Optimal Designs for a Class of Polynomials of Odd or Even Degree

Abstract

In the class of polynomials of odd (or even) degree up to the order $2r - 1 (2r)$ optimal designs are determined which minimize a product of the variances of the estimates for the highest coefficients weighted with a prior $\gamma = (\gamma_1,\ldots,\gamma_r)$, where the numbers $\gamma_j$ correspond to the models of degree $2j - 1 (2j)$ for $j = 1,\ldots,r$. For a special class of priors, optimal designs of a very simple structure are calculated generalizing the $D_1$-optimal design for polynomial regression of degree $2r - 1 (2r)$. The support of these designs splits up in three sets and the masses of the optimal design at the support points of every set are all equal. The results are derived in a general context using the theory of canonical moments and continued fractions. Some applications are given to the $D$-optimal design problem for polynomial regression with vanishing coefficients of odd (or even) powers.