  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.
