A geometric characterization of $c$-optimal designs for heteroscedastic regression

TL;DR

This paper extends the geometric characterization of $c$-optimal designs to heteroscedastic nonlinear regression models, where both mean and variance depend on parameters, generalizing Elfving's classical result.

Contribution

It introduces a new geometric framework for $c$-optimal designs in heteroscedastic models, involving convex hulls of ellipsoids, and generalizes Elfving's characterization.

Findings

The Elfving set becomes the convex hull of ellipsoids related to the model.

$c$-optimal designs are points where a line intersects the boundary of this set.

Illustrations include pharmacokinetic models with random effects.

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.