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)提供资金的。本项目的研究目标是首先证明Bootstrap方法作为半参数模型通用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。半参数模型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建模，而不需要对其他特征做任何假设。Bootstrap是统计分析中最流行的数据重采样方法，最近已被应用于各种情况下的半参数模型。因此，对半参数模型的Bootstrap推断进行系统的理论研究是非常重要的。在实际应用中，对于半参数模型，Bootstrap推理过程的计算成本特别高。因此，研究人员提出了一种近似Bootstrap方法，即k步Bootstrap，并将证明这种新方法在不牺牲任何程度的推理精度的情况下，导致了巨大的计算量节省。此外，研究者还将发展一组渐近结果来阐明半参数M-估计的渐近结构，这对未来的理论研究是至关重要的。M-估计是一种一般的估计方法，其中最大似然估计是一种特殊的估计方法，其主要作用是通过Bootstrap抽样为一般的半参数推断奠定坚实的理论基础。此外，建议的k步自举方法在几个方面都是实际有益的。例如，引导大型数据集的科学家将受益，因为将精确分析k步引导所需的最小计算成本以实现令人满意的推断精度。然而，拟议活动的更广泛影响是多方面的。例如，这个项目的一个关键方面是研究和教学的结合，这将通过在半参数推理和自举计算的课堂教学中为学生提出具体的项目来实现。这种教学方法也有助于代表人数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.