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.

翻译：在工程、工业和医学领域存在多种回归应用场景，其中结果在给定协变量时并非条件独立，而是协变量遵循自适应实验设计——即下一次测量依赖于先前观测值，从而引入依赖性。例如在估计材料对物理刺激的敏感度时选择测试值，就常采用此类设计。除了估计回归参数外，我们还关注假设检验和构建置信区间等任务，这些任务都依赖于最大似然估计量的渐近正态性。然而，当采用自适应设计时，最大似然估计量的大样本理论比标准回归设置（其中结果在给定协变量时被假定为条件独立）更为复杂。因此，必须逐例显式验证渐近正态性。对于某些经典的自适应设计（如Bruceton升降法），该问题已得到解决。但基于最小假设的自适应设计的一般性结果在很大程度上仍然缺乏。本文针对广泛的自适应设计建立了一套通用的大样本理论。受该理论启发，我们提出了一种新的马尔可夫化Langlie设计，并验证了该方案的渐近正态性。模拟结果表明，与原始Langlie设计相比，马尔可夫设计具有更高的稳定性，并能产生更优的置信区间。