Constrained $D$- and $D_1$-optimal designs for polynomial regression

Abstract

In the common polynomial regression model of degree m we consider the problem of determining the $D$- and $D_1$-optimal designs subject to certain constraints for the $D_1$-efficiencies in the models of degree $m - j, m - j + 1,\dots, m + k(m > j \geq 0, k \geq 0 \text{given})$.We present a complete solution of these problems, which on the one hand allow a fast computation of the constrained optimal designs and, on the other hand, give an answer to the question of the existence of a design satisfying all constraints. Our approach is based on a combination of general equivalence theory with the theory of canonical moments. In the case of equal bounds for the $D_1$-efficiencies the constrained optimal designs can be found explicitly by an application of recent results for associated orthogonal polynomials.