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 y = ( y 1 , … , y m ) . The components of y 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 T = ( T 1 , … , T m ) 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 x 0 , 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 x 0 , which minimizes the mean squared error of the estimator of x 0 . 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.