OPTIMAL DESIGNS FOR A CLASS OF POLYNOMIALS OF ODD OR EVEN DEGREE

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),..., gamma(r)), where the numbers gamma(j) correspond to the models of degree 2j - 1 (2j) for j = 1,...,r. For a special class of priors, optimal designs of a very simple structure are calculated generalizing the D1-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.