June 2022 Design admissibility and de la Garza phenomenon in multifactor experiments
Holger Dette, Xin Liu, Rong-Xian Yue
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Ann. Statist. 50(3): 1247-1265 (June 2022). DOI: 10.1214/21-AOS2147

Abstract

The determination of an optimal design for a given regression problem is an intricate optimization problem, especially for models with multivariate predictors. Design admissibility and invariance are main tools to reduce the complexity of the optimization problem and have been successfully applied for models with univariate predictors. In particular, several authors have developed sufficient conditions for the existence of minimally supported designs in univariate models, where the number of support points of the optimal design equals the number of parameters. These results generalize the celebrated de la Garza phenomenon (Ann. Math. Statistics 25 (1954) 123–130), which states that for a polynomial regression model of degree k1 any optimal design can be based on k points.

This paper provides—for the first time—extensions of these results for models with a multivariate predictor. In particular, we study a geometric characterization of the support points of an optimal design to provide sufficient conditions for the occurrence of the de la Garza phenomenon in models with multivariate predictors and characterize properties of admissible designs in terms of admissibility of designs in conditional univariate regression models.

Funding Statement

Dr. Liu and Dr. Yue were partially supported by the National Natural Science Foundation of China under Grants 11871143, 11971318. The work of Dr. Dette has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt C2) of the German Research Foundation (DFG).

Acknowledgments

The authors would like to thank the referees and the associate editor for their constructive comments on three earlier versions of the paper and Martina Stein, who typed parts of the paper with considerable technical expertise. All authors contributed equally to the manuscript.

Citation

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Holger Dette. Xin Liu. Rong-Xian Yue. "Design admissibility and de la Garza phenomenon in multifactor experiments." Ann. Statist. 50 (3) 1247 - 1265, June 2022. https://doi.org/10.1214/21-AOS2147

Information

Received: 1 March 2020; Revised: 1 October 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441119
zbMATH: 07547929
Digital Object Identifier: 10.1214/21-AOS2147

Subjects:
Primary: 62K05
Secondary: 62J05

Keywords: Admissibility , conditional model , dual problem , multifactor experiment , optimal design

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 3 • June 2022
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