Corrected confidence intervals for adaptive nonlinear regression models

A nonlinear regression model is considered in which the design variable may be a function of the previous responses. The aim is to construct confidence intervals for the parameter which are asymptotically valid to a high order. This is accomplished by using a tilting argument to construct a first approximation to a pivotal quantity, and then by using a version of Stein's identity and very weak expansions to determine the correction terms. The accuracy of the approximations is assessed by simulation for two well-known nonlinear regression models—the first-order growth or decay model and the Michaelis–Menten model, when one of the two parameters is known. Detailed proofs of the expansions are given.