  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.
