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We extend the approach in [Ann. Statist. 38 (2010) 2499-2524] for identifyinglocally optimal designs for nonlinear models. Conceptually the extension isrelatively simple, but the consequences in terms of applications are profound.As we will demonstrate, we can obtain results for locally optimal designs undermany optimality criteria and for a larger class of models than has been donehitherto. In many cases the results lead to optimal designs with the minimalnumber of support points.

Identifying locally optimal designs for nonlinear models: A simple extension with profound consequences