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February 2019 Approximate optimal designs for multivariate polynomial regression
Yohann De Castro, Fabrice Gamboa, Didier Henrion, Roxana Hess, Jean-Bernard Lasserre
Ann. Statist. 47(1): 127-155 (February 2019). DOI: 10.1214/18-AOS1683


We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semialgebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically the approximate optimal design problem. The geometry of the design is recovered via semidefinite programming duality theory. This article shows that the hierarchy converges to the approximate optimal design as the order of the hierarchy increases. Furthermore, we provide a dual certificate ensuring finite convergence of the hierarchy and showing that the approximate optimal design can be computed numerically with our method. As a byproduct, we revisit the equivalence theorem of the experimental design theory: it is linked to the Christoffel polynomial and it characterizes finite convergence of the moment-sum-of-square hierarchies.


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Yohann De Castro. Fabrice Gamboa. Didier Henrion. Roxana Hess. Jean-Bernard Lasserre. "Approximate optimal designs for multivariate polynomial regression." Ann. Statist. 47 (1) 127 - 155, February 2019.


Received: 1 June 2017; Revised: 1 January 2018; Published: February 2019
First available in Project Euclid: 30 November 2018

zbMATH: 07036197
MathSciNet: MR3909929
Digital Object Identifier: 10.1214/18-AOS1683

Primary: 62K05 , 90C25
Secondary: 15A15 , 41A10 , 49M29 , 90C90

Keywords: Christoffel polynomial , equivalence theorem , Experimental design , linear model , semidefinite programming

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 1 • February 2019
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