The Annals of Statistics

Optimum design accounting for the global nonlinear behavior of the model

Andrej Pázman and Luc Pronzato

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Abstract

Among the major difficulties that one may encounter when estimating parameters in a nonlinear regression model are the nonuniqueness of the estimator, its instability with respect to small perturbations of the observations and the presence of local optimizers of the estimation criterion.

We show that these estimability issues can be taken into account at the design stage, through the definition of suitable design criteria. Extensions of $E$-, $c$- and $G$-optimality criteria are considered, which when evaluated at a given $\theta^{0}$ (local optimal design), account for the behavior of the model response $\eta(\theta )$ for $\theta $ far from $\theta^{0}$. In particular, they ensure some protection against close-to-overlapping situations where $\|\eta(\theta )-\eta(\theta^{0})\|$ is small for some $\theta $ far from $\theta^{0}$. These extended criteria are concave and necessary and sufficient conditions for optimality (equivalence theorems) can be formulated. They are not differentiable, but when the design space is finite and the set $\Theta$ of admissible $\theta $ is discretized, optimal design forms a linear programming problem which can be solved directly or via relaxation when $\Theta$ is just compact. Several examples are presented.

Article information

Source
Ann. Statist. Volume 42, Number 4 (2014), 1426-1451.

Dates
First available in Project Euclid: 25 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1403715206

Digital Object Identifier
doi:10.1214/14-AOS1232

Mathematical Reviews number (MathSciNet)
MR3226162

Zentralblatt MATH identifier
1302.62174

Subjects
Primary: 62K05: Optimal designs
Secondary: 62J02: General nonlinear regression

Keywords
Optimal design nonlinear least-squares estimability curvature

Citation

Pázman, Andrej; Pronzato, Luc. Optimum design accounting for the global nonlinear behavior of the model. Ann. Statist. 42 (2014), no. 4, 1426--1451. doi:10.1214/14-AOS1232. https://projecteuclid.org/euclid.aos/1403715206.


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