Asymptotic properties of adaptive penalized optimal designs over a finite space

Adaptive optimal design with a cost constraint is considered, both for LS estimation in nonlinear regression and ML estimation in Bernoulli-type experiments, with possible applications in clinical trials. We obtain the strong consistency of the estimators for designs over a finite space, both when the cost level is fixed (and the adaptive design converges to an optimum constrained design) and when the objective is to minimize the cost. Moreover, the asymptotic normality of the estimators is obtained in the first situation, with an asymptotic covariance matrix given by the inverse of the usual information matrix, calculated as if the design were not constructed sequentially.