The Annals of Statistics

Optimal designs for rational models and weighted polynomial regression

Holger Dette, Linda M. Haines, and Lorens Imhof

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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.

Article information

Ann. Statist., Volume 27, Number 4 (1999), 1272-1293.

First available in Project Euclid: 4 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K05: Optimal designs
Secondary: 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)

D-optimal design weighted polynomial regression rational models Schrödinger equation


Dette, Holger; Haines, Linda M.; Imhof, Lorens. Optimal designs for rational models and weighted polynomial regression. Ann. Statist. 27 (1999), no. 4, 1272--1293. doi:10.1214/aos/1017938926.

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