## The Annals of Statistics

### Approximate optimal designs for multivariate polynomial regression

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 127-155.

Dates
Revised: January 2018
First available in Project Euclid: 30 November 2018

https://projecteuclid.org/euclid.aos/1543568584

Digital Object Identifier
doi:10.1214/18-AOS1683

Mathematical Reviews number (MathSciNet)
MR3909929

#### Citation

De Castro, Yohann; Gamboa, Fabrice; Henrion, Didier; Hess, Roxana; Lasserre, Jean-Bernard. Approximate optimal designs for multivariate polynomial regression. Ann. Statist. 47 (2019), no. 1, 127--155. doi:10.1214/18-AOS1683. https://projecteuclid.org/euclid.aos/1543568584

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#### Supplemental materials

• Supplementary material of approximate optimal designs for multivariate polynomial regression. We provide the proof of Theorem 1 and the detailed numerical results of the numerical examples in supplement article [2].