title: Optimal designs in regression with correlated errors.
abstract: This paper discusses the problem of determining optimal designs for regression models, when the observations are dependent and taken on an interval. A complete solution of this challenging optimal design problem is given for a broad class of regression models and covariance kernels. We propose a class of estimators which are only slightly more complicated than the ordinary least-squares estimators. We then demonstrate that we can design the experiments, such that asymptotically the new estimators achieve the same precision as the best linear unbiased estimator computed for the whole trajectory of the process. As a by-product we derive explicit expressions for the BLUE in the continuous time model and analytic expressions for the optimal designs in a wide class of regression models. We also demonstrate that for a finite number of observations the precision of the proposed procedure, which includes the estimator and design, is very close to the best achievable. The results are illustrated on a few numerical examples.
authors.affiliation: Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany.
authors.forename: Holger
authors.initials: H
authors.lastname: Dette
authors.affiliation: School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK.
authors.forename: Andrey
authors.initials: A
authors.lastname: Pepelyshev
authors.affiliation: School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK.
authors.forename: Anatoly
authors.initials: A
authors.lastname: Zhigljavsky
keywords: BLUE; Doob representation; Gaussian processes; correlated observations; linear regression; optimal design; signed measures