bayesian estimation and experimental design in linear regression models. Leipzig: Teubner (1983; Zbl 0533.62007)], and [\textit{K. Chaloner}, Ann. Stat. 12, 283--300 (1984; Zbl 0554.62063)].