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

Kurzfassung/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 {\it O. Krafft} and {\it M. Schaefer} in J. Multivariate Anal. 42, No. 1, 130-140 (1992; Zbl 0774.62081).\par 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 the author [Ann. Inst. Stat. Math. 44, No. 2, 229-238 (1992; Zbl 0781.62117)]; such a design $\tau\sp*$ remains $D$-optimal when the covariance matrix is changed, and additionally the Gauss-Markov estimator with respect to the design $\tau\sp*$ 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.