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August 1999 Optimal designs for rational models and weighted polynomial regression
Holger Dette, Linda M. Haines, Lorens Imhof
Ann. Statist. 27(4): 1272-1293 (August 1999). DOI: 10.1214/aos/1017938926

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.

Citation

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Holger Dette. Linda M. Haines. Lorens Imhof. "Optimal designs for rational models and weighted polynomial regression." Ann. Statist. 27 (4) 1272 - 1293, August 1999. https://doi.org/10.1214/aos/1017938926

Information

Published: August 1999
First available in Project Euclid: 4 April 2002

zbMATH: 0957.62062
MathSciNet: MR1735997
Digital Object Identifier: 10.1214/aos/1017938926

Subjects:
Primary: 62K05
Secondary: 34L40

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 4 • August 1999
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