Motivated by modern applications such as computerized adaptive testing, sequential rank aggregation, and heterogeneous data source selection, we study the problem of active sequential estimation, which involves adaptively selecting experiments for sequentially collected data. The goal is to design experiment selection rules for more accurate model estimation. Greedy information-based experiment selection methods, optimizing the information gain for one-step ahead, have been employed in practice thanks to their computational convenience, flexibility to context or task changes, and broad applicability. However, statistical analysis is restricted to one-dimensional cases due to the problem's combinatorial nature and the seemingly limited capacity of greedy algorithms, leaving the multidimensional problem open. In this study, we close the gap for multidimensional problems. In particular, we propose adopting a class of greedy experiment selection methods and provide statistical analysis for the maximum likelihood estimator following these selection rules. This class encompasses both existing methods and introduces new methods with improved numerical efficiency. We prove that these methods produce consistent and asymptotically normal estimators. Additionally, within a decision theory framework, we establish that the proposed methods achieve asymptotic optimality when the risk measure aligns with the selection rule. We also conduct extensive numerical studies on both simulated and real data to illustrate the efficacy of the proposed methods. From a technical perspective, we devise new analytical tools to address theoretical challenges. These analytical tools are of independent theoretical interest and may be reused in related problems involving stochastic approximation and sequential designs.

翻译：受计算机自适应测试、序贯排序聚合以及异构数据源选择等现代应用的启发，我们研究主动序贯估计问题，该问题涉及为序贯收集的数据自适应地选择实验。目标是设计更精确模型估计的实验选择规则。基于信息论的贪心实验选择方法通过优化一步前瞻的信息增益，因其计算便捷性、对上下文或任务变化的灵活性以及广泛适用性，已在实践中得到应用。然而，由于问题的组合特性以及贪心算法看似有限的能力，统计分析仅限于一维情形，多维问题尚未解决。在本项研究中，我们填补了多维问题的空白。具体而言，我们提出采用一类贪心实验选择方法，并为遵循这些选择规则的最大似然估计量提供统计分析。该类方法既包含现有方法，也引入了数值效率更高的新方法。我们证明这些方法产生一致且渐近正态的估计量。此外，在决策理论框架下，我们建立当风险度量与选择规则一致时，所提方法实现渐近最优性。我们还对模拟数据和真实数据进行了广泛的数值研究，以说明所提方法的有效性。从技术角度来看，我们设计了新的分析工具来应对理论挑战。这些分析工具具有独立的理论价值，并可能用于涉及随机逼近和序贯设计的相关问题中。