Optimality of Observed Information Adaptive Designs in Linear Models
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This work considers experimental design in linear models with additive errors. A traditional objective in design is to minimize the variance of the estimates of the model parameters. The optimal design, which is found by minimizing a convex function of the expected Fisher information, accomplishes this objective, approximately. The inverse of expected Fisher information is asymptotically equivalent to the variance of the maximum likelihood estimate. It is often remarked that observed Fisher information is a better measure of the variance of the maximum likelihood estimate than the expected Fisher information [Efron and Hinkley (1978)]. However, unlike expected Fisher information, observed Fisher information depends on the observed data and cannot be used to design an experiment in advance of data collection. In a sequential experiment the observed Fisher information from past observations is available to incorporate into the design of the current observation. In this work an adaptive design that incorporates observed Fisher information is proposed. It is shown that this proposed design is optimal, at the limit, with respect to inference and conditional mean square error. In a simulation study the proposed adaptive design performs nearly uniformly better than the optimal design.
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