General Semiparametric Inference via Bootstrap Sampling

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本项目的研究目标是首先证明Bootstrap方法作为半参数模型的一般推理工具的理论有效性，然后发明一种计算上有吸引力的Bootstrap推理过程，称为k-step Bootstrap。 半参数建模为现代复杂数据提供了一个很好的框架，因为它可以灵活地对数据的某些特征进行参数化建模，而无需对其他特征进行任何假设。自举法是统计分析中最常用的数据恢复方法，近年来已被广泛应用于半参数模型。因此，对半参数模型的自助推断进行系统的理论研究是十分重要的。在实践中，自举推理过程的计算成本是特别高的半参数模型。因此，研究者提出了一种近似的自举方法，即k步自举，并将表明这种新的方法可以节省大量的计算量，但不会牺牲任何程度的推理精度。此外，研究者还将得到一组渐近结果，以阐明半参数M-估计的渐近结构，这对未来的理论研究至关重要。M-估计是一种一般的估计方法，包括最大似然估计的特殊情况，本文的主要工作是为Bootstrap抽样的一般半参数推断奠定了理论基础。此外，所提出的k步自举方法在几个方面实际上是有益的。例如，引导一个大数据集的科学家将受益，因为在k步引导中实现令人满意的推理精度所需的最小计算成本将被精确分析。然而，拟议活动的广泛影响是多方面的。例如，该项目的一个关键方面是研究和教学的整合，这将通过在半参数推理和自助计算的课程教学中为学生提出具体项目来实现。这一教学方法也有利于代表性不足的学生群体的参与。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The research objectives of this project are first to prove the theoretical validity of the bootstrap method as a general inferential tool for the semiparametric models, and then invent a computationally attractive bootstrap inference procedure, called k-step bootstrap. Semiparametric modelling has provided an excellent framework for the modern complex data due to its flexibility to model some features of the data parametrically but without assuming anything for the other features. The bootstrap is the most popular data-resampling method used in statistical analysis, and has recently been applied to the semiparametric models arising from a wide variety of contexts. Therefore, the systematic theoretical studies on the bootstrap inferences for the semiparametric models are fundamentally important. In practice, the computational cost of the bootstrap inference procedure is particularly high for the semiparametric models. Thus, the investigator proposes an approximate bootstrap method, i.e. k-step bootstrap, and will show that this novel approach results in huge computational savings but without sacrificing any degree of inference accuracy. In addition, the investigator will develop a set of asymptotic results to elucidate the asymptotic structure of the semiparametric M-estimation, which is crucial for the future theoretical research. M-estimation refers to a general method of estimation including the maximum likelihood estimation as a special case.The primary impact of the proposed work is to lay solid theoretical foundation for the general semiparametric inferences via bootstrap sampling. In addition, the proposed k-step bootstrap approach is practically beneficial in several regards. For instance, the scientists who bootstrap a large data set will benefit, as the minimal computational cost needed in the k-step bootstrap to achieve the satisfactory inference accuracy will be precisely analyzed. However, the broader impacts of the proposed activities are multiple. For instance, a key aspect of this project is the integration of research and teaching, which will be achieved by proposing specific projects for students during the teaching of classes on semiparametric inferences and bootstrap computation. This pedagogical method also facilitates the participation of underrepresented groups of students.