  There exist multiple regression applications in engineering, industry and
medicine where the outcomes are not conditionally independent given the
covariates, but where instead the covariates follow an adaptive experimental
design in which the next measurement depends on the previous observations,
introducing dependence. Such designs are commonly employed for example for
choosing test values when estimating the sensitivity of a material under
physical stimulus. In addition to estimating the regression parameters, we are
also interested in tasks such as hypothesis testing and constructing confidence
intervals, both of which rely on asymptotic normality of the maximum likelihood
estimator. When adaptive designs are employed, however, the large-sample theory
of the maximum likelihood estimator is more involved than in the standard
regression setting, where the outcomes are assumed conditionally independent
given the covariates. Hence, asymptotic normality must be verified explicitly
case by case. For some classic adaptive designs like the Bruceton up-and-down
designs, this issue has been resolved. However, general results for adaptive
designs relying on minimal assumptions are to a large extent lacking. In this
paper we establish a general large-sample theory for a wide selection of
adaptive designs. Motivated by the theory, we propose a new Markovian version
of the Langlie design and verify asymptotic normality for this proposal. Our
simulations indicate that the Markovian design is more stable and yields better
confidence intervals than the original Langlie design.
