The Annals of Statistics

A geometric characterization of c-optimal designs for heteroscedastic regression

Holger Dette and Tim Holland-Letz

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Abstract

We consider the common nonlinear regression model where the variance, as well as the mean, is a parametric function of the explanatory variables. The c-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of c-optimal designs in this context is presented, which generalizes the classical result of Elfving [Ann. Math. Statist. 23 (1952) 255–262] for c-optimal designs. As in Elfving’s famous characterization, c-optimal designs can be described as representations of boundary points of a convex set. However, in the case where there appear parameters of interest in the variance, the structure of the Elfving set is different. Roughly speaking, the Elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space. The c-optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new Elfving set. The theory is illustrated in several examples including pharmacokinetic models with random effects.

Article information

Source
Ann. Statist., Volume 37, Number 6B (2009), 4088-4103.

Dates
First available in Project Euclid: 23 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1256303537

Digital Object Identifier
doi:10.1214/09-AOS708

Mathematical Reviews number (MathSciNet)
MR2572453

Zentralblatt MATH identifier
1191.62130

Subjects
Primary: 62K05: Optimal designs

Keywords
c-optimal design heteroscedastic regression Elfving’s theorem pharmacokinetic models random effects locally optimal design geometric characterization

Citation

Dette, Holger; Holland-Letz, Tim. A geometric characterization of c -optimal designs for heteroscedastic regression. Ann. Statist. 37 (2009), no. 6B, 4088--4103. doi:10.1214/09-AOS708. https://projecteuclid.org/euclid.aos/1256303537


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