Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least square estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality.

翻译：序贯数据收集已成为提升数据采集过程效率的一种广泛应用技术。尽管具有诸多优势，但这种数据收集机制往往给统计推断过程带来复杂性。例如，在自适应线性回归模型中，普通最小二乘估计量可能表现出非正态渐近行为，这给准确推断与解释带来了挑战。本文提出了一种构建去偏估计量的通用方法以解决该问题。该方法利用了自适应线性估计方程的思想，我们建立了渐近正态性的理论保证，并辅以实现近最优渐近方差的讨论。我们估计量的一个显著特点是：在多臂老虎机情境下，该估计量在保持最小二乘估计量非渐近性能的同时，获得了渐近正态性。因此，本研究有助于连接自适应推断的两大富有成果的范式：a）基于集中不等式的非渐近推断，以及b）基于渐近正态性的渐近推断。