Bayesian optimal one point designs for one parameter nonlinear models

For nonlinear one parameter models and concave optimality criteria there always exists a locally optimal one point design. This can be proved by an application of Caratheodory's theorem (Lauter, Math. Operationsforsch. Statist. Ser. Statist. 5 (1974a) 625-636). If prior distributions with densities are used, this theorem gives no useful bound on the number of support points of a Bayesian optimal design. Chaloner (J. Statist. Plann. Inference, 37 (1993) 229-236) gave a sufficient condition on the support of the prior distribution for the existence of a Bayesian optimal one point design. In this article, a condition on the shape of the prior density is given, which is also sufficient for the existence of a Bayesian optimal one point design in nonlinear models with one parameter.