Sequential experimental designs in regression: Theory for the Bruceton and Langlie designs

There exist multiple regression applications in engineering and industry where the outcomes are not conditionally independent given the covariates, but where instead the covariates follow a sequential experimental design in which the next measurement depends on the previous outcomes, introducing dependence. Such designs are commonly employed for example for choosing test values when estimating the sensitivity of a material under physical stimulus. Apart from the extensive study of the Robbins--Monro procedure, virtually no attention has been given to verifying asymptotic normality of the maximum likelihood estimator in the general sequential setting, despite the wide use of such designs in industry since at least the 1940s. This is a considerable gap in the literature, since said properties underlie the construction of confidence intervals and hypothesis testing. In this paper we close this gap by establishing a large-sample theory for sequential experimental designs other than the Robbins--Monro procedure. First, we use martingale theory to prove a general result for when such asymptotic normality may be asserted. Second, we consider the special case where the covariate process forms a Markov chain. In doing so, we verify asymptotic normality for the widely applied Bruceton design and a proposed Markovian version of the Langlie design.