There exist multiple regression applications in engineering, industry and medicine where the outcomes follow an adaptive experimental design in which the next measurement depends on the previous observations, so that the observations are not conditionally independent given the covariates. In the existing literature on such adaptive designs, results asserting asymptotic normality of the maximum likelihood estimator require regularity conditions involving the second or third derivatives of the log-likelihood. Here we instead extend the theory of differentiability in quadratic mean (DQM) to the setting of adaptive designs, which requires strictly fewer regularity assumptions than the classical theory. In doing so, we discover a new DQM assumption, which we call summable differentiability in quadratic mean (S-DQM). As applications, we first verify asymptotic normality for a classical adaptive designs, namely the Bruceton design, before moving on to a complicated problem, namely a Markovian version of the Langlie design.

翻译：在工程、工业和医学领域的众多回归应用中，结果遵循自适应实验设计，即下一次测量依赖于先前的观测值，从而导致观测值在给定协变量的条件下并非条件独立。现有关于此类自适应设计的文献中，关于最大似然估计量渐近正态性的结论通常需要涉及对数似然函数二阶或三阶导数的正则性条件。本文则将二次平均可微性理论推广至自适应设计场景，该理论所需的规范性假设严格少于经典理论。在此过程中，我们提出了一种新的二次平均可微性假设，称为可和二次平均可微性。作为应用，我们首先验证了经典自适应设计（即Bruceton设计）的渐近正态性，继而研究了一个复杂问题——Langlie设计的马尔可夫版本。