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.

翻译：在工程与工业领域存在多种回归应用，其中结果变量在给定协变量条件下并非条件独立，而是协变量遵循序贯实验设计——即下次测量值依赖于先前结果，从而引入相依性。此类设计被广泛用于例如评估材料在物理刺激下的灵敏度时选择测试值。尽管Robbins-Monro程序已得到深入研究，但自1940年代以来这类设计在工业界广泛应用，其最大似然估计量的渐近正态性验证在一般序贯设定中几乎未被关注。这构成了文献中的重要空白——因为所述性质是构建置信区间和假设检验的基础。本文通过建立Robbins-Monro程序以外的序贯实验设计大样本理论填补这一空白。首先，我们利用鞅理论证明了此类渐近正态性成立的通用结论。其次，我们考虑协变量过程构成马尔可夫链的特殊情形。由此验证了广泛应用的Bruceton设计及本文提出的Langlie设计马尔可夫版本的渐近正态性。