Many standard estimators, when applied to adaptively collected data, fail to be asymptotically normal, thereby complicating the construction of confidence intervals. We address this challenge in a semi-parametric context: estimating the parameter vector of a generalized linear regression model contaminated by a non-parametric nuisance component. We construct suitably weighted estimating equations that account for adaptivity in data collection, and provide conditions under which the associated estimates are asymptotically normal. Our results characterize the degree of "explorability" required for asymptotic normality to hold. For the simpler problem of estimating a linear functional, we provide similar guarantees under much weaker assumptions. We illustrate our general theory with concrete consequences for various problems, including standard linear bandits and sparse generalized bandits, and compare with other methods via simulation studies.

翻译：许多标准估计器在应用于自适应收集的数据时，无法保持渐近正态性，从而给置信区间的构建带来困难。我们在半参数框架下解决这一挑战：估计受非参数干扰分量污染的广义线性回归模型的参数向量。我们构建了适当加权的估计方程以考虑数据收集的自适应性，并给出了相关估计量保持渐近正态性的条件。我们的结果刻画了维持渐近正态性所需的“可探索性”程度。对于估计线性泛函这一更简单的问题，我们在更弱的假设下提供了类似的保证。我们通过具体问题（包括标准线性赌博机和稀疏广义赌博机）展示一般理论的实际应用，并通过模拟研究与其他方法进行比较。