CAREER: Bootstrap M-estimation in Semi-Nonparametric Models

PI讨论了半非参数模型下两类广泛的Bootstrap方法的Bootstrap推理策略。作为一种通用的统计推断方法，Bootstrap在半参数模型中得到了广泛的应用。不幸的是，关于Bootstrap推断的系统理论研究非常有限，特别是当非参数分量不可根估计的时候。考虑了两类Bootstrap方法：可交换加权Bootstrap(EWB)和基于模型的Bootstrap(也称为参数Bootstrap)。PI证明了EWB一致估计欧氏估计的渐近方差，并且在惩罚M估计的框架下得出半参数推断在理论上是有效的。但是，EWB在绘制非参数元件的推理时可能会变得无效。因此，PI考虑了基于模型的Bootstrap，并从理论上证明了它是半非参数模型中所有参数的普遍有效的推断过程。拟议的研究还涉及高级经验过程工具的开发。上述研究为通过各种Bootstrap抽样方案进行一般半参数推断奠定了理论基础，并建立了关于非参数分量的非标准渐近理论的一般框架。快速有效地从现代海量数据集的所有维度中提取信息的迫切需要导致了半非参数模型的日益流行。例如，为了理解最近的金融危机，半参数Copula模型被用来处理不同金融序列的冲击之间的尾部相关性，并恢复单个金融序列的影响曲线的形状。该研究通过开发一系列创新和有效的Bootstrap推理工具，促进了半非参数模型在现代复杂数据分析中的应用，并最终在不同学科之间获得了实质性的科学生产率。统计科学受益于越来越多的研究人员从统计和科学的角度接受半非参数建模培训。这将包括由这项工作资助的学生、更广泛的合作研究和教育活动。上述研究还为公众制作了易于实现的软件。

The PI deals with the bootstrap inferential strategies for two broad classes of bootstrap methods in the context of semi-nonparametric models. As a general-purpose approach to statistical inferences, the bootstrap has found wide applications in semi-nonparametric models. Unfortunately, systematic theoretical studies on the bootstrap inferences are extremely limited, especially when the nonparametric component is not root-n estimable. Two classes of bootstrap methods are considered: the exchangeably weighted bootstrap (EWB) and the model-based bootstrap (also known as the parametric bootstrap). The PI proves that the EWB consistently estimates the asymptotic variance of the Euclidean estimate and is theoretically valid in drawing semiparametric inferences in the framework of penalized M-estimation. However, the EWB may become invalid in drawing inferences for nonparametric components. Hence, the PI considers the model-based bootstrap, and theoretically justifies it as an universally valid inference procedure for all the parameters in semi-nonparametric models. The proposed research also involves the development of advanced empirical processes tools. The above research lays the theoretical foundation for the general semi-nonparametric inferences via various bootstrap sampling schemes, and establishes a general framework for non-standard asymptotic theory concerning the nonparametric components.The immediate need for fast and efficiently extracting information from all the dimensions of modern massive data sets gives rise to the increasing popularity of the semi-nonparametric models. For example, to understand the recent financial crisis, the semi-nonparametric copula models are applied to address tail dependence among shocks to different financial series and also to recover the shapes of the impact curve for individual financial series. The proposed research promotes the use of semi-nonparametric models in analyzing modern complex data by developing a series of innovative and valid bootstrap inferential tools, and eventually gain substantial scientific productivity across various disciplines. Statistical science benefits from the increasing number of researchers trained in semi-nonparametric modelling both from the statistical and scientific viewpoints. This would include the students funded by this work, broader collaborating research and educational activities. The above research also produces easy-to-implement software for the public.