Optimal designs for rational models and weighted polynomial regression

Abstract

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 $\mathbb{R}$ 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 $p + 1$ points 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 points $\pm 1$ 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.