On D-optimal designs for linear models under correlated observations with an application to a linear model with multiple response

Abstract

Abstract In the general linear model we set conditions under which an exact D-optimal design for uncorrelated observations with common variance is also D-optimal for correlated observations. Further we determine conditions under which approximate D-optimal designs can be considered as approximate D-optimal designs for correlated observations. Then these results are applied to a regression model with multiple response generalizing Theorem 1 of Krafft and Schaefer (J. Multivariate Anal. 42, 1992). In the above context, however, a serious problem may arise if the covariance matrix is not known; for the Gauss-Markov estimator with respect to a D-optimal design does not need to be calculable for the correlated case. This leads to D-optimal-invariant designs introduced by Bischoff (Ann. Inst. Statist. Math., 44, 1992); such a design τ∗ remains D-optimal when the covariance matrix is changed, and additionally the Gauss-Markov estimator with respect to the design τ∗ stays fixed. For regression models with multiple response we determine classes of covariance matrices for which a D-optimal design for uncorrelated observations with common variance is D-optimal-invariant. As examples we consider linear models where each response belongs to a regression model with intercept term.