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ON EXACT *D*-OPTIMAL DESIGNS FOR REGRESSION MODELS

WITH CORRELATED OBSERVATIONS

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WOLFGANG BISCHOFF

*Institute of Mathematical Stochastics, Department of Mathematics,*

University of Karlsruhe, D-7500 Karlsruhe 1, Germany
(Received January 8, 1990; revised October 22, 1990)

**Abstract.**
Let *tau*^{*} be an exact *D*-optimal design for a given
regression model *Y*_{tau} = *X*_{tau}*beta* + *Z*_{tau}. In this paper sufficient
conditions
are given for designing how the covariance matrix of *Z*_{tau} may be
changed so that
not only *tau*^{*} remains *D*-optimal but also that the best linear unbiased
estimator (BLUE) of *beta* stays fixed for the design *tau*^{*}, although the
covariance matrix of *Z*_{tau*} is changed. Hence under these conditions
a best,
according to *D*-optimality, BLUE of *beta* is known for the model with the
changed covariance matrix. The results may also be considered as
determination of
exact *D*-optimal designs for regression models with special correlated
observations
where the covariance matrices are not fully known. Various examples are given,
especially for regression with intercept term, polynomial regression, and
straight-line regression. A real example in electrocardiography is treated
shortly.

*Key words and phrases*:
*D*-optimality, exact designs, correlated
observations, linear regression, robustness against disturbances.

**Source**
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