Optimal designs for rational models and weighted polynomial regression

Abstract

¨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. 1. Introduction. Weighted polynomial regression models with variance functions which depend on the explanatory variable have played, and continue to play, an important role in the development of classical optimal design theory. The reasons for this are essentially twofold. First, there is a wealth of elegant mathematics associated with the construction of D-optimal designs for many of these models. In particular for certain classes of variance functions, it is possible to show that the D-optimal designs put equal masses on p 1 points of support, where p is the degree of the polynomial embedded in the model, and, furthermore, it is possible to use tools from the theory of differential equations and of canonical moments in order to establish that these support points coincide with the zeros of classical orthogonal polynomiŽ. Ž .