Extended optimality criteria for optimum design in nonlinear regression

Abstract : Among the major difficulties that one may encounter when estimating parameters in a nonlinear regression model are the non-uniqueness of the estimator, its instability with respect to small perturbations of the observations and the presence of local optimizers of the estimation criterion. We show that these estimability issues can be taken into account at the design stage, through the definition of suitable design criteria. Extensions of E, c and G-optimality criteria will be considered, which, when evaluated at a given $\theta_0$ (local optimal design), account for the behavior of the model response $\eta(\theta)$) for $\theta$ far from $\theta_0$. In particular, they ensure some protection against close-to-overlapping situations where $\|\eta(\theta)-\eta(\theta_0)\|$ is small for some $\theta$ far from $\theta_0$. These extended criteria are concave, their directional derivative can be computed and necessary and sufficient conditions for optimality (Equivalence Theorems) can be formulated. They are not differentiable, but a relaxation based on maximum-entropy regularization is proposed to obtain concave and differentiable alternatives. When the design space is finite and the set of admissible $\theta$ is discretized, their optimization forms a linear programming problem.
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Communication dans un congrès
D. Ucinski, A.C. Atkinson, M. Patan. mODa 10, Jun 2013, Lagow Lubuski, Poland. Springer, pp.195-202, 2013, Contributions to Statistics. <10.1007/978-3-319-00218-7_23>
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Andrej Pazman, Luc Pronzato. Extended optimality criteria for optimum design in nonlinear regression. D. Ucinski, A.C. Atkinson, M. Patan. mODa 10, Jun 2013, Lagow Lubuski, Poland. Springer, pp.195-202, 2013, Contributions to Statistics. <10.1007/978-3-319-00218-7_23>. <hal-00934877>

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