Optimal designs for estimating the control values in multi-univariate regression models

This paper considers a linear regression model with a one-dimensional control variable x and an m-dimensional response vector . The components of are correlated with a known covariance matrix. Based on the assumed regression model, it is of interest to obtain a suitable estimation of the corresponding control value for a given target vector on the expected responses. Due to the fact that there is more than one target value to be achieved in the multiresponse case, the m expected responses may meet their target values at different respective control values. Consideration on the performance of an estimator for the control value includes the difference of the expected response E(y_i) from its corresponding target value T_i for each component and the optimal value of control point, say x₀, is defined to be the one which minimizes the weighted sum of squares of those standardized differences within the range of x. The objective of this study is to find a locally optimal design for estimating x₀, which minimizes the mean squared error of the estimator of x₀. It is shown that the optimality criterion is equivalent to a c-criterion under certain conditions and explicit solutions with dual response under linear and quadratic polynomial regressions are obtained.