Constrained design strategies for improving normal approximations in nonlinear regression problems

In nonlinear regression problems, the assumption is usually made that parameter estimates will be approximately normally distributed. The accuracy of the approximation depends on the sample size and also on the intrinsic and parameter-effects curvatures. Based on these curvatures, criteria are defined here that indicate whether or not an experiment will lead to estimates with distributions well approximated by a normal distribution. An approach is motivated of optimizing a primary design criterion subject to satisfying constraints based on these nonnormality measures. The approach can be used either to I) find designs for a fixed sample size or to II) choose the sample size for the optimal design based on the primary objective so that the constraints are satisfied. This later objective is useful as the nonnormality measures decrease with the sample size. As the constraints are typically not concave functions over a set of design measures, the usual equivalence theorems of optimal design theory do not hold for the first approach, and numerical implementation is required. Examples are given, and a new notation using tensor products is introduced to define tractable general notation for the nonnormality measures.