Corrected Confidence Levels for Adaptive Nonlinear Regression

SYNOPTIC ABSTRACT

Consider a nonlinear regression model of the form y_k = g(x_k;θ)+e_k, where θ is an unknown parameter, x_k = x_k (y₁,…,y_k-1 is a design variable which depends on previous repsonses for k = 1, 2, …, and e₁, e₂, … are normal errors. Then the functional form of the maximum likelihood estimator is the same as for deterministic designs, but the sampling distributions may be quite different. Asymptotic expansions for the sampling distributions of normalized estimation error are found, and their use in setting confidence bounds is illustrated. The expansions are unconventional in that a Bayesian approach is used to obtain frequentist properties.

Key words and phrases:

• average confidence levels
• martingale central limit theorem
• maximum likelihood estimator
• prior and posterior distributions