Optimal designs for rational models and weighted polynomial regression

In this paper D-optimal designs for the weighted polynomial regression model of degree p with efficiency function (1 + x(2))(-n) 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 R 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 paints are developed. In this case the problem of 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 paints +/- 1 and at the p - 1 zeros of a sum of n + 1 ultraspherical polynomials. A conjecture which extends these analytical results ta cases with n an integer greater than 2 is given and is examined empirically.