Optimal Designs for Rational Models and Weighted Polynomial Regression by Holger Dette,

In this paper D-optimal designs for the weighted polynomial regresŽ 2 . n sion model of degree p with efficiency function 1 x are presented. Interest in these designs stems from the fact that they are equivalent to locally D-optimal designs for inverse quadratic polynomial models. For the unrestricted design space and p n, the D-optimal designs put equal masses on p 1 points which coincide with the zeros of an ultraspherical polynomial, while for p n they are equivalent to D-optimal designs for certain trigonometric regression models and exhibit all the curious and interesting features of those designs. For the restricted design space 1, 1 sufficient, but not necessary, conditions for the D-optimal designs to be based on p 1 points are developed. In this case the problem of Ž . constructing p 1 -point D-optimal designs is equivalent to an eigenvalue problem and the designs can be found numerically. For n 1 and 2, the problem is solved analytically and, specifically, the D-optimal designs put equal masses at the points 1 and at the p 1 zeros of a sum of n 1 ultraspherical polynomials. A conjecture which extends these analytical results to cases with n an integer greater than 2 is given and is examined empirically.