Abstract
In adaptive optimal procedures, the design at each stage is an estimate of the optimal design based on all previous data. Asymptotics for regular models with fixed number of stages are straightforward if one assumes the sample size of each stage goes to infinity with the overall sample size. However, it is not uncommon for a small pilot study of fixed size to be followed by a much larger experiment. We study the large sample behavior of such studies. For simplicity, we assume a nonlinear regression model with normal errors. We show that the distribution of the maximum likelihood estimates converges to a scale mixture family of normal random variables. Then, for a one parameter exponential mean function we derive the asymptotic distribution of the maximum likelihood estimate explicitly and present a simulation to compare the characteristics of this asymptotic distribution with some commonly used alternatives.
Highlights
Elfving 1 introduced a geometric approach for determining a c-optimal design for linear regression models
Kiefer and Wolfowitz 2 developed the celebrated equivalence theorem which provides an efficient method for verifying if a design is D-optimal, again for a linear model. These two results were generalized by Chernoff 3 and White 4 to include nonlinear models, respectively
Researchers in optimal design have built an impressive body of theoretical and practical tools for linear models based on these early results
Summary
Elfving 1 introduced a geometric approach for determining a c-optimal design for linear regression models. Kiefer and Wolfowitz 2 developed the celebrated equivalence theorem which provides an efficient method for verifying if a design is D-optimal, again for a linear model. These two results were generalized by Chernoff 3 and White 4 to include nonlinear models, respectively. Current literature has characterized the adaptive optimal design procedure under the assumption that both stage-one and stage-two sample sizes are large. The distribution for a nonlinear regression model with normal errors and a one parameter exponential mean function is derived explicitly. 2. Adaptive Optimal Procedure for a Two-Stage Nonlinear Regression Model with Normal Errors. S1 S2, where Si represents the score function for the ith stage
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