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August 2014 E-optimal designs for second-order response surface models
Holger Dette, Yuri Grigoriev
Ann. Statist. 42(4): 1635-1656 (August 2014). DOI: 10.1214/14-AOS1241

Abstract

$E$-optimal experimental designs for a second-order response surface model with $k\geq1$ predictors are investigated. If the design space is the $k$-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1 (1977a) 121–132] determined optimal designs in a restricted class of designs (defined by the multiplicity of the minimal eigenvalue) and stated their universal optimality as a conjecture. In this paper, we prove this claim and show that these designs are in fact $E$-optimal in the class of all approximate designs. Moreover, if the design space is the unit ball, $E$-optimal designs have not been found so far and we also provide a complete solution to this optimal design problem.

The main difficulty in the construction of $E$-optimal designs for the second-order response surface model consists in the fact that for the multiplicity of the minimum eigenvalue of the “optimal information matrix” is larger than one (in contrast to the case $k=1$) and as a consequence the corresponding optimality criterion is not differentiable at the optimal solution. These difficulties are solved by considering nonlinear Chebyshev approximation problems, which arise from a corresponding equivalence theorem. The extremal polynomials which solve these Chebyshev problems are constructed explicitly leading to a complete solution of the corresponding $E$-optimal design problems.

Citation

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Holger Dette. Yuri Grigoriev. "E-optimal designs for second-order response surface models." Ann. Statist. 42 (4) 1635 - 1656, August 2014. https://doi.org/10.1214/14-AOS1241

Information

Published: August 2014
First available in Project Euclid: 7 August 2014

zbMATH: 1310.62097
MathSciNet: MR3262463
Digital Object Identifier: 10.1214/14-AOS1241

Subjects:
Primary: 62K05
Secondary: 41A50

Keywords: $E$-optimality , Duality , extremal polynomial , nonlinear Chebyshev approximation , Optimal designs , Response surface models

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 4 • August 2014
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