The Annals of Statistics

E-optimal designs for second-order response surface models

Holger Dette and Yuri Grigoriev

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$E$-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.

Article information

Ann. Statist., Volume 42, Number 4 (2014), 1635-1656.

First available in Project Euclid: 7 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K05: Optimal designs
Secondary: 41A50: Best approximation, Chebyshev systems

Response surface models optimal designs $E$-optimality extremal polynomial duality nonlinear Chebyshev approximation


Dette, Holger; Grigoriev, Yuri. E -optimal designs for second-order response surface models. Ann. Statist. 42 (2014), no. 4, 1635--1656. doi:10.1214/14-AOS1241.

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