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：在工程与工业中存在多种回归应用场景，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——即下一次测量取决于先前结果，从而引入依赖性。此类设计常被用于例如在物理刺激下估算材料灵敏度时选择测试值。尽管Robbins-Monro程序已获广泛研究，但自20世纪40年代以来工业界虽普遍采用这类设计，却几乎无人关注一般序贯框架下最大似然估计渐近正态性的验证。这构成了文献中的重大空白，因为所述性质是构建置信区间与假设检验的基础。本文通过建立除Robbins-Monro程序外序贯实验设计的大样本理论填补此空白。首先，我们运用鞅理论证明此类渐近正态性可成立的一般性结论。其次，我们考虑协变量过程构成马尔可夫链的特殊情形。通过这一分析，我们验证了广泛应用的Bruceton设计及所提出的Langlie设计马尔可夫版本的渐近正态性。