Abstract: Maximum likelihood recursions were proposed in Wu (1985, 1986) to obtain recursive procedures for nonlinear sequential design problems associated with many commonly used generalized linear models. It was argued empirically and heuristically there that these recursions should lead to asymptotically consistent and efficient designs. We prove that such recursions are consistent and asymptotically normal, at least for the location models including logistic, Poisson, gamma and inverse Gaussian. We show that a simple truncation leads to robust designs so that even if the models are incorrectly specified, the recursions still converge to the desired optimal design points. Asymptotic results concerning the sequential designs for the location-scale models are also obtained.
Key words and phrases: Asymptotic efficiency, asymptotic normality, generalized linear models, maximum likelihood estimator, Stochastic approximation, strong consistency.